research article fractal analysis of laplacian pyramidal...

Research ArticleFractal Analysis of Laplacian Pyramidal Filters Applied toSegmentation of Soil Images

J de Castro1 F Ballesteros1 A Meacutendez2 and A M Tarquis3

1 Matematica Aplicada a las Tecnologıas de la Informacion ETS de Ingenieros de Telecomunicacion Technical University of MadridAvenida Complutense 30 28040 Madrid Spain

2Matematica Aplicada a la Ingenierıa Tecnica de Telecomunicacion ETS de Ingenierıa y Sistemas de TelecomunicacionTechnical University of Madrid Carretera de Valencia km 7 28031 Madrid Spain

3Matematica Aplicada a la Ingenierıa Agronomica and CEIGRAM ETS de Ingenieros Agronomos Technical University of MadridAvenida Complutense 3 28040 Madrid Spain

Correspondence should be addressed to F Ballesteros franciscoballesterosupmes

Received 25 March 2014 Accepted 3 June 2014 Published 10 July 2014

Academic Editor Antonio Paz Gonzalez

Copyright copy 2014 J de Castro et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The laplacian pyramid is a well-known technique for image processing in which local operators of many scales but identical shapeserve as the basis functions The required properties to the pyramidal filter produce a family of filters which is unipara metrical inthe case of the classical problem when the length of the filter is 5 We pay attention to gaussian and fractal behaviour of these basisfunctions (or filters) and we determine the gaussian and fractal ranges in the case of single parameter 119886 These fractal filters looseless energy in every step of the laplacian pyramid and we apply this property to get threshold values for segmenting soil imagesand then evaluate their porosity Also we evaluate our results by comparing them with the Otsu algorithm threshold values andconclude that our algorithm produce reliable test results

1 Introduction

Image analysis involves many different tasks such as identi-fying objects into images (segmentation) assigning labels toindividual pixels by taking into account relevant information(classification) or extracting some meaning from the imageas a whole (interpretation) The segmentation of soil imagesappears into Soil Science as a tool for the measurement ofproperties as well as for detecting and recognizing objects insoil [1ndash3]

Different methods have been used to segment soil imagessuch as a simple binary threshold method [4] or multiplethreshold method [5] and thresholds for typical and criticalregions Wang et al [6] did a wide review of different seg-mentation methods applied in Geoscience Other methodsthat appear to be most promising for soil applications areclustering methods and entropy-based methods [7ndash9]

Soil is not a continuous medium because soil is sus-ceptible to changes from many influences wetting drying

compaction plant growth and so forth So the continuoussoil models lead to approximate results only and anomalousphenomena cannot be easily handled It is known that poresin porous material are highly complex [10] Their study andanalysis have been usually avoided because of their difficulty

Soil is formed from many constituents and to representit as a two-phase material solid and pore is often an over-simplification The behaviour of water gas and organismscan affect it A classification of pore models could be [11] (i)nonspatial (ii) schematic (iii) random set (iv) fractal and (v)other stochastic models The fractal group has more modelsproposed and publications about

Models of soil physical structure have been developedsince the 1950s Childs and Collisgeorge [12] introduced thecut-and-rejoin models of soil capillaries which were modi-fied by Marshall [13] While many models of soil structurehave been developed since then most relate the structureto physical processes generally ignoring heterogeneity orassume simple pore-size distribution models

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 212897 13 pageshttpdxdoiorg1011552014212897

2 The Scientific World Journal

More sophisticated approaches are [14] using a one-dimensional Markov chain model for horizontal soil [15]proposing a two-dimensional fuzzy random model of soilpore structure and [16] describing a network model topredict physical properties from topological parameters andfractal-based approaches like [17]

Our goal is to calculate the porosity of soil images Theproposed procedure for segmentation of soil micromorpho-logical images is based on Laplacian pyramid algorithm[18] from which we compute a threshold that will binarizethe original image resulting with an image composed ofcontinuous regions of pores (shown in black) and soil (shownin white) From this binarized soil image we compute anestimation to its porosity

Another objective of this study is also to compare theresults with those provided by the commonly used Otsursquosalgorithm [19] that is widely accepted as a good method toget an appropriate threshold

2 Materials and Methods

21 Laplacian Pyramid A multiresolution model consists ofgenerating different versions of a given image by decreasingthe initial resolution which also means decreasing the initialsize This is achieved by a downsampling operator whichmust be associated with an appropriate filtering to avoidaliasing phenomena (downsampling theorem) Multiresolu-tion approaches have been investigated for different purposessuch as image segmentation and image compression [18] Interms of image analysis low resolution representations areconvenient for global detection and recognition of imagefeatures while minute details can only be seen on highresolution images

One usual property of images is that neighboring pixelsare highly correlated This property is inefficient to rep-resent the complete image directly in terms of its pixelvalues because most of the encoded information wouldbe redundant Burt and Adelson designed a techniquenamed Laplacian pyramid for removing image correlationwhich combines characteristics of predictive and transformmethods [18]This technique is noncausal and computationsare relatively simple and local The predicted value for eachnew pixel is computed as a local weighted average using aunimodal weighting function centered on the pixel itself

This pyramidal representation is useful for two importantclasses of computer graphics problems The first class iscomposed of those tasks that involve analysis of existingimages such as merging images or interpolating to fill inmissing data smoothly become much more intuitive whenwe canmanipulate easily visible local image features at severalspatial resolutions And the second whenwe are synthesizingimages the pyramid becomes a multiresolution sketch padWe can fill in the local spatial information at increasingly finedetail by specifying successive levels of a pyramid

The first time that pyramidal structures were applied tomultiresolution decompositions was at [18] Later the rela-tionship to wavelets was realized shortly thereafter becauseboth decompositions are based on the idea of successiverefinement the image is obtained as a sum of an initial coarse

version plus detail signals One interesting thing to noteabout the pyramidal approach is that perfect reconstructionis possible therefore it is a lossless data algorithm

Pyramidal methods for multiresolution image analysishave been used since the 1970s Early work inmultiresolutionimage description was primarily motivated by a desire toreduce the computational cost of methods for image descrip-tion and image matching Later multiresolution processingwas generalized to computing multiple copies of an imageby repeatedly summing nonoverlapping blocks of pixels andresampling until the image is reduced to a small number ofpixels Such a structure became known as a multiresolutionpyramid [20]

Interest in multiresolution techniques for signal process-ing and analysis is increasing steadily [21] An importantinstance of such a technique is the so-called pyramidaldecomposition scheme Our work uses a general axiomaticpyramidal decomposition scheme for soil image analysisThis scheme comprises the following ingredients

(i) The pyramid consists of a finite number of levelssuch that the information content decreases towardshigher levels

(ii) Each step towards a higher level is constituted byan information-reducing analysis operator whereaseach step towards a lower level is modeled by aninformation-preserving synthesis operator One basicassumption is necessary synthesis followed by anal-ysis yields the identity operator meaning that noinformation is lost by these two consecutive steps

22 Fractal Dimension The techniques based on fractalsshow promising results in the field of image understandingand visualization of high complexity data

The high complexity of some images demands newtechniques for understanding and analyzing them The sim-ilarity of fractals and real world objects has been observedand studied from the very beginning The fractal geometrybecame a tool for computer graphics and data visualizationin the simulation of the real world In order to perform visualanalysis and comparisons between natural and syntheticscenes several techniques have been developed After a periodof qualitative experiments fractal geometry began to beused for objective and accurate purposes modeling imagesevaluating their characteristics analyzing their textures andso forth

Nowadays there are a lot of fields where fractals appear[22] First of all we present some of the elementary ideasnecessary to understand applications of fractal geometry ingeo-information processing [23]

Fractal geometry theory deals with the behaviors of sets ofpoints 119878 in the 119899-dimensional spaceR119899 Images particularlysoil images are sets of points in R2

Mandelbrot defined a fractal as a shape made of partssimilar to the whole in some way [24] That definition isqualitative but not ambiguous as it looks at the first glanceThe main behavior of a fractal is its self-similarity A set iscalled self-similar if it can be expressed as a union of sets eachof which is a reduced copy of the full set More generally a set

The Scientific World Journal 3

is said to be self-affine if it can be decomposed into subsetsthat can be linearly mapped into the full set If the linearmapping is a rotation translation or isotropic dilatation theset is self-similar The self-similar objects are particular casesof self-affine ones

A fractal object is self-similar or self-affine at any scale Ifthe similarity is not described by deterministic laws stochasticresemblance criteria can be found Such an object is saidto be statistical self-similar The natural fractal objects arestatistically self-similar A statistically self-similar fractal isby definition isotropic To have a more precise quantitativedescription of the fractal behavior of a set a measure anda dimension are introduced The rigorous mathematicaldescription is done by Hausdorff rsquos measure and dimension[25]

Let 119878 sub R119899 and 119903 gt 0 and a 120575-cover of 119878 is a collection ofsets 119880

119894 119894 isin 119868 with diameter which is smaller than 120575 such

that

119878 sub ⋃

119894isin119868

119880119894sub R119899 with 0 lt

1003816100381610038161003816119880119894

1003816100381610038161003816lt 120575 (1)

where 119868 is a finite or countable index set and | sdot | representsthe diameter of the 119899-dimension set defined as

|119880| = sup 1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 119909 119910 isin 119880 (2)

Also let R120575(119878) be the collection of all 120575-covers of 119878 we can

define

119867119903

120575(119878) = inf

RisinR120575

sum

119894isin119868

1003816100381610038161003816119880119894

1003816100381610038161003816

119903

R = ⋃

119894isin119868

119880119894 (3)

Now if in (3) we let 120575 decrease to zero we get theHausdorff measure of the set 119878119867119903(119878)

119867119903

(119878) = lim120575rarr0

119867119903

120575(119878) (4)

The Hausdorff measure generalizes the definition of lengtharea volume and so on119867119903

120575(119878) gives the volume of a set 119878 as

measured with a ruler of 120575 unitsThere is an interesting property of theHausdorffmeasure

If the Hausdorff dimension of the set 119878 is 119904 then

119867119901

(119878) =

infin if 119901 lt 1199040 if 119901 gt 119904

(5)

So the Hausdorff dimension of the set 119878 sub R119899 could bedefined as

dim119867119878 = sup 119903 119867119903 (119878) = infin = inf 119903 119867119903 (119878) = 0 (6)

as we can see in Figure 1Then the value of the parameter 119903 for which the 119903-

dimensional Hausdorff measure of the set jumps from zeroto infinite is said to be the Hausdorff dimension dim

119867 of the

set 119878A set is said to be fractal if itsHausdorff dimension strictly

exceeds its topological dimension dim119867119878 gt 119899

Numerical evaluation of Hausdorff dimension is difficultbecause of the necessity to evaluate the infimum of the

infin

s p

Hp(S)

Figure 1 Hausdorff dimension of the set 119878

measure over all the coverings belongings to the set ofinterest That is the reason to look for another definition forthe dimension of a set The box counting dimension allowsthe evaluation of the dimension of sets of points spread inan 119899-dimensional space and also gives possibilities for easyalgorithmic implementation

Given a set of points 119878 in a 119899-dimensional space R119899 and119873120575is the least number of sets of diameter at most 120575 that cover

119878 the box counting dimension dim119861 is defined as

dim119861119878 = lim120575rarr0

log119873120575

log 1120575 (7)

Depending on the geometry of the box and the modalityto cover the set several box counting dimensions can bedefined using closed balls cubes and so on [25]

The equivalence of these definitions was proved Also itwas proved that these dimensions are inferior bounded by theHausdorff dimension [26]

Fractal geometry provides a mathematical model formany complex objects found in nature such as coastlinesmountains and clouds [22 24] These objects are too com-plex to possess characteristic sizes and to be described bytraditional Euclidean geometry Fractal dimension has beenapplied in texture analysis and segmentation [27 28] Thereare differentmethods that have been proposed to estimate thefractal dimension The three major categories are the box-counting methods the variance methods and the spectralmethodsThe box-counting dimension is themost frequentlyused for measurements in various application fields Thereason for its dominance lies in its simplicity and automaticcomputability

23 Segmentation of Soil Images Image segmentation is theprocess of partitioning an image into several regions in orderto be easier to analyze and work with

In image segmentation the level to which the subdivisionof an image into its constituent regions or objects is carrieddepending on the problem being solved In other wordswhen the object of focus is separated image segmentationshould stop [29] The main goal of segmentation is to dividean image into parts having strong correlation with areas ofinterest in the image

We study the simplest problem dividing the image intojust only two parts foreground and background or objectpixels and background pixelsThe intensity values continuity


or discontinuity color texture and other image character-istics are the origin of the different image segmentationtechniques Reference [9] is an exhaustive performance com-parison of 40 selected methods put into groups according tohistogram shape informationmeasurement space clusteringhistogram entropy information image attribute informationspatial information and local characteristics

So some of the most important groups in image segmen-tation techniques are the threshold-based the histogram-based the edge-based and the region based

The threshold-based methods are based on pixels inten-sity values The main goal here is to decide a threshold value120582 to apply the rule

119892119894+1

(119909 119910) =

0 if 119892119894(119909 119910) lt 120582

255 if 119892119894(119909 119910) ge 120582

(8)

where 119892119894+1(119909 119910) is the new pixel value and 119892

119894(119909 119910) is the

old pixel value In other words after choosing a thresholdthen every pixel in the image is compared with this thresholdand if the pixel lies above the threshold it will be marked asforeground and if it is below the threshold it will bemarked asbackgroundThe histogram-basedmethods are also based onpixelsrsquo intensity values Here histogram bars help to find theclusters of pixels values One of the most famous threshold-based methods is Otsursquos method [19]

The edge-based methods show boundaries in the imagedetermining different regions where we have to decide if theyare foreground or backgroundThe boundaries are calculatedanalyzing high contrasts in intensity color or texture On theother hand an opposed point of view are the region-basedmethods divide the image into regions searching for sametextures colors or intensity values

In soil science the porosity of a porousmedium is definedby the ratio of the void area and the total bulk areaThereforeporosity is a fraction whose numerical value is between 0 and1 typically ranging from 0005 to 0015 for solid granite to 02to 035 for sand It may also be represented in percent termsbymultiplying the number by 100 Porosity is a dimensionlessquantity and can be reported either as a decimal fraction oras a percentage

The total porosity of a porous medium is the ratio of thepore volume to the total volume of a representative sampleof the medium Assuming that the soil system is composedof three phasesmdashsolid liquid (water) and gas (air)mdashwhere119881119904is the volume of the solid phase 119881

119897is the volume of the

liquid phase119881119892is the volume of the gaseous phase119881

119901= 119881119897+

119881119892is the volume of the pores and 119881

119905= 119881119904+ 119881119897+ 119881119892is the

total volume of the sample then the total porosity of the soilsample 119901

119905 is defined as follows

119901119905=

119881119901

119881119905

=

119881119897+ 119881119892

119881119904+ 119881119897+ 119881119892

(9)

Table 1 lists representative porosity ranges for variousgeologicmaterials [30] In general porosity values for uncon-solidated materials lie in the range of 025ndash07 (ie 25ndash70) Coarse-textured soil materials (such as gravel andsand) tend to have a lower total porosity than fine-textured

Table 1 Range of porosity values

Unconsolidateddeposits Porosity Rocks Porosity

Gravel 025ndash040 Fractured basalt 005ndash050Sand 025ndash050 Karst limestone 005ndash050Silt 035ndash050 Sandstone 005ndash030Clay 040ndash070 Limestone dolomite 000ndash020

Shale 000ndash010Fractured crystallinerock 000ndash010

Dense crystalline rock 000ndash005

soils (such as silts and clays) Porosity values in soils are nota constant quantity because the soil particularly clayey soilalternately swells shrinks compacts and cracksTheporosityof our test image shown in Figure 10(a) is 0284

Our work applies image segmentation techniques tocalculate the porosity of soil images Also we have comparedour results with the Otsu image segmentation algorithm

3 Methodology

31 Laplacian Pyramid The Laplacian pyramid representa-tion expresses the original image as a sum of spatially band-passed images while retaining local spatial information ineach band The Gaussian pyramid is created by low-pass-filtering an image 119866

0with a two-dimensional compact filter

The filtered image is then subsampled by removing everyother pixel and every other row to obtain a reduced image1198661 Graphical representations of these processes in one and

two dimensions are given in Figures 2 and 3This process is repeated to form a Gaussian pyramid

1198660 1198661 1198662 119866

119873

119866119896(119894 119895) = sum

119898

sum

119899

119866119896minus1

(2119894 + 119898 2119895 + 119899) 119896 = 1 119873 (10)

Expanding 1198661to the same size as 119866

0and subtracting

yields the band-passed image 1198710 a Laplacian pyramid

1198710 1198711 119871

119899minus1can be built containing band-passed images

of decreasing size and spatial frequency

119871119896= 119866119896minus 119866119896+1

119896 = 0 119873 minus 1 (11)

where the expanded image 119866119896is given by

119866119896(119894 119895) = 4sum

119898

sum

119899

119908 (119898 119899) 119866119896minus1

(

2119894 + 119898

2

2119895 + 119899

2

) (12)

The original image can be reconstructed from the expandedbandpass images

1198660= 1198710+ 1198661

= 1198710+ 1198711+ 1198662

= 1198710+ 1198711+ 1198712+ sdot sdot sdot + 119871

119873minus1+ 119866119873

(13)


G2

G1

G0

Figure 2 Representation of the one-dimensional Gaussian pyramidprocess

G1

G0

Figure 3 Representation of the two-dimensional Gaussian pyramidprocess

The Gaussian pyramid contains low-passed versions of theoriginal 119866

0at progressively lower spatial frequencies while

the Laplacian pyramid consists of band-passed copies of theoriginal image 119866

0 Each Laplacian level contains the edges of

a certain size and spans approximately an octave in spatialfrequency

32 Fractal Dimension Box-Counting Dimension Fractaldimension is a useful feature for texture segmentation shapeclassification and graphic analysis in many fields The box-counting approach is one of the frequently used techniquesto estimate the fractal dimension of an image

There are several methods available to estimate thedimension of fractal sets The Hausdorff dimension is theprincipal definition of fractal dimension However there areother definitions like box-counting or box dimension thatis popular due to its relative ease of mathematical calculationand empirical estimation The main idea to most definitionsof fractal dimension is the idea ofmeasurement at scale 120575 Foreach 120575 we measure a set ignoring irregularities of size lessthan 120575 and then we see how these measurements behave as120575 rarr 0 For example if 119865 is a plane curve (one of our filters)then 119873

120575(119865) might be the number of steps required by a pair

of dividers set at length 120575 to traverse 119865 Then the dimensionof 119865 is determined by the power law if any exists obeyed by119873120575(119865) as 120575 rarr 0 So we might say that 119865 has dimension 119904 if

a constant 119904 exists so that

120575119904

sdot 119873120575(119865) ≃ 1 (14)

where taking logarithms and limits when 120575 tends to 0 we get(7)

These formulae are appealing for computational or exper-imental purposes since 119904 can be estimated as the gradient ofa log-log graph plotted over a suitable range of 120575

33 Kolmogorov-Smirnov Normality Test In order to deter-mine the normality interval we use the Kolmogorov-Smirnovnormality test [31 32] which is the most usual empiricaldistribution function test for normality

For a data set119860 of 119899wemake the contrast of a distributionfunction 119865

119899from a theoretical distribution function 119865 using

the statistic119863119899

119863119899= 119863119899(119865119899 119865) = max

119909isin119860

1003816100381610038161003816119865119899(119909) minus 119865 (119909)

1003816100381610038161003816 (15)

that represents the distance between 119865119899and 119865

For 119899 large enough the statistical distribution of 119863119899is

close to the Kolmogorov-Smirnov distribution K which istabulated for some significant values Obviously the assump-tion of normality is rejected with significance level 1 minus 120572 if119863119899gt 119863119899120572 with 119875(K le 119863

119899120572) = 120572

Let 119863119899120572

be the K-S distribution percentiles We reject atlevel 1 minus 120572 because if 119899 is big enough 119863

119899= K and 120572 = 001

Then we reject small values of119863119899 so if |119865

119899minus119865| is smaller than

the percentile 120572 we accept the hypothesis

34 Otsursquos Thresholding Method The most common imagesegmentationmethods are the histogram thresholding basedsince thresholding is easy fast and economical in compu-tation For performing the image segmentation we need tocalculate a threshold which will separate the objects and thebackground in our image Since soil images are relativelysimple when we just pay attention to void and bulk so we aregoing to apply the global threshold technique instead ofmoreadvanced variations (band thresholding local thresholdingand multithresholding) The global thresholding techniqueconsists of selecting one threshold value and applying it tothe whole image

The resultant image is a binary image where pixels thatcorrespond to objects and background have values of 255and 0 respectively Quick and simple calculation is the mainadvantage of global thresholding

Otsursquos method searches for the threshold that minimizesthe intraclass variance (or within class variance) 120590

2

119882(119905)

defined as the weighted sum of variances of the two classes

1205902

119908(119905) = 120596

0(119905) 1205902

0(119905) + 120596

1(119905) 1205902

1(119905) (16)

where 120596119894are the probabilities of the two classes (foreground

and background) separated by a threshold 119905 and 1205902119894are the

variances of these both classesOtsu [19] proofed that minimizing the intraclass variance

is the same as maximizing the interclass variance 1205902

119887(119905)

defined as follows

1205902

119861(119905) = 120590

2

minus 1205902

119882(119905) = 120596

0(119905) 1205961(119905) [1205830(119905) minus 120583

1(119905)]2

(17)

where 120596119894are the class probabilities and 120583

119894the class means

The class probabilities 1205960(119905) and 120596

1(119905) and the class

means 1205830(119905) and 120583

1(119905) are computed as

1205960(119905) =

119905

sum

0

119901 (119894) 1205961(119905) = sum

119894gt119905

119901 (119894)

1205830(119905) =

119905

sum

0

119901 (119894) 119909 (119894) 1205831(119905) = sum

119894gt119905

119901 (119894) 119909 (119894)

(18)

where 119909(119894) is the centered value of the 119894th histogram bin


The class probabilities and class means can be computediteratively This idea yields an effective algorithm

Otsursquos algorithm assumes just only two sets of pixelintensities the foreground and the background or void andbulk for soil imagesThe main idea of the Otsursquos method is tominimize the weighted sum of within-class variances of theforeground and background pixels to establish an optimumthreshold It can be formulated as

120582Otsu = arg min 1205960(120582) 1205902

0(120582) + 120596

1(120582) 1205902

1(120582) (19)

where theweights120596119894(120582) are the probabilities of the two classes

separated by the threshold 120582 and 1205902119894(120582) are the corresponding

variances of these classes Otsursquos method gives satisfactoryresults when the values of pixels in each class are close to eachother as in soil images

Let the pixels of a given picture be represented in 256 graylevels 0 1 2 255 Let the number of pixels at level 119894 bedenoted by 119899

119894and the total number of pixels by119873 If we define

119901119894as 119901119894= 119899119894119873 then 119901

119894ge 0 and sum119901

119894= 1 So we have a

probability distribution point of viewThe threshold at level 119896 defines two classes the fore-

ground (1198620) and the background (119862

1)Then the probabilities

of these classes and their means are

1205960= Pr (119862

0) =

119896

sum

119894=0

119901119894= 120596 (119896)

1205961= Pr (119862

1) =

255

sum

119894=119896+1

119901119894= 1 minus 120596 (119896)

1205830=

119896

sum

119894=0

119894Pr (119894 | 1198620) =

119896

sum

119894=0

119894119901119894

1205960

=

120583 (119896)

120596 (119896)

1205831=

255

sum

119894=119896+1

119894Pr (119894 | 1198621) =

255

sum

119894=119896+1

119894119901119894

1205961

=

120583119879minus 120583 (119896)

1 minus 120596 (119896)

(20)

where

120596 (119896) =

119896

sum

119894=0

119901119894

120583 (119896) =

119896

sum

119894=0

119894119901119894 120583

119879=

255

sum

0

119894119901119894 (21)

We can easily verify that for any 119896 1205960+ 1205961= 1 and 120596

01205830+

12059611205831= 120583119879 Now we define the class variances as

1205902

0=

119896

sum

119894=0

(119894 minus 1205830)2Pr (119894 | 119862

0) =

119896

sum

119894=0

(119894 minus 1205830)2

119901119894

1205960

1205902

1=

255

sum

119894=119896+1

(119894 minus 1205831)2Pr (119894 | 119862

1) =

255

sum

119894=119896+1

(119894 minus 1205831)2

119901119894

1205961

(22)

Now we are going to apply the discriminant analysis toevaluate and quantify the threshold at level 119896 using the

measures of class separability 120582 120581 and 120578 based on the within-class variance1205902

119882 the between-class variance1205902

119861 and the total

variance 1205902119879 defined as

120582 =

1205902

119861

1205902

119882

120581 =

1205902

119879

1205902

119882

120578 =

1205902

119861

1205902

119879

1205902

119882= 12059601205902

0+ 12059611205902

1

1205902

119861= 1205960(1205830minus 120583119879)2

+ 1205961(1205831minus 120583119879)2

= 12059601205961(1205831minus 1205830)2

1205902

119879=

255

sum

119894=0

(119894 minus 120583119879)2

119901119894

(23)

4 Results

41 Classification of 1D Filters Our 1D filters are defined bythe weighting function119908(119898) and the pyramidal constructionis equivalent to convolving repeatedly the original signalwith this weighting functions Some of these Gaussian-likeweighting functions are shown in Figure 5

Note that the functions double in width with each levelThe convolution acts as a low-pass filter with the bandlimit reduced correspondingly by one octave with eachlevel Because of this resemblance to the Gaussian densityfunction we refer to the pyramid of low-pass images asthe Gaussian pyramid Just as the value of each node inthe Gaussian pyramid could have been obtained directly byconvolving a Gaussian-like equivalent weighting functionwith the original image each value of this bandpass pyramidcould be obtained by convolving a difference of twoGaussianswith the original imageThese functions closely resemble theLaplacian operators commonly used in image processing Forthis reason the bandpass pyramid is known as a Laplacianpyramid An important property of the Laplacian pyramid isthat it is a complete image representation the steps used toconstruct the pyramidmay be reversed to recover the originalimage exactly The top pyramidal level 119871

119873 is first expanded

and added to 119871119873minus1

to form119866119873minus1

Then this array is expandedand added to 119871

119873minus2to recover 119866

119873minus2 and so on

The weighting function 119908(119898) is determined by theseconstraints

Symmetry 119908 (119898) = 119908 (minus119898)

Normalization2

sum

119898=minus2

119908 (119898) = 1

Equal contribution sum

119898 odd119908 (119898) = sum

119898 even119908 (119898)

(24)

Normalization symmetry and equal contribution are satis-fied when

119908 (0) = 119886

119908 (1) = 119908 (minus1) =

1

4

119908 (2) = 119908 (minus2) =

1

4

minus

119886

2

(25)


If the size is 5 we have the filter 119908

(1199080[119896]) = [

1

4

minus

119886

2

1

4

119886

1

4

1

4

minus

119886

2

] 119896 = 1 5 (26)

which represents a uniparametric family of weighting func-tions with parameter 119886 Observe that if the size is biggerthan 5 constraints (24) generate a multiparametric family ofweighting functions

Convolution is a basic operation of most signal analysissystemsWhen the convolution and decimation operators areapplied repeatedly 119899 times they generate a new equivalentfilter 119908

119899 whose length is

1198720= 5

119872119899= 2119872

119899minus1+ 3 119899 ge 1

(27)

where1198720is the length of the initial filter 119908

0

Figure 4 shows an example of several iterations of thefilter for 119886 = 04 We can observe that there is a quickconvergence to a stable shape And in this case the shape ofthe plot of the limit filter is Gaussian

We have tested different values 119886 from 119886 = 01 to 119886 = 12The shape of the filter the equivalent weighting functiondepends on the choice of parameter 119886 There are severaldifferent shapes for different values of 119886 Gaussian-like andfractal-like Figure 5 shows some examples of Gaussian andfractal shapes

The first two filters are Gaussian-like and the last two arefractal-like It is possible to confirm these early conclusionsWe have successfully applied normality tests to verify thenormality of the filters obtained with the lowest 119886 valuesOn the other hand in fractal geometry the box-countingdimension is a way of determining the fractal dimension ofa set To calculate the box-counting dimension for a set 119878we draw an evenly-spaced grid over the set and count howmany boxes are required to cover the set The box-countingdimension is calculated by applying (7)

We can see the results of fractal dimension (FD) of filters1D whose values are shown in Table 2 and Figure 9(a) Fromthese results and this figurewe can observe a fractal behaviourwhen 119886 lt 0 and when 119886 gt 06

The generation of bidimensional filters 119908(119898 119899) alsocalled generating kernels or mask filters is based on thecondition

Separability 119908 (119898 119899) = 119908 (119898)119908 (119899) (28)

So a filter119908(119898 119899) is called separable if it can be broken downinto the convolution of two filtersThis property is interestingbecause if we can separate a filter into two smaller filters thenusually it is computationally more efficient and quicker toapply both of them instead the original one We work with2D filters that can be separated into horizontal and verticalfilters 2D filters have been obtained by sequentially applyingthe same one-dimensional filter on rows and columns

When we calculate the bidimensional filters we obtainfilters like Figure 6 These are obtained when 119886 = 04 and119886 = 08

Table 2 Fractal dimension of filters

119886 Fractal dimensionminus02 1578

minus01 1240

00 1011

01 1020

02 1009

03 1009

04 1004

05 1009

06 1022

07 1142

08 1141

09 1209

10 1248

11 1276

12 1289

42 Normality Interval There is a relevant result whenwe study the normality of filters the Gaussian function isseparable if variables are independent Burt and Adelsonshow that if we choose 119899 = 5 and 119886 = 04 then the equivalentweight function is Gaussian [18] Indeed there is an intervalfor the parameter 119886 where we can see the Gaussian behavior

Once we have a chosen value for 119886 and a pyramidal depthvalue 119896 we need to select the best normal distribution119873(120583 )that fits our weights We estimate 120583 by the arithmetic meanand by the minim of all 119863

119899in an empirical confidence

interval calculated asCI ()

= (max (min (1 2) minus 1 0) max (min (

1 2) + 1 0))

(29)

with 1and

2two initial estimations of

Because the symmetry property of the filter 119908 we havethat the arithmetic mean is the central point so

120583 =

119872119895+ 1

2

= 2119895+1

minus 1 (30)

because the 119895th level is119872119895= 2119895+2

minus 3 which is the solutionof the linear recurrence relation (27)

Conditions to calculate 1and

2are the adjust to the

histogram of the filter 119908 and the normal distribution relatedto two known percentiles specifically

Pr (119873 (120583 120590) le 120583 minus 1199111120590) =

1 minus 1199011

2

Pr (119873 (120583 120590) le 120583 minus 1199112120590) =

1 minus 1199012

2

(31)

so we have the system of equations

1= 120583 minus 119911

11198881199011

2=

120583 minus 1198881199012

1199112

(32)


1 2 3 4 50

005

01

015

02

025

03

035

04

045

05

2 4 6 8 10 12

002

004

006

008

01

012

014

016

018

10 20 30 40 50 60

0005

001

0015

002

0025

003

0035

004

k

kk

w[k]

w2[k]

w4[k]

Figure 4 First second and fourth iteration of the filter 1D (119886 = 04)

where 1198881199011

and 1198881199012

are the corresponding abscissas to per-centiles 119901

1and 119901

2

Specifically the numerical simulation with 119899 = 5 1199011=

0682 and 1199012= 095 (that correspond to the normalized

abscissas 1199111= 1 and 119911

2= 196) generates the normality

intervals which determinates the estimation for of thenormal distribution applied Corresponding values for thiscase are shown in Figure 7(a) As we can see 119886 = 039 is theminimal value and so it has the best adaptation to a normaldistribution as shown in Figure 7(b)

Figure 7(a) shows the value of the statistic 119863119899as a

function of the parameter 119886 and the normal threshold fora confidence level of 99 For values of 119886 in the interval[minus008 057] the assumption of normality is not rejected119863119899attains the minimum for 119886 = 039 corresponding

to estimated Gaussian distribution with mean 127 andstandard deviation 3674 Figure 7(b) shows the filter 119908

6

and the estimated density where we can see the adjustmentgoodness

43 Image Segmentation and Performance Evaluation Afterthese results we have generated the Gaussian and Lapla-cian pyramids corresponding to one Gaussian value for 119886and another fractal value for 119886 applying the methodologypreviously shown getting the corresponding Gaussian andLaplacian pyramids and then we have compared the resultsobtained Figure 8 shows pyramidal sets for 119886 = 04 and 119886 =12 corresponding to a Gaussian and a fractal respectively

When we have applied our method to segment imageswith different values for the parameter 119886 we have obtainedthe threshold values shown in Figure 9(b) and results asshown in the Figure 10 We can compare these results withOtsursquos method threshold value 0259


100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12

Figure 5 Sixth iteration of 1D Laplacian pyramid filters 1199086[119896] for different 119886 values

The application of our methods with different values 119886produces the results shown in Figure 10(b) together withOtsursquos value Aswe can see ourmethod obtains similar resultsfor fractal threshold values

44 Pore Size Distribution We have presented thresholdvalues obtained from Laplacian pyramid and the comparisonwith Otsursquos values On the other hand if we compare thepore size frequency distribution obtained by Otsursquos method

and the threshold obtained based on Laplacian filter structuresome difference is observed as we can see in Figure 11 In thesmallest size range between 5 to 20 pixels the formermethodpresents higher porosity and the decrease in frequency ismuch smoother than with the latest method Even thedifference in porosity is not significant Otsursquos method gives275 and this method estimates 311 the difference in sizescould affect several percolation models These results areshowed in Figure 11 Finally the porosity obtained is this


(a) (b)

Figure 6 Bidimensional filters corresponding to 119886 = 04 (a) and 119886 = 08 (b)

Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k

(b) Density function of normal distribution (black) andweighting function for 119886 = 039 (red)

Figure 7 Gaussian filter adjustment

study approaches Otsursquos result when 119886 value increases fromthe range 0 till 12

5 Conclusions

Thefield of fractals has been developed as an interdisciplinaryarea between branches ofmathematics and physics and foundapplications in different sciences and engineering fields Ingeo-information interpretation the applications developedfrom simple verifications of the fractal behavior of naturalland structures simulations of artificial landscapes andclassification based on the evaluation of the fractal dimensionto advanced remotely sensed image analysis scene under-standing and accurate geometric and radiometric modelingof land and land cover structures

Referring to the computational effort fractal analysisgenerally asks high complexity algorithms Both wavelets andhierarchic representation allow now the implementation offast algorithms or parallel ones As a consequence a devel-opment of new experiments and operational applications isexpected

We have seen that the different choice of the parameter119886 gives two kinds of filters the Gaussian-like and the fractal-likeThis is demonstrated applying normality tests (Gaussian)and fractal dimension techniques (fractal) analytical andgraphical in both cases

The different shape of filters Gaussianfractal has per-ceptible effects when we generate new levels of the Gaussianand Laplacian pyramids getting blurred or accentuated new


(a) (b)

Figure 8 Gaussian and Laplacian pyramids of the soil image ((a) 119886 = 04 (b) 119886 = 12)

Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a

(a) Fractal dimension of filters 1D

Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01

(b) Threshold values from Laplacian pyramid filters and Otsursquos tech-nique

Figure 9 Fractal dimensions and threshold values

images at every new level Filters generated with lowest 119886values produce blurred edges and images On the other handfilters obtained from highest 119886 values generate new imageswith higher contrasts and sharp edges

Also there is a different behaviour of the energy of thedifferent levels of the Laplacian pyramid if we choose different119886 values that is if we choose a Gaussian or fractal filter TheGaussian-like filters always make a lower energy image Thisfall is slowbut there is always a fall On the other hand fractal-like filters also fall but this happens after several iterations

and then the fall is bigger than the fall of the Gaussian-likefilters

These filters can be applied to image segmentation of soilimages with a simple computation and good results quitesimilar or even better to some famous techniques such asOtsursquos method

Moreover results concerning porosity are similar butthere are differences in pore size distribution that couldimprove percolation simulationsThe implementation of thismethod in three dimensions is straightforward


(a) Original (b) Otsu (c) 119886 = 02

(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12

Figure 10 Soil image and several segmentations

5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y

Pore size distribution

OtsuLaplacian pyramid

Figure 11 Pore size distribution

Future work could add other image segmentation tech-niques and neural networks methods to select the optimalthreshold values from information and characteristics of theimage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P C Baveye M Laba W Otten et al ldquoObserver-dependentvariability of the thresholding step in the quantitative analysis ofsoil images and X-ray microtomography datardquo Geoderma vol157 no 1-2 pp 51ndash63 2010

[2] P Lehmann P Wyss A Flisch et al ldquoTomographical imagingand mathematical description of porous media used for theprediction of fluid distributionrdquoVadose Zone Journal vol 5 no1 pp 80ndash97 2006

[3] H-J Vogel U Weller and S Schluter ldquoQuantification ofsoil structure based on Minkowski functionsrdquo Computers andGeosciences vol 36 no 10 pp 1236ndash1245 2010

[4] J S Perret S O Prasher and A R Kacimov ldquoMass fractaldimension of soil macropores using computed tomographyfrom the box-counting to the cube-counting algorithmrdquo Euro-pean Journal of Soil Science vol 54 no 3 pp 569ndash579 2003

[5] A M Tarquis R J Heck D Andina A Alvarez and J MAnton ldquoPore network complexity and thresholding of 3D soilimagesrdquo Ecological Complexity vol 6 no 3 pp 230ndash239 2009

[6] WWang A N Kravchenko A J M Smucker andM L RiversldquoComparison of image segmentation methods in simulated


2D and 3D microtomographic images of soil aggregatesrdquoGeoderma vol 162 no 3-4 pp 231ndash241 2011

[7] P Iassonov and M Tuller ldquoApplication of segmentation forcorrection of intensity bias in x-ray computed tomographyimagesrdquo Vadose Zone Journal vol 9 no 1 pp 187ndash191 2010

[8] P Iassonov T Gebrenegus and M Tuller ldquoSegmentation ofX-ray computed tomography images of porous materials acrucial step for characterization and quantitative analysis ofpore structuresrdquoWater Resources Research vol 45 no 9 2009

[9] M Sezgin and B Sankur ldquoSurvey over image thresholdingtechniques and quantitative performance evaluationrdquo Journal ofElectronic Imaging vol 13 no 1 pp 146ndash168 2004

[10] A E Scheidegger The Physics of Flow Through Porous MediaUniversity of Toronto Press 1974

[11] G W Horgan A Review of Soil Pore Models 1996[12] E C Childs and N Collisgeorge ldquoThe permeability of porous

materialsrdquo Proceedings of the Royal Society of London Series Avol 201 no 1066 pp 392ndash405 1950

[13] T J Marshall ldquoA relation between permeability and sizedistribution of poresrdquo Journal of Soil Science vol 9 no 1 pp1ndash8 1958

[14] A R Dexter ldquoInternal structure of tilled soilrdquo Journal of SoilScience vol 27 no 3 pp 267ndash278 1976

[15] C J Moran and A B McBratney ldquoA two-dimensional fuzzyrandom model of soil pore structurerdquo Mathematical Geologyvol 29 no 6 pp 755ndash777 1997

[16] H J Vogel ldquoA numerical experiment on pore size poreconnectivity water retention permeability and solute transportusing network modelsrdquo European Journal of Soil Science vol 51no 1 pp 99ndash105 2000

[17] J W Crawford N Matsui and I M Young ldquoThe relationbetween the moisture-release curve and the structure of soilrdquoEuropean Journal of Soil Science vol 46 no 3 pp 369ndash375 1995

[18] P J Burt and E H Adelson ldquoThe laplacian pyramid as acompact image coderdquo IEEE Transactions on Communicationsvol 31 no 4 pp 532ndash540 1983

[19] N Otsu ldquoA threshold selection method from gray-level his-togramsrdquo IEEE Transactions on Systems Man and Cyberneticsvol 9 no 1 pp 62ndash66 1979

[20] S Tanimoto and T Pavlidis ldquoA hierarchical data structure forpicture processingrdquo Computer Graphics and Image Processingvol 4 no 2 pp 104ndash119 1975

[21] H J A M Heijmans and J Goutsias ldquoNonlinear mul-tiresolution signal decomposition schemes II Morphologicalwaveletsrdquo IEEE Transactions on Image Processing vol 9 no 11pp 1897ndash1913 2000

[22] H O Peitgen H Jurgens and D Saupe Chaos and FractalsNew Frontiers of Science Springer 2004

[23] D L Turcotte Fractals and Chaos in Geology and GeophysicsCambridge University Press 1997

[24] B B Mandelbrot The Fractal Geometry of Nature Henry Holtand Company San Francisco Calif USA 1982

[25] K J Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons 1990

[26] J Belair and S Dubuc ldquoFractal geometry and analysisrdquo in Pro-ceedings of the NATOAdvanced Study Institute and Seminaire deMathematiques Superieures on Fractal Geometry and AnalysisKluwer Academic Publishers Montreal Canada July 1989

[27] B B Chaudhuri and N Sarkar ldquoTexture segmentation usingfractal dimensionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 17 no 1 pp 72ndash77 1995

[28] T Ida and Y Sambonsugi ldquoImage segmentation and contourdetection using fractal codingrdquo IEEE Transactions on Circuitsand Systems for Video Technology vol 8 no 8 pp 968ndash9751998

[29] R C Gonzalez R E Woods and S L Eddins Matlab Digital Image Processing Using MATLAB Publishing House ofElectronics Industry Beijing China 2009

[30] R A Freeze and J A CherryGroundwater Prentice-Hall 1979[31] I M Chakravarti R G Laha and J RoyHandbook of Methods

of Applied Statistics John Wiley amp Sons 1967[32] J R Brown andM E Harvey ldquoArbitrary precisionmathematica

functions to evaluate the one-sided one sample K-S cumulativesampling distributionrdquo Journal of Statistical Software vol 26no 3 pp 1ndash55 2008

Submit your manuscripts athttpwwwhindawicom

Forestry ResearchInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of



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MeteorologyAdvances in

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Marine BiologyJournal of


Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

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Atmospheric SciencesInternational Journal of



Waste ManagementJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Geological ResearchJournal of

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OceanographyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


ClimatologyJournal of


More sophisticated approaches are [14] using a one-dimensional Markov chain model for horizontal soil [15]proposing a two-dimensional fuzzy random model of soilpore structure and [16] describing a network model topredict physical properties from topological parameters andfractal-based approaches like [17]

Our goal is to calculate the porosity of soil images Theproposed procedure for segmentation of soil micromorpho-logical images is based on Laplacian pyramid algorithm[18] from which we compute a threshold that will binarizethe original image resulting with an image composed ofcontinuous regions of pores (shown in black) and soil (shownin white) From this binarized soil image we compute anestimation to its porosity

Another objective of this study is also to compare theresults with those provided by the commonly used Otsursquosalgorithm [19] that is widely accepted as a good method toget an appropriate threshold

2 Materials and Methods

21 Laplacian Pyramid A multiresolution model consists ofgenerating different versions of a given image by decreasingthe initial resolution which also means decreasing the initialsize This is achieved by a downsampling operator whichmust be associated with an appropriate filtering to avoidaliasing phenomena (downsampling theorem) Multiresolu-tion approaches have been investigated for different purposessuch as image segmentation and image compression [18] Interms of image analysis low resolution representations areconvenient for global detection and recognition of imagefeatures while minute details can only be seen on highresolution images

One usual property of images is that neighboring pixelsare highly correlated This property is inefficient to rep-resent the complete image directly in terms of its pixelvalues because most of the encoded information wouldbe redundant Burt and Adelson designed a techniquenamed Laplacian pyramid for removing image correlationwhich combines characteristics of predictive and transformmethods [18]This technique is noncausal and computationsare relatively simple and local The predicted value for eachnew pixel is computed as a local weighted average using aunimodal weighting function centered on the pixel itself

This pyramidal representation is useful for two importantclasses of computer graphics problems The first class iscomposed of those tasks that involve analysis of existingimages such as merging images or interpolating to fill inmissing data smoothly become much more intuitive whenwe canmanipulate easily visible local image features at severalspatial resolutions And the second whenwe are synthesizingimages the pyramid becomes a multiresolution sketch padWe can fill in the local spatial information at increasingly finedetail by specifying successive levels of a pyramid

The first time that pyramidal structures were applied tomultiresolution decompositions was at [18] Later the rela-tionship to wavelets was realized shortly thereafter becauseboth decompositions are based on the idea of successiverefinement the image is obtained as a sum of an initial coarse

version plus detail signals One interesting thing to noteabout the pyramidal approach is that perfect reconstructionis possible therefore it is a lossless data algorithm

Pyramidal methods for multiresolution image analysishave been used since the 1970s Early work inmultiresolutionimage description was primarily motivated by a desire toreduce the computational cost of methods for image descrip-tion and image matching Later multiresolution processingwas generalized to computing multiple copies of an imageby repeatedly summing nonoverlapping blocks of pixels andresampling until the image is reduced to a small number ofpixels Such a structure became known as a multiresolutionpyramid [20]

Interest in multiresolution techniques for signal process-ing and analysis is increasing steadily [21] An importantinstance of such a technique is the so-called pyramidaldecomposition scheme Our work uses a general axiomaticpyramidal decomposition scheme for soil image analysisThis scheme comprises the following ingredients

(i) The pyramid consists of a finite number of levelssuch that the information content decreases towardshigher levels

(ii) Each step towards a higher level is constituted byan information-reducing analysis operator whereaseach step towards a lower level is modeled by aninformation-preserving synthesis operator One basicassumption is necessary synthesis followed by anal-ysis yields the identity operator meaning that noinformation is lost by these two consecutive steps

22 Fractal Dimension The techniques based on fractalsshow promising results in the field of image understandingand visualization of high complexity data

The high complexity of some images demands newtechniques for understanding and analyzing them The sim-ilarity of fractals and real world objects has been observedand studied from the very beginning The fractal geometrybecame a tool for computer graphics and data visualizationin the simulation of the real world In order to perform visualanalysis and comparisons between natural and syntheticscenes several techniques have been developed After a periodof qualitative experiments fractal geometry began to beused for objective and accurate purposes modeling imagesevaluating their characteristics analyzing their textures andso forth

Nowadays there are a lot of fields where fractals appear[22] First of all we present some of the elementary ideasnecessary to understand applications of fractal geometry ingeo-information processing [23]

Fractal geometry theory deals with the behaviors of sets ofpoints 119878 in the 119899-dimensional spaceR119899 Images particularlysoil images are sets of points in R2

Mandelbrot defined a fractal as a shape made of partssimilar to the whole in some way [24] That definition isqualitative but not ambiguous as it looks at the first glanceThe main behavior of a fractal is its self-similarity A set iscalled self-similar if it can be expressed as a union of sets eachof which is a reduced copy of the full set More generally a set






that

119878 sub ⋃

119894isin119868

119880119894sub R119899 with 0 lt

1003816100381610038161003816119880119894

1003816100381610038161003816lt 120575 (1)


|119880| = sup 1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 119909 119910 isin 119880 (2)


define

119867119903

120575(119878) = inf

RisinR120575

sum

119894isin119868

1003816100381610038161003816119880119894

1003816100381610038161003816

119903

R = ⋃

119894isin119868

119880119894 (3)


119867119903

(119878) = lim120575rarr0

119867119903

120575(119878) (4)





119867119901

(119878) =

infin if 119901 lt 1199040 if 119901 gt 119904

(5)


dim119867119878 = sup 119903 119867119903 (119878) = infin = inf 119903 119867119903 (119878) = 0 (6)



119867 of the




infin

s p

Hp(S)





dim119861119878 = lim120575rarr0

log119873120575

log 1120575 (7)











119892119894+1

(119909 119910) =

0 if 119892119894(119909 119910) lt 120582

255 if 119892119894(119909 119910) ge 120582

(8)


119894(119909 119910) is the







119901= 119881119897+


119905= 119881119904+ 119881119897+ 119881119892is the



119901119905=

119881119901

119881119905

=

119881119897+ 119881119892

119881119904+ 119881119897+ 119881119892

(9)









3 Methodology





1198660 1198661 1198662 119866

119873

119866119896(119894 119895) = sum

119898

sum

119899

119866119896minus1

(2119894 + 119898 2119895 + 119899) 119896 = 1 119873 (10)


0and subtracting


1198710 1198711 119871



119871119896= 119866119896minus 119866119896+1

119896 = 0 119873 minus 1 (11)


119866119896(119894 119895) = 4sum

119898

sum

119899

119908 (119898 119899) 119866119896minus1

(

2119894 + 119898

2

2119895 + 119899

2

) (12)


1198660= 1198710+ 1198661

= 1198710+ 1198711+ 1198662

= 1198710+ 1198711+ 1198712+ sdot sdot sdot + 119871

119873minus1+ 119866119873

(13)


G2

G1

G0


G1

G0












120575119904

sdot 119873120575(119865) ≃ 1 (14)







119863119899= 119863119899(119865119899 119865) = max

119909isin119860

1003816100381610038161003816119865119899(119909) minus 119865 (119909)

1003816100381610038161003816 (15)




119899120572) = 120572

Let 119863119899120572


119899= K and 120572 = 001







2

119882(119905)


1205902

119908(119905) = 120596

0(119905) 1205902

0(119905) + 120596

1(119905) 1205902

1(119905) (16)





119887(119905)

defined as follows

1205902

119861(119905) = 120590

2

minus 1205902

119882(119905) = 120596

0(119905) 1205961(119905) [1205830(119905) minus 120583

1(119905)]2

(17)





means 1205830(119905) and 120583


1205960(119905) =

119905

sum

0

119901 (119894) 1205961(119905) = sum

119894gt119905

119901 (119894)

1205830(119905) =

119905

sum

0

119901 (119894) 119909 (119894) 1205831(119905) = sum

119894gt119905

119901 (119894) 119909 (119894)

(18)





120582Otsu = arg min 1205960(120582) 1205902

0(120582) + 120596

1(120582) 1205902

1(120582) (19)






119901119894as 119901119894= 119899119894119873 then 119901

119894ge 0 and sum119901






1205960= Pr (119862

0) =

119896

sum

119894=0

119901119894= 120596 (119896)

1205961= Pr (119862

1) =

255

sum

119894=119896+1

119901119894= 1 minus 120596 (119896)

1205830=

119896

sum

119894=0

119894Pr (119894 | 1198620) =

119896

sum

119894=0

119894119901119894

1205960

=

120583 (119896)

120596 (119896)

1205831=

255

sum

119894=119896+1

119894Pr (119894 | 1198621) =

255

sum

119894=119896+1

119894119901119894

1205961

=

120583119879minus 120583 (119896)

1 minus 120596 (119896)

(20)

where

120596 (119896) =

119896

sum

119894=0

119901119894

120583 (119896) =

119896

sum

119894=0

119894119901119894 120583

119879=

255

sum

0

119894119901119894 (21)


01205830+


1205902

0=

119896

sum

119894=0

(119894 minus 1205830)2Pr (119894 | 119862

0) =

119896

sum

119894=0

(119894 minus 1205830)2

119901119894

1205960

1205902

1=

255

sum

119894=119896+1

(119894 minus 1205831)2Pr (119894 | 119862

1) =

255

sum

119894=119896+1

(119894 minus 1205831)2

119901119894

1205961

(22)






120582 =

1205902

119861

1205902

119882

120581 =

1205902

119879

1205902

119882

120578 =

1205902

119861

1205902

119879

1205902

119882= 12059601205902

0+ 12059611205902

1

1205902

119861= 1205960(1205830minus 120583119879)2

+ 1205961(1205831minus 120583119879)2

= 12059601205961(1205831minus 1205830)2

1205902

119879=

255

sum

119894=0

(119894 minus 120583119879)2

119901119894

(23)

4 Results










Symmetry 119908 (119898) = 119908 (minus119898)

Normalization2

sum

119898=minus2

119908 (119898) = 1


119898 odd119908 (119898) = sum

119898 even119908 (119898)

(24)


119908 (0) = 119886

119908 (1) = 119908 (minus1) =

1

4

119908 (2) = 119908 (minus2) =

1

4

minus

119886

2

(25)



(1199080[119896]) = [

1

4

minus

119886

2

1

4

119886

1

4

1

4

minus

119886

2

] 119896 = 1 5 (26)




1198720= 5

119872119899= 2119872

119899minus1+ 3 119899 ge 1

(27)


0






Separability 119908 (119898 119899) = 119908 (119898)119908 (119899) (28)





minus01 1240

00 1011

01 1020

02 1009

03 1009

04 1004

05 1009

06 1022

07 1142

08 1141

09 1209

10 1248

11 1276

12 1289






1 2) + 1 0))

(29)

with 1and



120583 =

119872119895+ 1

2

= 2119895+1

minus 1 (30)






Pr (119873 (120583 120590) le 120583 minus 1199111120590) =

1 minus 1199011

2

Pr (119873 (120583 120590) le 120583 minus 1199112120590) =

1 minus 1199012

2

(31)


1= 120583 minus 119911

11198881199011

2=

120583 minus 1198881199012

1199112

(32)


1 2 3 4 50

005

01

015

02

025

03

035

04

045

05

2 4 6 8 10 12

002

004

006

008

01

012

014

016

018

10 20 30 40 50 60

0005

001

0015

002

0025

003

0035

004

k

kk

w[k]

w2[k]

w4[k]


where 1198881199011

and 1198881199012


1and 119901

2



abscissas 1199111= 1 and 119911






6





100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12






(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics



















that

119878 sub ⋃

119894isin119868

119880119894sub R119899 with 0 lt

1003816100381610038161003816119880119894

1003816100381610038161003816lt 120575 (1)


|119880| = sup 1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 119909 119910 isin 119880 (2)


define

119867119903

120575(119878) = inf

RisinR120575

sum

119894isin119868

1003816100381610038161003816119880119894

1003816100381610038161003816

119903

R = ⋃

119894isin119868

119880119894 (3)


119867119903

(119878) = lim120575rarr0

119867119903

120575(119878) (4)





119867119901

(119878) =

infin if 119901 lt 1199040 if 119901 gt 119904

(5)


dim119867119878 = sup 119903 119867119903 (119878) = infin = inf 119903 119867119903 (119878) = 0 (6)



119867 of the




infin

s p

Hp(S)





dim119861119878 = lim120575rarr0

log119873120575

log 1120575 (7)











119892119894+1

(119909 119910) =

0 if 119892119894(119909 119910) lt 120582

255 if 119892119894(119909 119910) ge 120582

(8)


119894(119909 119910) is the







119901= 119881119897+


119905= 119881119904+ 119881119897+ 119881119892is the



119901119905=

119881119901

119881119905

=

119881119897+ 119881119892

119881119904+ 119881119897+ 119881119892

(9)









3 Methodology





1198660 1198661 1198662 119866

119873

119866119896(119894 119895) = sum

119898

sum

119899

119866119896minus1

(2119894 + 119898 2119895 + 119899) 119896 = 1 119873 (10)


0and subtracting


1198710 1198711 119871



119871119896= 119866119896minus 119866119896+1

119896 = 0 119873 minus 1 (11)


119866119896(119894 119895) = 4sum

119898

sum

119899

119908 (119898 119899) 119866119896minus1

(

2119894 + 119898

2

2119895 + 119899

2

) (12)


1198660= 1198710+ 1198661

= 1198710+ 1198711+ 1198662

= 1198710+ 1198711+ 1198712+ sdot sdot sdot + 119871

119873minus1+ 119866119873

(13)


G2

G1

G0


G1

G0












120575119904

sdot 119873120575(119865) ≃ 1 (14)







119863119899= 119863119899(119865119899 119865) = max

119909isin119860

1003816100381610038161003816119865119899(119909) minus 119865 (119909)

1003816100381610038161003816 (15)




119899120572) = 120572

Let 119863119899120572


119899= K and 120572 = 001







2

119882(119905)


1205902

119908(119905) = 120596

0(119905) 1205902

0(119905) + 120596

1(119905) 1205902

1(119905) (16)





119887(119905)

defined as follows

1205902

119861(119905) = 120590

2

minus 1205902

119882(119905) = 120596

0(119905) 1205961(119905) [1205830(119905) minus 120583

1(119905)]2

(17)





means 1205830(119905) and 120583


1205960(119905) =

119905

sum

0

119901 (119894) 1205961(119905) = sum

119894gt119905

119901 (119894)

1205830(119905) =

119905

sum

0

119901 (119894) 119909 (119894) 1205831(119905) = sum

119894gt119905

119901 (119894) 119909 (119894)

(18)





120582Otsu = arg min 1205960(120582) 1205902

0(120582) + 120596

1(120582) 1205902

1(120582) (19)






119901119894as 119901119894= 119899119894119873 then 119901

119894ge 0 and sum119901






1205960= Pr (119862

0) =

119896

sum

119894=0

119901119894= 120596 (119896)

1205961= Pr (119862

1) =

255

sum

119894=119896+1

119901119894= 1 minus 120596 (119896)

1205830=

119896

sum

119894=0

119894Pr (119894 | 1198620) =

119896

sum

119894=0

119894119901119894

1205960

=

120583 (119896)

120596 (119896)

1205831=

255

sum

119894=119896+1

119894Pr (119894 | 1198621) =

255

sum

119894=119896+1

119894119901119894

1205961

=

120583119879minus 120583 (119896)

1 minus 120596 (119896)

(20)

where

120596 (119896) =

119896

sum

119894=0

119901119894

120583 (119896) =

119896

sum

119894=0

119894119901119894 120583

119879=

255

sum

0

119894119901119894 (21)


01205830+


1205902

0=

119896

sum

119894=0

(119894 minus 1205830)2Pr (119894 | 119862

0) =

119896

sum

119894=0

(119894 minus 1205830)2

119901119894

1205960

1205902

1=

255

sum

119894=119896+1

(119894 minus 1205831)2Pr (119894 | 119862

1) =

255

sum

119894=119896+1

(119894 minus 1205831)2

119901119894

1205961

(22)






120582 =

1205902

119861

1205902

119882

120581 =

1205902

119879

1205902

119882

120578 =

1205902

119861

1205902

119879

1205902

119882= 12059601205902

0+ 12059611205902

1

1205902

119861= 1205960(1205830minus 120583119879)2

+ 1205961(1205831minus 120583119879)2

= 12059601205961(1205831minus 1205830)2

1205902

119879=

255

sum

119894=0

(119894 minus 120583119879)2

119901119894

(23)

4 Results










Symmetry 119908 (119898) = 119908 (minus119898)

Normalization2

sum

119898=minus2

119908 (119898) = 1


119898 odd119908 (119898) = sum

119898 even119908 (119898)

(24)


119908 (0) = 119886

119908 (1) = 119908 (minus1) =

1

4

119908 (2) = 119908 (minus2) =

1

4

minus

119886

2

(25)



(1199080[119896]) = [

1

4

minus

119886

2

1

4

119886

1

4

1

4

minus

119886

2

] 119896 = 1 5 (26)




1198720= 5

119872119899= 2119872

119899minus1+ 3 119899 ge 1

(27)


0






Separability 119908 (119898 119899) = 119908 (119898)119908 (119899) (28)





minus01 1240

00 1011

01 1020

02 1009

03 1009

04 1004

05 1009

06 1022

07 1142

08 1141

09 1209

10 1248

11 1276

12 1289






1 2) + 1 0))

(29)

with 1and



120583 =

119872119895+ 1

2

= 2119895+1

minus 1 (30)






Pr (119873 (120583 120590) le 120583 minus 1199111120590) =

1 minus 1199011

2

Pr (119873 (120583 120590) le 120583 minus 1199112120590) =

1 minus 1199012

2

(31)


1= 120583 minus 119911

11198881199011

2=

120583 minus 1198881199012

1199112

(32)


1 2 3 4 50

005

01

015

02

025

03

035

04

045

05

2 4 6 8 10 12

002

004

006

008

01

012

014

016

018

10 20 30 40 50 60

0005

001

0015

002

0025

003

0035

004

k

kk

w[k]

w2[k]

w4[k]


where 1198881199011

and 1198881199012


1and 119901

2



abscissas 1199111= 1 and 119911






6





100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12






(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics


















119892119894+1

(119909 119910) =

0 if 119892119894(119909 119910) lt 120582

255 if 119892119894(119909 119910) ge 120582

(8)


119894(119909 119910) is the







119901= 119881119897+


119905= 119881119904+ 119881119897+ 119881119892is the



119901119905=

119881119901

119881119905

=

119881119897+ 119881119892

119881119904+ 119881119897+ 119881119892

(9)









3 Methodology





1198660 1198661 1198662 119866

119873

119866119896(119894 119895) = sum

119898

sum

119899

119866119896minus1

(2119894 + 119898 2119895 + 119899) 119896 = 1 119873 (10)


0and subtracting


1198710 1198711 119871



119871119896= 119866119896minus 119866119896+1

119896 = 0 119873 minus 1 (11)


119866119896(119894 119895) = 4sum

119898

sum

119899

119908 (119898 119899) 119866119896minus1

(

2119894 + 119898

2

2119895 + 119899

2

) (12)


1198660= 1198710+ 1198661

= 1198710+ 1198711+ 1198662

= 1198710+ 1198711+ 1198712+ sdot sdot sdot + 119871

119873minus1+ 119866119873

(13)


G2

G1

G0


G1

G0












120575119904

sdot 119873120575(119865) ≃ 1 (14)







119863119899= 119863119899(119865119899 119865) = max

119909isin119860

1003816100381610038161003816119865119899(119909) minus 119865 (119909)

1003816100381610038161003816 (15)




119899120572) = 120572

Let 119863119899120572


119899= K and 120572 = 001







2

119882(119905)


1205902

119908(119905) = 120596

0(119905) 1205902

0(119905) + 120596

1(119905) 1205902

1(119905) (16)





119887(119905)

defined as follows

1205902

119861(119905) = 120590

2

minus 1205902

119882(119905) = 120596

0(119905) 1205961(119905) [1205830(119905) minus 120583

1(119905)]2

(17)





means 1205830(119905) and 120583


1205960(119905) =

119905

sum

0

119901 (119894) 1205961(119905) = sum

119894gt119905

119901 (119894)

1205830(119905) =

119905

sum

0

119901 (119894) 119909 (119894) 1205831(119905) = sum

119894gt119905

119901 (119894) 119909 (119894)

(18)





120582Otsu = arg min 1205960(120582) 1205902

0(120582) + 120596

1(120582) 1205902

1(120582) (19)






119901119894as 119901119894= 119899119894119873 then 119901

119894ge 0 and sum119901






1205960= Pr (119862

0) =

119896

sum

119894=0

119901119894= 120596 (119896)

1205961= Pr (119862

1) =

255

sum

119894=119896+1

119901119894= 1 minus 120596 (119896)

1205830=

119896

sum

119894=0

119894Pr (119894 | 1198620) =

119896

sum

119894=0

119894119901119894

1205960

=

120583 (119896)

120596 (119896)

1205831=

255

sum

119894=119896+1

119894Pr (119894 | 1198621) =

255

sum

119894=119896+1

119894119901119894

1205961

=

120583119879minus 120583 (119896)

1 minus 120596 (119896)

(20)

where

120596 (119896) =

119896

sum

119894=0

119901119894

120583 (119896) =

119896

sum

119894=0

119894119901119894 120583

119879=

255

sum

0

119894119901119894 (21)


01205830+


1205902

0=

119896

sum

119894=0

(119894 minus 1205830)2Pr (119894 | 119862

0) =

119896

sum

119894=0

(119894 minus 1205830)2

119901119894

1205960

1205902

1=

255

sum

119894=119896+1

(119894 minus 1205831)2Pr (119894 | 119862

1) =

255

sum

119894=119896+1

(119894 minus 1205831)2

119901119894

1205961

(22)






120582 =

1205902

119861

1205902

119882

120581 =

1205902

119879

1205902

119882

120578 =

1205902

119861

1205902

119879

1205902

119882= 12059601205902

0+ 12059611205902

1

1205902

119861= 1205960(1205830minus 120583119879)2

+ 1205961(1205831minus 120583119879)2

= 12059601205961(1205831minus 1205830)2

1205902

119879=

255

sum

119894=0

(119894 minus 120583119879)2

119901119894

(23)

4 Results










Symmetry 119908 (119898) = 119908 (minus119898)

Normalization2

sum

119898=minus2

119908 (119898) = 1


119898 odd119908 (119898) = sum

119898 even119908 (119898)

(24)


119908 (0) = 119886

119908 (1) = 119908 (minus1) =

1

4

119908 (2) = 119908 (minus2) =

1

4

minus

119886

2

(25)



(1199080[119896]) = [

1

4

minus

119886

2

1

4

119886

1

4

1

4

minus

119886

2

] 119896 = 1 5 (26)




1198720= 5

119872119899= 2119872

119899minus1+ 3 119899 ge 1

(27)


0






Separability 119908 (119898 119899) = 119908 (119898)119908 (119899) (28)





minus01 1240

00 1011

01 1020

02 1009

03 1009

04 1004

05 1009

06 1022

07 1142

08 1141

09 1209

10 1248

11 1276

12 1289






1 2) + 1 0))

(29)

with 1and



120583 =

119872119895+ 1

2

= 2119895+1

minus 1 (30)






Pr (119873 (120583 120590) le 120583 minus 1199111120590) =

1 minus 1199011

2

Pr (119873 (120583 120590) le 120583 minus 1199112120590) =

1 minus 1199012

2

(31)


1= 120583 minus 119911

11198881199011

2=

120583 minus 1198881199012

1199112

(32)


1 2 3 4 50

005

01

015

02

025

03

035

04

045

05

2 4 6 8 10 12

002

004

006

008

01

012

014

016

018

10 20 30 40 50 60

0005

001

0015

002

0025

003

0035

004

k

kk

w[k]

w2[k]

w4[k]


where 1198881199011

and 1198881199012


1and 119901

2



abscissas 1199111= 1 and 119911






6





100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12






(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics















G2

G1

G0


G1

G0












120575119904

sdot 119873120575(119865) ≃ 1 (14)







119863119899= 119863119899(119865119899 119865) = max

119909isin119860

1003816100381610038161003816119865119899(119909) minus 119865 (119909)

1003816100381610038161003816 (15)




119899120572) = 120572

Let 119863119899120572


119899= K and 120572 = 001







2

119882(119905)


1205902

119908(119905) = 120596

0(119905) 1205902

0(119905) + 120596

1(119905) 1205902

1(119905) (16)





119887(119905)

defined as follows

1205902

119861(119905) = 120590

2

minus 1205902

119882(119905) = 120596

0(119905) 1205961(119905) [1205830(119905) minus 120583

1(119905)]2

(17)





means 1205830(119905) and 120583


1205960(119905) =

119905

sum

0

119901 (119894) 1205961(119905) = sum

119894gt119905

119901 (119894)

1205830(119905) =

119905

sum

0

119901 (119894) 119909 (119894) 1205831(119905) = sum

119894gt119905

119901 (119894) 119909 (119894)

(18)





120582Otsu = arg min 1205960(120582) 1205902

0(120582) + 120596

1(120582) 1205902

1(120582) (19)






119901119894as 119901119894= 119899119894119873 then 119901

119894ge 0 and sum119901






1205960= Pr (119862

0) =

119896

sum

119894=0

119901119894= 120596 (119896)

1205961= Pr (119862

1) =

255

sum

119894=119896+1

119901119894= 1 minus 120596 (119896)

1205830=

119896

sum

119894=0

119894Pr (119894 | 1198620) =

119896

sum

119894=0

119894119901119894

1205960

=

120583 (119896)

120596 (119896)

1205831=

255

sum

119894=119896+1

119894Pr (119894 | 1198621) =

255

sum

119894=119896+1

119894119901119894

1205961

=

120583119879minus 120583 (119896)

1 minus 120596 (119896)

(20)

where

120596 (119896) =

119896

sum

119894=0

119901119894

120583 (119896) =

119896

sum

119894=0

119894119901119894 120583

119879=

255

sum

0

119894119901119894 (21)


01205830+


1205902

0=

119896

sum

119894=0

(119894 minus 1205830)2Pr (119894 | 119862

0) =

119896

sum

119894=0

(119894 minus 1205830)2

119901119894

1205960

1205902

1=

255

sum

119894=119896+1

(119894 minus 1205831)2Pr (119894 | 119862

1) =

255

sum

119894=119896+1

(119894 minus 1205831)2

119901119894

1205961

(22)






120582 =

1205902

119861

1205902

119882

120581 =

1205902

119879

1205902

119882

120578 =

1205902

119861

1205902

119879

1205902

119882= 12059601205902

0+ 12059611205902

1

1205902

119861= 1205960(1205830minus 120583119879)2

+ 1205961(1205831minus 120583119879)2

= 12059601205961(1205831minus 1205830)2

1205902

119879=

255

sum

119894=0

(119894 minus 120583119879)2

119901119894

(23)

4 Results










Symmetry 119908 (119898) = 119908 (minus119898)

Normalization2

sum

119898=minus2

119908 (119898) = 1


119898 odd119908 (119898) = sum

119898 even119908 (119898)

(24)


119908 (0) = 119886

119908 (1) = 119908 (minus1) =

1

4

119908 (2) = 119908 (minus2) =

1

4

minus

119886

2

(25)



(1199080[119896]) = [

1

4

minus

119886

2

1

4

119886

1

4

1

4

minus

119886

2

] 119896 = 1 5 (26)




1198720= 5

119872119899= 2119872

119899minus1+ 3 119899 ge 1

(27)


0






Separability 119908 (119898 119899) = 119908 (119898)119908 (119899) (28)





minus01 1240

00 1011

01 1020

02 1009

03 1009

04 1004

05 1009

06 1022

07 1142

08 1141

09 1209

10 1248

11 1276

12 1289






1 2) + 1 0))

(29)

with 1and



120583 =

119872119895+ 1

2

= 2119895+1

minus 1 (30)






Pr (119873 (120583 120590) le 120583 minus 1199111120590) =

1 minus 1199011

2

Pr (119873 (120583 120590) le 120583 minus 1199112120590) =

1 minus 1199012

2

(31)


1= 120583 minus 119911

11198881199011

2=

120583 minus 1198881199012

1199112

(32)


1 2 3 4 50

005

01

015

02

025

03

035

04

045

05

2 4 6 8 10 12

002

004

006

008

01

012

014

016

018

10 20 30 40 50 60

0005

001

0015

002

0025

003

0035

004

k

kk

w[k]

w2[k]

w4[k]


where 1198881199011

and 1198881199012


1and 119901

2



abscissas 1199111= 1 and 119911






6





100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12






(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics

















120582Otsu = arg min 1205960(120582) 1205902

0(120582) + 120596

1(120582) 1205902

1(120582) (19)






119901119894as 119901119894= 119899119894119873 then 119901

119894ge 0 and sum119901






1205960= Pr (119862

0) =

119896

sum

119894=0

119901119894= 120596 (119896)

1205961= Pr (119862

1) =

255

sum

119894=119896+1

119901119894= 1 minus 120596 (119896)

1205830=

119896

sum

119894=0

119894Pr (119894 | 1198620) =

119896

sum

119894=0

119894119901119894

1205960

=

120583 (119896)

120596 (119896)

1205831=

255

sum

119894=119896+1

119894Pr (119894 | 1198621) =

255

sum

119894=119896+1

119894119901119894

1205961

=

120583119879minus 120583 (119896)

1 minus 120596 (119896)

(20)

where

120596 (119896) =

119896

sum

119894=0

119901119894

120583 (119896) =

119896

sum

119894=0

119894119901119894 120583

119879=

255

sum

0

119894119901119894 (21)


01205830+


1205902

0=

119896

sum

119894=0

(119894 minus 1205830)2Pr (119894 | 119862

0) =

119896

sum

119894=0

(119894 minus 1205830)2

119901119894

1205960

1205902

1=

255

sum

119894=119896+1

(119894 minus 1205831)2Pr (119894 | 119862

1) =

255

sum

119894=119896+1

(119894 minus 1205831)2

119901119894

1205961

(22)






120582 =

1205902

119861

1205902

119882

120581 =

1205902

119879

1205902

119882

120578 =

1205902

119861

1205902

119879

1205902

119882= 12059601205902

0+ 12059611205902

1

1205902

119861= 1205960(1205830minus 120583119879)2

+ 1205961(1205831minus 120583119879)2

= 12059601205961(1205831minus 1205830)2

1205902

119879=

255

sum

119894=0

(119894 minus 120583119879)2

119901119894

(23)

4 Results










Symmetry 119908 (119898) = 119908 (minus119898)

Normalization2

sum

119898=minus2

119908 (119898) = 1


119898 odd119908 (119898) = sum

119898 even119908 (119898)

(24)


119908 (0) = 119886

119908 (1) = 119908 (minus1) =

1

4

119908 (2) = 119908 (minus2) =

1

4

minus

119886

2

(25)



(1199080[119896]) = [

1

4

minus

119886

2

1

4

119886

1

4

1

4

minus

119886

2

] 119896 = 1 5 (26)




1198720= 5

119872119899= 2119872

119899minus1+ 3 119899 ge 1

(27)


0






Separability 119908 (119898 119899) = 119908 (119898)119908 (119899) (28)





minus01 1240

00 1011

01 1020

02 1009

03 1009

04 1004

05 1009

06 1022

07 1142

08 1141

09 1209

10 1248

11 1276

12 1289






1 2) + 1 0))

(29)

with 1and



120583 =

119872119895+ 1

2

= 2119895+1

minus 1 (30)






Pr (119873 (120583 120590) le 120583 minus 1199111120590) =

1 minus 1199011

2

Pr (119873 (120583 120590) le 120583 minus 1199112120590) =

1 minus 1199012

2

(31)


1= 120583 minus 119911

11198881199011

2=

120583 minus 1198881199012

1199112

(32)


1 2 3 4 50

005

01

015

02

025

03

035

04

045

05

2 4 6 8 10 12

002

004

006

008

01

012

014

016

018

10 20 30 40 50 60

0005

001

0015

002

0025

003

0035

004

k

kk

w[k]

w2[k]

w4[k]


where 1198881199011

and 1198881199012


1and 119901

2



abscissas 1199111= 1 and 119911






6





100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12






(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics
















(1199080[119896]) = [

1

4

minus

119886

2

1

4

119886

1

4

1

4

minus

119886

2

] 119896 = 1 5 (26)




1198720= 5

119872119899= 2119872

119899minus1+ 3 119899 ge 1

(27)


0






Separability 119908 (119898 119899) = 119908 (119898)119908 (119899) (28)





minus01 1240

00 1011

01 1020

02 1009

03 1009

04 1004

05 1009

06 1022

07 1142

08 1141

09 1209

10 1248

11 1276

12 1289






1 2) + 1 0))

(29)

with 1and



120583 =

119872119895+ 1

2

= 2119895+1

minus 1 (30)






Pr (119873 (120583 120590) le 120583 minus 1199111120590) =

1 minus 1199011

2

Pr (119873 (120583 120590) le 120583 minus 1199112120590) =

1 minus 1199012

2

(31)


1= 120583 minus 119911

11198881199011

2=

120583 minus 1198881199012

1199112

(32)


1 2 3 4 50

005

01

015

02

025

03

035

04

045

05

2 4 6 8 10 12

002

004

006

008

01

012

014

016

018

10 20 30 40 50 60

0005

001

0015

002

0025

003

0035

004

k

kk

w[k]

w2[k]

w4[k]


where 1198881199011

and 1198881199012


1and 119901

2



abscissas 1199111= 1 and 119911






6





100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12






(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics















1 2 3 4 50

005

01

015

02

025

03

035

04

045

05

2 4 6 8 10 12

002

004

006

008

01

012

014

016

018

10 20 30 40 50 60

0005

001

0015

002

0025

003

0035

004

k

kk

w[k]

w2[k]

w4[k]


where 1198881199011

and 1198881199012


1and 119901

2



abscissas 1199111= 1 and 119911






6





100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12






(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics















100 200 300 400 500 600 700 800 900 1000

2

4

6

8

10

12

14

16

k

w6[k]

times10minus4

(a) 119886 = 02

100 200 300 400 500 600 700 800 900 1000

05

1

15

2

25

kw6[k]

times10minus3

(b) 119886 = 04

100 200 300 400 500 600 700 800 900 1000

0

0005

001

0015

002

0025

003

k

w6[k]

minus001

minus0005

(c) 119886 = 08

100 200 300 400 500 600 700 800 900 1000

0

05

1

k

w6[k]

minus05

(d) 119886 = 12






(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics















(a) (b)


Dn099

0075

0050

0025

minus025 0 025 050

a

Dn

(a) 119863119899and119863

119899099

0010

0008

0006

0004

0002

0

minus0002

50 100 150 200 250

k




5 Conclusions






(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics















(a) (b)


Frac

tal d

imen

sion

(FD

)

15

1

05

0minus02 0 02 04 06 08 1 12

a


Thre

shol

d (120582)

minus02 00

02 04 06 08 1 12

a

03

02

01










(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics
















(d) 119886 = 04 (e) 119886 = 08 (f) 119886 = 12


5 10 15 20 25

10

20

30

40

50

60

70

80

90

Pore size

Freq

uenc

y







References







































Journal of












Volume 2014

Advances in









Geophysics














































Journal of












Volume 2014

Advances in









Geophysics














research article fractal analysis of laplacian pyramidal...

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