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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 6, NO. 7, JULY 1997 1025

A New Approach to Vector MedianFiltering Based on Space Filling Curves

Carlo S. Regazzoni,Member, IEEE,and Andrea Teschioni

Abstract—The availability of a wide set of multidimensionalinformation sources in different application fields (e.g., colorcameras, multispectral remote sensing imagery devices, etc.) is thebasis for the interest of image processing research on extensionsof scalar nonlinear filtering approaches to multidimensional datafiltering.

In this paper, a new approach to multidimensional medianfiltering is presented. The method is structured into two steps.Absolute sorting of the vectorial space based on Peano spacefilling curves is proposed as a preliminary step in order tomap vectorial data onto an appropriate one-dimensional (1-D)space. Then, a scalar median filtering operation is applied. Themain advantage of the proposed approach is the computationalefficiency of the absolute sorting step, which makes the methodglobally faster than existing median filtering techniques. Thisis particularly important when dealing with a large amount ofdata (e.g., image sequences). Presented results also show that thefiltering performances of the proposed approach are comparablewith those of vector median filters presented in the literature.

I. INTRODUCTION

T HE development of image processing methods for filter-ing and segmentation of vectorial information is becom-

ing more and more important, thanks to the availability ofa large amount of vectorial data. Noise filtering and enhance-ment of color images and of multispectral remote-sensing dataare examples of applications where vectorial information mustbe processed.

Many sophisticated methods for scalar image processingare available that have been developed for grey-level imageprocessing. A direct extension of such methods to the multi-variate case can be performed by separately applying a scalarmethod to each component of a vectorial image. However, inmany cases, this generalization exhibits several drawbacks, themost important of which lies in not exploiting interchannelcorrelation. A further consequence of this approach is thatcomputational complexity increases linearly with the numberof channels. Therefore, the study of new methods becomesnecessary to realize efficient vectorial counterparts of scalarmethods.

In this paper, an image filtering method is proposed. Theproposed filter aims at extending median filters to the vectorialcase by using a more flexible and more computationallyefficient method.

Manuscript received April 18, 1996; revised February 28, 1997. Theassociate editor coordinating the review of this manuscript and approving itfor publication was Prof. Jan Allebach.

The authors are with the Department of Biophysical and ElectronicEngineering (DIBE), University of Genoa, I-16145 Genoa, Italy (e-mail:[email protected]).

Publisher Item Identifier S 1057-7149(97)04730-1.

In particular, the reduced vector median filter (RVMF) isintroduced that has been derived directly from the concept ofvector median filter (VMF). The VMF [1] is defined as thegeneralization of the scalar median filter [2] to the case ofvector-valued signals.

The RVMF operates a vectorial-scalar transformation fol-lowed by scalar data ordering instead of directly realizing amedian on the vectorial data, as for the VMF. The proposedtransformation is based on the concept of space filling curves[3], [4].

In Section V, it is shown that our method exhibits a muchlower computational complexity than other VMF implementa-tions, and it is demonstrated that the RVMF performances arecomparable to those of other VMF realizations.

The paper is organized as follows. Sections II and IIIprovide a general description of the VMF and an introductionto the problem. Section IV focuses the attention on the chosenspace filling curve and Section V shows the performances ofthe proposed algorithm. Section VI concludes the paper.

II. V ECTOR MEDIAN FILTERS

The use of the median operator in image processing was in-troduced by Tukey [5]. The median filter performs a nonlinearfiltering operation where a window moves over a signal, and,at each point, the median value of the data within the windowis taken as the output.

The median of a scalar set wasdefined as the value such that

(1)

and it is known that can always be chosen as one ofthe .

A statistical analysis of the median filter [2] revealed that itoutperforms the moving average filter in the case of additivelong-tailed noise, and that it is very suitable for the removalof impulsive noise. These facts make the median filter veryattractive for digital image filtering applications.

Astola et al. [1] proposed an extension of the median oper-ation to vector-valued signals, introducing some requirementsfor the resulting vector median.

The median value of a vectorial set, consisting of -variate samples, i.e.,

, is defined as

(2)

1057–7149/97$10.00 1997 IEEE

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1026 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 6, NO. 7, JULY 1997

The extended definition implies that the vectorial operator hasproperties similar to those of the median operation in the scalarcase, and also implies that the vector median is reduced tothe scalar median when the vector dimension is one. Such adefinition of the VMF requires that the VMF output be oneof the input vectors.

One can observe that this definition of the VMF heavilydepends on the choice of an appropriate norm for thevectorial values; the related distance is used to make a sortingof the quantities corresponding to each vectorwithin the set .

The vector median can be computed by two different classesof algorithms, depending on the way in which the sorting isperformed: absolute sorting or relative sorting.

The methods belonging to the first class use the samestrategy for the median choice as in the scalar case: A scalarrank value is first associated with each vector on which thesorting is made.

For example, it is possible to consider the Euclidean distanceof each vector from the origin of the -dimensionalspace and then to make a sorting of the vectors within the set

according to this distance and to choose the median valuefrom the distances, that is

(3)

where and is an norm.A different choice (relative sorting) to implement the VMF

may involve decomposing the minimum operation in (2) intothe following two steps.

1) Find Prototype vector within the set

2) (4)

The VMF implementations based on reduced ordering (r-ordering) [6] adopt this strategy in order to compute the VMFwithin the vectorial set .

According to the r-ordering principle, multivariate orderingis reduced to a scalar one, where the scalar is a distancefunction of the multivariate samples to a central location (i.e.,the prototype).

The VMF implementations based on this concept may differin the choice of the prototype, which can be, for example, themean of the vectors (r-ordering) orthe marginal median (m-ordering [7]), which is the result of ascalar median operation independently performed along eachcomponent of the multivariate samples.

The presented methods usually choose the prototype vectorusing the modulus information of each vector. Further imple-mentations of VMF based on relative sorting are presentedin [8] and [9]. In such algorithms, the prototype vectors arechosen on the basis of the angular distance between the vectorsinstead of rank information like that used by the previouslypresented vectors.

The main advantage of absolute sorting methods is that it ispossible to compute the VMF in only one step, so they exhibita much higher computational efficiency than relative sortingmethods.

Fig. 1. Space filling curve used in image scanning algorithms.

A further consideration concerns the fact that absolutesorting can be easily extended to other kinds of filters, suchas rank-order filters or morphological filters.

In this paper, we propose a new vector sorting approach. Tothis end, the concept of space filling curves is exploited. Theproposed approach is of the absolute type, so the necessity forchoosing a reference vector for sorting can be avoided.

The problem of mapping a multidimensional space intoone-dimensional (1-D) space has long been considered animportant one in the image processing literature. Space fillingcurves represent a possible solution to this problem. Spacefilling curves can be defined as a set of discrete curves thatmake it possible to cover all the points of a-dimensionalvectorial space. In particular, a space filling curve must passthrough all the points of the space only once, and makes itpossible to realize a mapping of a-dimensional space intoa scalar interval.

In Fig. 1 it is possible to observe quite a complicated two-dimensional (2-D) filling curve useful for an image scanning:

By means of a space filling curve, for example, a setof bidimensional elements may be reduced to a list of 1-D elements, representing the curvilinear abscissa of the 2-Dpoints along the curve itself.

The space filling curves have been used in cryptographyproblems; Alexopouloset al. [10] generated a family of scan-ning patterns for the protection of picture data in transmission.An image is scanned in a different way, as compared withthe raster-scan technique: The result of this new scanning isthe generation of a list of image pixels to which the inversescanning can be applied in the detection phase in order torecover the original image. An image scanning method basedon space filling curves has also been employed by Quweiderand Salari [11] and by Kamataet al. [12] in order to obtainan algorithm for image compression and coding: This methodallows one to fully exploit the correlation between adjacentpixels. Image coding algorithms based on space filling curveshave been proven to be much better than raster methods.

III. PROBLEM DEFINITION

We used the concept of space filling curves as a startingpoint in order to derive a vector sorting algorithm. Let usconsider, for example, the filling curve described above. Asmentioned earlier, the curve makes it possible to cover all thepoints of a -dimensional space continuously and once. It isthen possible to associate with each point in the-dimensionalspace a scalar value that is directly proportional to the length


REGAZZONI AND TESCHIONI: NEW APPROACH TO VECTOR MEDIAN FILTERING 1027

of the curve necessary to reach the point itself starting from theorigin of the coordinates. Vector sorting may be reduced to thesimple sorting of the scalar values associated with each vector.

The problem is stated as follows: Let be the set of vectorsthat have to be sorted. As previously said, the VMF can bedefined as the vector that minimizes the costfunction

(5)

That is

(6)

The proposed filter is the RVMF, and its output is the vectorthat minimizes the cost function defined as

(7)

where is a biunivocal transformation that allowsa vectorial-scalar transformation, and in our case

(8)

The only requirement to be met is that the functionbeinvertible, i.e., that exist.

The main advantage of the choice of such a cost functionis that it is not necessary to define an appropriate vector norm

for the -dimensional space, as the norm can beunivocally chosen.

The algorithm by which we can find the vector maybe decomposed into three steps:

1) Compute for all

2)

3)

(9)

The first step is the application of the function toeach vector belonging to the set , followed by a scalarmedian operation performed on the set of transformed values

: The result provided by the RVMF is eventuallygiven by the antitransformed value of the median among thetransformed values.

Before carrying out the minimum operation that leads tothe vector median, we introduce into the proposed definitiona vectorial-scalar transformation that allows us to simplifythe solution of the problem itself. Through the use of fillingcurves, we also obtain a remarkable reduction in computationalcomplexity, as compared with previous algorithms for vectorsorting, and a considerable decrease in the processing time,thus obtaining comparable results, even though with a loss interms of signal-to-noise ratio (SNR) performances.

IV. THE CHOSEN SPACE FILLING CURVE

A space filling curve , associated with a 2-D latticeallows the association of a scalar value with a-dimensional

vector in a discrete vectorial space as

1)

2)

3)

(10)

For each , the arc length is defined as

(11)

where exists, whereas the parameter represents thecurvilinear abscissaof the curve.

A filling curve makes it possible to cover, as the parametervaries, all the points of the discrete vectorial space, so

that each point is crossed only once, i.e.,

If and then(12)

In accordance with (12), a filling curve substantially makes ascanning operation of the space and it generates a list ofvectors in which there is no repetition of the sameelement .

An important consequence of the definition of filling curveis that the curve is invertible, i.e.,

If then(13)

So far, the characteristics of a general space filling curve havebeen introduced. Let us now examine the required propertiesof the specific class of filling curves used in this paper todefine the RVMF.

The curve we are searching for should reflect a “natural”ordering of vectors in the space. In particular, it shouldbe regular, and it should avoid irregular jumps similar to theones in the curve of Fig. 1.

In fact, a filling curve like the one of Fig. 1 presents twomain disadvantages, as follows, due to the fact that jumpsbetween close vectors in the space are not uniform:

• There is a “suboptimal” mapping of closeness propertiesin the space onto the 2-D filling space (i.e., orderingdoes not follow a natural, continuous rule, but it is subjectto “jumps”).

• Computation of the arc length could be complex.

The first point expresses a local constraint on the absoluteordering we are searching for. In particular, we can define anoptimal mapping in the following way.

Let us defineto be a neighborhood set of the pointin the space and

be a neighborhood setof points in the space.

Then we say that a space filling curve is optimal if all thepoints belonging to the neighborhood set ofin the spaceare close to (i.e., such that ) in the space, too.

In a formal way it can be expressed as

card

card (14)



For example, if , then card , so that it ispossible to say that the optimality constraint implies that thereexist two points and that are close to in the

space and which map onto the two elements of .This clearly does not hold in the curve of Fig. 1.

There does not exist a unique space filling curve thatsatisfies the above requirements, so that it is necessary tointroduce some additional criteria. To this end, we note thatthe desired curve should be able to generate a sorting of thespace points, which maintains, at best, modulus and angularinformation.

In this paper, we define a strategy to design a curve wherea greater weight is given to the first requirement (i.e., rankpreservation according to modulus information).

Let us define the of as

(15)

According to (15), the following property has to be valid.Given two vectors

If then(16)

In such a way, a higher value of the parameterwill beassociated with vectors having a larger distancefrom the origin, and vice versa. The resulting curve canbe imagined as consisting of successive layers, increasinglyordered according to the value.

Within each layer, the curve is chosen in such a way asto preserve angular information at the best. Among the set ofspace filling curves satisfying (14) and (16), a particular fillingcurve has been chosen, according to the latter observation.

The case of a bidimensional space will be firstillustrated for two reasons. The first one is that some conceptsand demonstrations are easier to be understood in the 2-D case,and they can be extended to the three–dimensional (3-D) casewith a direct generalization.

The second reason is that the 2-D case can be very in-teresting by itself because it has several applications (e.g.,displacement field motion field filtering).

A. The 2-D Case

In the 2-D case, the filling curve is reduced to a functionwhich associates with a scalar value a vector in the 2-D

space, that is

(17)The graphic representation of the chosen filling curve in the2-D space (see also [13]) is the following.

The dependence of the components of each vector on theparameter is expressed in a graphic way by these relations:The direct relations joining the components and of

Fig. 2. Graphic representation of the chosen 2-D filling curve .

to the parameter are

(18)

The decomposition into function series of the two vector com-ponents reveals that they can be interpreted as a superpositionof the same basis function translated along the axis.

Let us define the basis functions and as

(19a)

(19b)

Fig. 4(a) and (b) illustrates in a graphic way the behavior ofand .

The shape of the generic-order basis function isdefined starting from the first order function in thefollowing way ( can be interpreted as an offset termdepending on the chosen starting point of the curve):

(19c)

Using (19c) it is possible to observe that the shape of the basisfunction is periodically repeated in (18) scaled up anddilated along the axis.

From (18) and (19c) it is possible to demonstrate that theregularity of the reiterations of the basis functions stronglydepends on the parameter.

A further interesting property of the proposed filling curveis

(20)



(a)

(b)

Fig. 3 (a)–(b) 2-D components of the chosen filling curve .

(a)

(b)

Fig. 4. (a)–(b) Graphic representation of the 2-D basis functions.

where we define and that

(21)

In the case of the chosen filling curve, the arc length ofthe curve is coincident with its curvilinear abscissa. Fig. 5illustrates in a graphic way such properties.

It is now necessary to demonstrate that (14) and (16) aresatisfied in the case of the chosen space filling curve. Thechosen filling curve performs an optimal mapping for aneighborhood system with , so that card

.Let us see how, starting from a chosen vector, it

is possible to calculate the corresponding valueof thecurvilinear abscissa.

Fig. 5. Graphic representation of the derivatives ofx1k(t) andx2k(t).

If , then, by definition of andby construction

and (22)

The value could be even or odd; it means that, ifand

or (23)

Let us suppose, without a loss of generality, that ;if we define

(24)

we will have

(25)

By construction and by definition of

(26)

According to (20) and (26) it must be andand the following statement will hold:

(27)

The obtained functions allow us to associate with each scalarvalue a point in the 2-D space ; conversely, itis possible to transform a point in the 2-D space into thecorresponding scalar value.

There is quite a complicated analytic relation that binds, inthis case, the parameter to the components andof : div is the integer division,mod is the remainder of theinteger division, and , as follows:

even odd

hist

(28)

Expression (28) can be decomposed into four main terms,i.e., even, odd, hist, and : In Appendix A the relationsbetween these terms and (18) and (19) definitions and the



Fig. 6. C-like algorithm for the computation of �1 in 2-D case.

demonstration of validity of (16) for the chosen filling curveare presented.

Consequently, the ranking of vectors can be performedaccording to thearc length of the filling curve at different2-D space points.

The C-like algorithm for the computation of the arc-lengthis described in Fig. 6.

The algorithm is simple and computationally efficient.

B. The 3-D Case

The generic expression of the 3-D space filling curve isexpressed by

(29)

To design a space filling curve able to be used for color imageprocessing it is necessary to extend results obtained in the2-D case to the 3-D case.

The extension of the 2-D filling curve to the 3-D spaceis performed in order to preserve, as much as possible, thecharacteristics of the described curve for the 2-D case: inparticular, in a similar way with respect to the 2-D case, the 3-D filling curve can be imagined as an expansion of successiveincreasing layers, ordered according to the value ofeach 3-D vector.

A possible strategy is to impose that the 3-D filling curvecrosses all points at the same value in a continuousway, e.g., by covering in a ordered way the three sides of acube.

In this way, it is possible to think that the behavior of the 3-D filling curve on a single side of the cube havingwill reflect the behavior that the 2-D filling curve follows inthe 2-D space.

Moreover, it was shown in the 2-D case that it is possibleto find a relation which allows to associate a scalar value,representing the arc length of the filling curve, with a vector

.In Appendix A it is also shown that such a relation is

composed by a term representing the history of the curve up tothe considered layer and by an “updating” term, which givesthe shift of the vector within the layer itself and that thehist term increases in a quadratic way according to .

This is mainly due to the fact that, as it can be easily seenfrom Fig. 2, the arc length of the curve from the origin to oneof the points or is equal to the number of pointsof the square of side , that is .

From Fig. 2 it is also possible to note that the behavior ofthe curve in the 2-D space can be considered as an oscillatorypath from one coordinate axis to the other one.

This behavior is reflected into the basis function as ascaled-up mirroring of the semitrapezoidal shape correspond-ing to the path itself.

A curve with a similar behavior in the 3-D case is shownin Fig. 7.

On this basis, we can define the curve of Fig. 7 in an analyticway as

(30)

where a basis function is used.



(a)

(b)

(c)

Fig. 7. Graphic representation of the chosen 3-D filling curve : the curvestarts inA in (a) and ends inB; the curve starts inB in (b) and ends inC;the curve starts inC in (c) and ends inD.

The parametric behaviors of the components of the 3-Dvector scanned by the curve are represented in a graphic wayas shown in Fig. 8(a)–(c).

These relations can be considered as a generalization ofexpression (18).

(a)

(b)

(c)

Fig. 8. (a)–(c) 3-D components of the chosen filling curve .

A regularity is also present in the 3-D components, as in the2-D case: This corresponds to a well-defined path followed bythe 3-D curve on the different sides of the cube.

The possible behaviors of the coordinates expressedby the basis function are a periodic repetition, scaling up,mirroring and dilation along the axis.

In particular, we can observe that the coordinateremains constant when the side of the cube correspondingto the plane is crossed.

To better understand the behavior of , it is possibleto fix a certain layer of the 3-D cube and to examine thedependence of each component of the 3-D vector on theparameter within the layer.

If the curve of Fig. 8 is chosen, the following graphicexpression is obtained, for example, for at the layer

.In general, each 3-D component can be expressed as a

series function depending on the basis function : Thisis consistent with the fact that, on a single 3-D layer, thebehavior of the curve on each side of the cube is the samefollowed by the 2-D curve in order to cover all the points ofthe 2-D space.

Along the same face, the other coordinates correspond to ascaled-up and truncated version of either a normal ora mirrored version of the basis shape [see Fig. 4(b)]depending on the direction by which the curve crosses theplane (see Fig. 10). In particular, if we consider the path for

shown in Fig. 9(a), the three versions , andcan be individuated:



Fig. 9. Graphic representation of the profile ofx1k(t) at layerp = 3.

(a)

(b)

(c)

Fig. 10. C; M, andN profiles individuated on the profile of Fig. 9.

On the basis of this description it can be shown that acoordinate crossed by the filling curve periodically describesa well-defined cycle in which it can be found in one of threestatuses (i.e., or ).

It is also possible to observe from Fig. 7 that the samebehavior of the curve in the 3-D space is repeated after anumber of layers equal to the dimension of the input vector(i.e., three).

This justifies the periodicity by which the different coordi-nates are expressed in (30).

It is also possible to demonstrate that, also in the 3-D case,properties (20) and (21) can be generalized to

(31)

(32a)

(32b)

(32c)

Also in this case, the arc length of the curve is coincidentwith the curvilinear abscissa.

The analytic relation that gives the arc-length in the 3-Dcase is

(33)

Also in this case,div is the integer division,mod is itsremainder, and .

The is nonzero only if is equal to zero,and so on for and .

The hist term contains the previous history and

(34)

is nonzero if and iff and andare equal to zero.

We can see that (31) is of the same kind of (28), even if,in the 3-D case, thehist term obeys to a cubic law becauseof the fact that in this case all the points of a cube will to becovered by the filling curve. From Fig. 7, one can easily seethat the arc length of the curve from the origin to one of thepoints or is equal to the numberof points of the cube of side , that is, .

In the 3-D case, in order to calculate the arc length of thechosen filling curves, three operations will be necessary: Amaximum computing in order to find which hasto be selected, another maximum computing, made only ontwo numbers (i.e., the two coordinates that do not correspondto the maximum previously calculated), which corresponds tothe maximum calculation made in the case of the 2-D fillingcurve, and finally a shift calculation which, as in the 2-D case,normally consists in an addition operation.

We could expect that in the case of the relationbinding the parameter to the -D components of a fillingcurve appropriately designed would be composed by a histterm (e.g., is the side of the layer of the-D hypercube)and by a term which gives the shift of the considered vectorwithin the layer with side .

More in general, in order to calculate the arc length of a-dimensional filling curve designed by extending the 2-D and3-D presented curves, it will be necessary to progressivelytransform a -dimensional vectorial space by computing

maxima within sets whose dimension is progressivelydecreased: At the th calculation, the vectorial spaceis reduced to a scalar one and the “updating” term to be addedto hist can be calculated.

In a more complicated way than in the 2-D case, it can bedemonstrated that such a choice of the filling curve will satisfythe properties (14) and (16). In particular, in order to demon-strate the validity of (16), we can refer to the demonstrationof Appendix A, because if we choose the particular side of



the 3-D on which we are, we come back, as we already said,to the 2-D situation, and then the demonstration of AppendixA is valid.

The -like algorithm to compute the arc-length in the 3-Dcase is shown in Fig. 11.

C. The 3-D RVMF

Immediately successive to the choice of the filling curve isthe introduction of the transformation, which is defined for avector and for the chosen 3-D filling curve as

(35)

After having computed, for each vector the value,it is possible to rank the vectors within a mask and tocompute the scalar median as in [2].

The RVMF output is given by

(36)

where .

V. RESULTS

In this section, an application of the proposed method tocolor image filtering is presented.

RVMF filter performances are evaluated, and RVMF iscompared with

1) marginal median[7] (i.e., ordering performed along eachcomponent of the multivariate samples);

2) r-ordering about the mean[6] (i.e., ordering of themultivariate samples according to their distances to apreselected central location, in such case the mean ofthe multivariate samples);

3) (Conditional)-ordering[6] (i.e., ordering of only oneof the components—other components are simply listedaccording to the position of the ranked component);

4) VMF with Square Euclidean Distance[6] (i.e., orderingperformed according to the square modulus of eachvector);

in terms of signal-to-noise ratio (SNR). The test image selectedfor the comparison is the color version of Lena. The test imagehas been contaminated using Gaussian and “salt and pepper”impulsive noise source models: A correlation factorbetween the components of the vectorial noise is used in theexperiments.

The SNR has been used as quantitative measure for evalu-ation purposes. It is computed as

SNR

(37)

where and are the image dimensions, andand denote the original image vector and the estimation

at pixel , respectively.Table II summarizes the results obtained for the test image

Lena for a filter window 3 3: The noise corruption typesare illustrated in Table I.

TABLE INOISE DISTRIBUTIONS

One can observe that the filter produces comparable resultswith the r-ordering implementation, which is currently con-sidered the filter that gives better performances, in the caseof impulsive noise; as expected, the performances are a fewworse for Gaussian noise, since the proposed transformationis based on a nonlinear distance.

Moreover, r-ordering performs a relative sorting, which usu-ally produces better results than absolute sorting; a comparisonwith methods based on an absolute sorting has been also made.

For example, RVMF gives almost the same performancesin terms of SNR as the marginal median, while performancesof RVMF are usually better than c-ordering and VMF ones.

A comparison also has been made between RVMF and mar-ginal median in terms of computational complexity; marginalmedian has been chosen because of its simplicity and becauseof the fact that it provides optimal computational performancesamong absolute sorting-based methods.

For each pixel of a -dimensional image a windowwith points is considered. Marginal median requiressorting operations: Since each sorting requires, as an averageterm, comparisons [16] ( being the cardinality ofthe set to be sorted), then globally for the whole image, themarginal median will requirecomparisons).

If a Sun SparcStation 20 is chosen, for each comparison oneclock cycle is required [15]. Then marginal median operationwill require clock cycles.

For what concerns RVMF, it has been shown that thefollowing operations are necessary for an image:

1) maximum calculations oneaddition , as an average;

2) scalar median operation.

One also has to consider that, for each maximum calculationthe set of elements to be sorted is decreased each time of oneelement itself. The Sun SparcStation 20 workstation requiresone clock cycle also for the addition, so the global number ofclock cycles required by RVMF are

(38)

Fig. 12 shows the behavior of the computational complexity(measured as clock cycles necessary for each pixel of animage) of the two methods, by using as independent variablethe dimension of the space and after having fixed a maskdimension : One can see that the complexity is muchlower in the case of the RVMF, even if there is the possibility



TABLE IISNR (dB) FOR THE LENA IMAGE, WINDOW 3 � 3

Fig. 11. C-like algorithm for the computation of �1 in 3-D case.

Fig. 12. Computational complexity of RVMF and marginal median.

that for a high value of marginal median could becomebetter.

Finally, qualitative results concerning the application ofRVMF, marginal median, and r-ordering MF to three samplenoisy images are presented; Lena in Fig. 13 has been corruptedby 4% impulsive noise, the squirrel image of Fig. 14 by 6%impulsive noise, and the house of Fig. 15 by 8% impulsivenoise. From the visual point of view, r-ordering gives the bestperformances, but one can also see that performances achievedby RVMF are always comparable and in some cases (e.g., inFig. 14) better than ones obtained by marginal median filter.

VI. CONCLUSION

In this paper, an image filtering method, called the reducedvector median filter (RVMF), has been proposed. The filter

implements a new approach for extending median filters tovectorial data.

The ranking operation for multivariate data is performed bya step in which the filter operates a vectorial scalar transfor-mation followed by scalar data ordering instead of directlyrealizing a median on the vectorial data. The transformation isbased on the concept of a space filling curve.

The general characteristics of space filling curves have beenexamined and particular curves chosen for the transformationare presented for the 2-D and 3-D curves. Peculiarities of thiscurve have been presented and an application of the methodto color image filtering has been discussed.

Results have shown the good computational efficiency ofthe proposed method.

Since the presented method is based on absolute sortingof vectorial data, it seems interesting to investigate possibleextensions of the proposed approach to new rank-order filtersand morphological operators for vectorial data processing.

APPENDIX A

In order to make easier the readability of the paper, themathematical expressions of the termsevenand odd in (28)are reported in this Appendix. At this end, let us examinethe superposition of the graphical expressions of and

; there are four different possibilities to be considered:



Fig. 13. (a) Noisy (impulsive noise 4%) Lena image. (b) Lena image filteredby RVMF. (c) Lena image filtered by marginal median. (d) Lena image filteredby r-ordering median.

1) and even;2) and even;3) and odd;4) and odd.

Fig. 14. (a) Noisy (impulsive noise 6%) squirrel image. (b) Squirrel imagefiltered by RVMF. (c) Squirrel image filtered by marginal median. (d) Squirrelimage filtered by r-ordering median.

The evenfunction is composed by contributions coming fromevents 1) and 2), whileodd function is composed by contri-butions coming from events 3) and 4).Even is nonzero onlyif is even, and so on forodd.



Fig. 15. (a) Noisy (impulsive noise 8%) house image. (b) House imagefiltered by RVMF. (c) House image filtered by marginal median. (d) Houseimage filtered by r-ordering median.

Let us find how it is possible to calculate the mathematicalexpressions of the termseven is composed by. If we are insituation 1), then (22) and (24) hold. Since within theintervalonly one value of has to be found, the profile has to be

considered and, in particular, the basis function which has tobe examined for if is, as it is also shown by Fig.A1, , which, if , holds

(A.1.1)

and the contribution to theevenfunction given by the event1) is then equal to .

By following the same reasoning for the events 2), 3), and4), the following expressions ofevenandodd are obtained:

even

odd

The global expression that allows to compute the arc length,which is composed by a “history” term and by an “updating”term, will be then finally given by (28). Let us now demon-strate the validity of (16) for a choice of filling curve whosethe arc length is expressed by (28) itself.

Let and be two vectors and let us impose, withouta loss of generality, and with

. It is now possible todemonstrate that, if and .In fact, as we have shown in this Appendix, we can write

(A.1.2)

and we have and . In theworst case, it will be, for (A.1.2), and .Then, we will have

(A.1.3)

and finally .This is equal to say that (16) holds for the chosen 2-D

filling curve.

ACKNOWLEDGMENT

The authors wish to thank Prof. A. N. Venetsanopoulosand Dr. K. Plataniotis for useful discussions and for provid-ing software tools for multidimensional noise generation andperformance evaluation.

REFERENCES

[1] J. Astola, P. Haavisto, and Y. Neuvo, “Vector median filters,” inProc.IEEE, vol. 78, pp. 678–689, Apr. 1990.

[2] I. Pitas and A. N. Venetsanopoulos,Nonlinear Digital Filters: Principlesand Applications, Norwell, MA: Kluwer, 1990.

[3] T. Bially, “Space filling curves: Their generation and their applicationto bandwidth reduction,”IEEE Trans. Inform. Theory, vol. IT-15, pp.658–664, 1969.

[4] A. R. Butz, “Alternative algorithm for Hilbert’s space filling curve,”IEEE Trans. Comput., vol. 20, pp. 424–426, Apr. 1971.

[5] J. M. Tukey, “Nonlinear (nonsuperposable) methods for smoothingdata,” in Proc. Congr. EASCON, 1974, p. 673.



[6] K. Tang, J. Astola, and Y. Neuvo, “Nonlinear multivariate image fil-tering techniques,”IEEE Trans. Image Processing, vol. 4, pp. 788–798,June 1995.

[7] I. Pitas and C. Kotropoulos, “MultichannelL filters based on marginaldata ordering,”IEEE Trans. Signal Processing, vol. 42, Oct. 1994.

[8] J. Scharcanski and A. N. Venetsanopoulos, “Color edge detection usingdirectional operators,” inProc. 1995 IEEE Workshop Nonlinear Signaland Image Processing, Neos Marmaras, Greece, June 1995.

[9] K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Color im-age processing using fuzzy vector directional filters,” inProc. 1995 IEEEWorkshop on Nonlinear Signal and Image Processing, Neos Marmaras,Greece, June 1995.

[10] C. Alexopoulos, N. Bourbakis, N. Ioannou, “Image encryption methodusing a class of fractals,”J. Electron. Imaging, pp. 251–259, July 1995.

[11] M. K. Quweider and E. Salari, “Peano scanning partial distance searchfor vector quantization,”IEEE Signal Processing Lett., vol. 2, pp.169–171, Sept. 1995.

[12] S. Kamata, A. Perez, and E. Kawaguchi, “N -dimensional Hilbertscanning and its application to data compression,” inProc. SPIE: Vol.1452, Image Processing Algorithms and Technology, 1991, pp. 430–441.

[13] K. N. Plataniotis, C. S. Regazzoni, A. Teschioni, and A. N. Venet-sanopoulos, “A new distance measure for vectorial rank order filtersbased on space filling curves,” inProc. IEEE Conf. Image Processing,ICIP ’96, Lausanne, Switzerland, vol. 1, pp. 411–414.

[14] M. Barni, V. Cappellini, and A. Mecocci, “The use of different metrics invector median filtering: Application to fine arts and paintings,” inProc.EUSIPCO-92, 6th Europ. Signal Processing Conf., Brussels, Belgium,Aug. 25–28, 1992, pp. 1485–1488.

[15] P. Baglietto, M. Maresca, M. Migliardi, and N. Zingirian, “Imageprocessing on high-performance RISC systems,” inProc. IEEE, Aug.1996, pp. 917–930.

[16] B. Codenotti and M. Leoncini,Fondamenti di Calcolo Parallelo.Reading, MA: Addison-Wesley, 1990.

Carlo S. Regazzoni (S’90–M’92) was born inSavona, Italy, in 1963. He received the Laureadegree in electronic engineering and the Ph.D.degree in telecommunications and signal processingfrom the University of Genoa, Italy, in 1987 and1992, respectively.

He is an Assistant Professor in Telecommuni-cations, Department of Biophysical and ElectronicEngineering (DIBE), University of Genoa, since1995. In 1987, he joined the Signal Processing andUnderstanding Group (SPUG) of DIBE, where he

is responsible of the Industrial Signal and Image Processing (ISIP) area since1990. The ISIP area participated in several EU research and developmentprojects (ESPRIT (P7809 DIMUS, P8433 PASSWORDS, P6068 ATHENA)).From 1993 to 1995, he was a visiting scientist at the University of Toronto,Toronto, Ont., Canada. His main research interests are probabilistic nonlineartechniques for signal and image processing, nonconventional signal detectionand estimation techniques, and distributed data fusion in multisensor systems.

Dr. Regazzoni is a referee of several international journals and is a reviewerfor EU projects (ESPRITBRA, 1993, LTR 1994–1997, MAST 1996) and ItalianHigher Education evaluation projects (CAMPUS 1997). He is a member ofIAPR and AIIA.

Andrea Teschioni was born in Genoa, Italy, in1969. He obtained the Laurea degree in electronicengineering from the University of Genoa in 1994,with a thesis on nonlinear and adaptive imageprocessing for multilevel segmentation. Since 1996,he has been a Ph.D. student in electronic engineeringat the University.

In 1995, he joined the research team on SignalProcessing and Understanding, Department of Bio-physical and Electronic Engineering, University ofGenoa, where he is working in the industrial signal

and image processing area. He is currently involved in the research fields ofcolor image processing, image sequence analysis, and signal processing formobile communications.


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