digital image processing lecture 11,12 – representation ...fit.mta.edu.vn/files/filemonhoc/lecture...

Representation and Description

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Digital Image Processing

Lecture 11,12 – Representation and Description

Lecturer: Ha Dai DuongFaculty of Information Technology

Digital Image Processing 2

I. Introduction

After an image has been segmented into regions by methods such as those discussed in previous lectures, segmented pixels usually is represented àn described in a form suiable for further computer processing.A segmented region can be represented by: external (boundary) pixels or internal pixels

When shape is important, a boundary representation is usedWhen colour or texture is important, an internal representation is used


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I. Introduction

The description of a region is based on its representation, for example a boundary can be described by its lengthThe features selected as descriptors are usually required to be as insensitive as possible to variations in:

Scale;Translation;Rotation;

=> That is the features should be scale, translation and rotation invariant


I. Introduction

Contents:RepresentationBoundary DescriptorsRegion Descriptors


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II.1. Chain code

Chain codes are used to represent a boundary by a connected sequence of straight-line segments of specified length and direction.Typically, this presentation is based on 4- or 8-connectivity of the segments, the direction of each segment is coded by using a numbering scheme such as the ones shown in Fig 11.1


II.1. Chain code

Generation of chain code: follow the boundary in an clockwise direction and assign a direction to the segment between successive pixelsDifficulties:

Code generally very longNoise changes the code

Solution: Resample the boundary using a larger grid spacing


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II.1. Chain code

Example


II.1 . Chain code

Chain code depends on starting pointNormalization: consider the code to be circular and choose the starting point in such a way that the sequence represents the smallest integerExample:


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II.1. Chain code

Rotation invariance: consider the first difference in the code: This code is obtained by counting the number of direction changes (go counter -clockwise)

Example:4-directional code: 1 0 1 0 3 3 2 2first difference: 3 3 1 3 3 0 3 0

Scale invariance: chance grid sizeNote: when objects differ in scale and orientation (rotation), they will be sampled differently

1

01

0

3

3

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II.2. Polygonal Approximations

A digital boundary can be approximated by a pologon. For closed curve, the approximation is exact when the number of segments in polygon is equal to the number of points in the boundary so that each pair of adjacent points defines a segment in the polygon.

In practice, the goal of polygonal approximation is to capture the “essence” of boundary shape with the fewest possible polygonal segmentsSeveral polygonal approximation techniques of modest complexity and processing requirements are well suited for image processing applications.


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Minimum perimeter polygons Imagine the boundary as a “rubber band” and let it shrink...

The maximum error per grid cell is √2d, where d is the dimension of a grid cell



Merging techniques1. Consider an arbitrary point on the boundary2. Consider the next point and fit a line through these two points:

E = 0 (least squares error is zero)3. Now consider the next point as well, and fit a line through all

three these points using a least squares approximation. Calculate E

4. Repeat until E > T5. Store a and b of y = ax + b, and set E = 06. Find the following line and repeat until all the edge pixels were

considered7. Calculate the vertices of the polygon, that is where the lines

intersect


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Merging techniquesLeast Squares Error

Calculate E as: where


II.2 . Polygonal Approximations

Splitting techniquesJoint the two furthest points on the boundary → line abObtain a point on the upper segment, that is c and a point on the lower segment, that is d, such that the perpendicular distance from these points to ab is as large as possibleNow obtain a polygon by joining c and d with a and bRepeat until the perpendicular distance is less than some predefined fraction of ab


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Splitting techniques - example


II.3. Signatures

A signature is a 1-D representation of boundary. It might be generated by various waysSimplest approach: plot r(θ)

r: distance from centroid of boundary to boundary pointθ: angle with the positive x-axis


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II.3. Signatures

Translation invariant, but not rotation or scale invariantNormalization for rotation:

(1) Choose the starting point as the furthest point from the centroid OR(2) Choose the starting point as the point on the major axis that is the furthest from the centroid

Normalization for scale:Note: ↑ scale => ↑ amplitude of signature(1) Scale signature between 0 and 1

Problem: sensitive to noise(2) Divide each sample by its variance - assuming it is not zero


II.3. Signatures

Alternative approach: plot Φ(θ)Φ: angle between the line tangent to the boundary and a reference lineθ: angle with the positive x-axisΦ(θ) carry information about basic shape characteristicsAlternative approach: use the so-called slope density function as a signature, that is a histogram of the tangent-angle valuesRespond strongly to sections of the boundary with constant tangent angles (straight or nearly straight segments)Deep valleys in sections producing rapidly varying angles (corners or other sharp inflections)


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II.4. Boundary segments

Boundary segments are usually easier to describe than the boundary as a wholeWe need a robust decomposition: convex hullA convex set (region) is a set (region) in which any two elements (points) A and B in the set (region) can be joined by a line AB, so that each point on AB is part of the set (region)The convex hull H of an arbitrary set (region) S is the smallest convex set (region) containing S


II.4. Boundary segments

Convex deficiency: D = H − S

The region boundary is partitioned by following the contour of S and marking the points at which a transition is made into or out of a component of the convex deficiency


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II.5. Skeletons

An important approach to representing the structural shape of a plane region is to reduce it to a graph. This reduction may be ascomplished by obtaining the skeleton of the region via a thinning algorithmThe skeleton of a region may be defined via the medial axis transformation proposed by Blum in 1967


II.5. Skeletons

Brute force method: Medial axis transformation (MAT - 1967)Consider a region R with boundary B...For each point p in R find its closest neighbour in B

If p has more than one closest neighbour, then p is part of the medial axisExample

Problem: we have to calculate the distance between every internal point and every point on the edge of the boundary!


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II.5. Skeletons

Thinning algorithm:Edge points are deleted in an iterative way so that

1. end points are not removed,2. connectivity is not broken, and3. no excessive erosion is caused to the region

This algorithm thins a binary region, where an edge point = 1 and a background point = 0Contour point: Edge point (= 1) with at least one neighbour with a value of 0


II.5. Skeletons

Step 1: A contour point is flagged for deletion ifa) 2 < N(p1) < 6b) T(p1) = 1c) p2 · p4 · p6 = 0d) p4 · p6 · p8 = 0Where:

N(p1) ≡ number of non-zero neighbours of p1T(p1) ≡ number of 0 − 1 transitions in the sequence{p2, p3, p4, p5, p6, p7, p8, p9, p2}


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II.5. Skeletons

Example- N(p1) ≡ number of non-zero neighbours of p1- T(p1) ≡ number of 0 − 1 transitions in the sequence

{p2, p3, p4, p5, p6, p7, p8, p9, p2}


II.5. Skeletons

Now delete all the flagged contour points and consider the remaining contour points...Step 2: A contour point is flagged for deletion if

a) 2 < N(p1) < 6b) T(p1) = 1c) p2 · p4 · p8 = 0d) p2 · p6 · p8 = 0

Delete all the flagged contour pointsRepeat steps 1 and step 2 until no contour point is deleted during an iteration


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II.5. Skeletons

Reasons for each of these conditions...


II.5. Skeletons

Example


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III.1. Simple boundary descriptors

Length: The number of pixels along a boundaryDiameter: Diameter of B is defined as

Where D is a distance measure

Major AxisMinor AxisBasic rectangleEccentricity: the ratio of the major to the minor axis.


III.2. Shape Numbers

Shape number: Difference of chain code that represents smallest integerOrder n of shape number: number of digits in its representationSuppose that we use a 4-directional chain code, then...

Closed boundary => n evenn limits the number of possible shapes


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Example



Note: the shape number is rotation invariant, but the coded boundary depends on the orientation of the grid!One way to normalize the grid orientation is by aliging the chain code grid with the sides of the basic ractangle defined in the previous sectionExample: Suppose that n=18 is specified for the boundary shown in Fig 11.12(a)

1. Find the basic rectangle - Fig 11.12(b)2. Subdivision of the basic rectangle with grid 3x6 - Fig 11.12(c)3. Chain code directions are aligned with the resulting grid, find

the shape number Fig 11.12 (d)


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Example


III.3. Fourier Discriptors

Suppose that a boundary is represented by K coordinate pairs in the xy-plane, (x0, y0), (x1, y1), (x2, y2), . . . , (xK−1, yK−1)


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When we traverse this boundary in an anti-clockwise direction the boundary can be represented as the sequence of coordinates sk= [xk, yk ] for k = 0, 1, 2, . . . , K − 1Each coordinate pair can be treated as a complex number so that sk = xk + i*yk => 2-D problem → 1-D problem



Sn (n=0…K-1) are called Fourier descritors of the boundary


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Note that the same number of points exist in the approximate boundary ˜sk. Also note that the smaller P becomes, the more detail in the boundary is lost...



Some basic properties of Fourier descriptors


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III.4. Statistical moments

Statistical moments can be used to describe the shape of a boundary segmentA boundary segment can be represented by a 1-D discrete function g(r) ...


III.4. Statistical moments


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IV.1. Simple Regional Descriptors

Area: Number of pixels in regionPerimeter: Length of boundaryCompactness: Perimeter2/AreaMean and median gray levelsMin and max gray level valuesNumber of pixels with values above or below mean


IV.2. Topological Descriptors

Topological properties are useful for global descriptions of regions in image plane.Topology is the study of properties of a fingure that are unaffected by any deformation.That properties may be:

Number of holesNumber of connected conponents


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Example



Euler number:E denotes Euler number, it is denined as:

E = C – H

Where C: The number of connected componentsH: The number of holes

Euler number is also topological property.


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Example


IV.3. Texture

An important approach to region description is to quantify its texture content.Although no formal definition of texture exist, intuitively this description provides measures of properties such as

SmoothnessCoarseness andRegularity


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IV.3. Texture


IV.3. Texture

Statistical approachesUsing the statistical moments of gray-level histogram of an image or regionLet z be a random variable denoting gray levels and let p(zi), i=0,1,.., L-1, be the corresponding histogram, the nth moment of z about mean is


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IV.3. Texture

Statistical approaches


IV.3. Texture

Statistical approaches


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IV.3. Texture

Mean: Average gray level of each region, not really texture.Standard deviation: First texture has significantly less variability in gray level than the other two textures. The same comment hold for RThird moment: generally is useful for determining the degree

of symmetry of histogram: negative - left, positive – rightUniformity: First subimage is smoother (more uniform than the rest).Entropy: same idea with standard deviation


V. Use Principal Components

Applicable for boundaries and regionsCan also describle sets of images that were registered diffirently, for example the three images of a color RGB image…Treat three images as unit by expressing each group of corresponding pixels as a vector

when we have n registered images ..


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For K vector samples from a random population, the mean vector can be approximated as

The covariance matri Cx can be approximated as follows:



Note


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Example




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Instead of storing all six images, we only store two transformed images, along with mx and the first two rows of A.




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VI. Relational Descriptors


VI. Relational Descriptors

digital image processing lecture 11,12 – representation ...fit.mta.edu.vn/files/filemonhoc/lecture...

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