DIGITAL IMAGE PROCESSING Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Kachooeian Kachooeian.mina@gmail.com

DIGITAL IMAGE PROCESSING Chapter 11 - Representation and Description Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Kachooeian Kachooeian.mina@gmail.com

Road map of chapter 11 Relational Descriptors 3.1 3.1 3.2 3.2 3.3 3.3 3.4 3.4 3.5 3.5 Relational Descriptors Use of Principal Components for Description Boundary Descriptors Regional Descriptors Representation 11.1- Representation 11.2- Boundary Descriptors 11.3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

11.1 Representation

Representation Chain Codes Chain Codes Polygonal Approximations 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Chain Codes Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

Chain Codes Chain codes are used to represent a boundary by a connected sequence of straight-line segments of specified length and direction. a chain code can be generated by following a boundary (in a clockwise direction) and assigning a direction to the segments connecting every pair of pixels. This method generally is unacceptable for two principal reasons: The resulting chain tends to be quite long any small disturbances along the boundary due to noise or imperfect segmentation cause changes in the code that may not be related to the principal shape features of the boundary. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Chain Codes 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Chain Codes 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors The chain code of a boundary depends on the starting point.It can be normalized with respect to the starting point. We simply treat the chain code as a circular sequence of direction numbers and redefine the starting point so that the resulting sequence of numbers forms an integer of minimum magnitude. We can normalize also for rotation (in angles that are integer multiples of the directions in Fig. 11.3) by using the firs difference of the chain code instead of the code itself.

Chain Codes Normalization For Rotation This difference is obtained by counting the number of direction changes that separate two adjacent elements of the code. Example Chain code is 10103322 Normalized with first difference 3133030 If we treat the code as a circular sequence to normalize with respect to the starting point,the result is 33133030. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Chain Codes 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Chain Codes The 8-directional Freeman chain code : 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors The 8-directional Freeman chain code : 0 0 0 0 6 0 6 6 6 6 6 6 6 6 4 4 4 4 4 4 2 4 2 2 2 2 2 0 2 2 0 2 The integer of minimum magnitude of the code: The first difference of either code is 0 0 0 6 2 6 0 0 0 0 0 0 0 6 0 0 0 0 0 6 2 6 0 0 0 0 6 2 0 6 2 6   Using any of these codes to represent the boundary results in a significant reduction in the amount of data needed to store the boundary. In addition, keep in mind that the subsampled boundary can be recovered from any of the preceding codes.

Representation Chain Codes Polygonal Approximations 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Chain Codes Polygonal Approximations Polygonal Approximations Signatures Boundary Segments Skeletons

Polygonal Approximations A digital boundary can be approximated with arbitrary accuracy by a polygon. Polygonal approximations: to represent a boundary by straight line segments, and a closed path becomes a polygon. to be exact,we should have: Number of segments of polygon=number of points in boundary 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations The size of the cells determines the accuracy of the polygonal approximation. if the size of each (square) cell corresponds to a pixel in the boundary, error in each cell between the boundary and the MPP approximation at most would be 2𝑑 , where d is the minimum possible distance between pixels. The objective is to use the largest possible cell size acceptable in a given application producing MPPs with the fewest number of vertices 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations Concave vertex Convex vertex 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors 1. The MPP bounded by a simply connected cellular complex is not self intersecting. 2. Every convex vertex of the MPP is a W vertex, but not every W vertex of a boundary is a vertex of the MPP 3. Every mirrored concave vertex of the MPP is a B vertex, but not every B vertex of a boundary is a vertex of the MPP 4. All B vertices are on or outside the MPR and all W vertices are on or in side the MPP 5. The uppermost, leftmost vertex in a sequence of vertices contained in a cellular complex is always a W vertex of the MPP

Polygonal Approximations orientation of triplets of points: the triplet of points, (a, b, c) a = (x1, y1), b = (x2,y2), and c = (x3,y3) 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations Algorithm: form a list whose rows are the coordinates of each vertex and an additional element denoting whether the vertex is W or B. V0 the first vertex be the uppermost leftmost assume that vertices are arranged in the counterclockwise direction. WC crawls along convex (W) vertices Bc crawls along mirrored concave (B) vertices VL last MPP vertex found Vk  current vertex being examined. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations SetWC = BC = V0 Vk lies to the positive side of the line through pair (VL, WC); that is, sgn(VL, WC, Vk) > 0. the next MPP vertex is WC, and we let VL = WC; then we reinitialize the algorithm by setting WC = BC = VL, and continue with the next vertex left (b) Vk lies on the negative side of the line though pair (VL, WC) or is collinear with it; that is sgn(VL, WC, Vk) 0. At the same time, VK lies to the positive side of the line through (VL, BC) or is collinear with it; that is, sgn(VL,BC, Vk) = 0. VL becomes a candidate MPP vertex. In this case, we set WC = Vk if Vk is convex (i.e., it is a W vertex); otherwise we set BC = Vk. We then continue with the next vertex in the list. . 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations (c) Vk lies on the negative side of the line though pair (VL, Bc): that is, sgn(VL, BC, VK) < 0. the next MPP vertex is BC and we let VL = BC; then we reinitialize the algorithm by setting WC = BC = VL and continue with the next vertex after VL 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Algorithm terminates when it reaches the first vertex again, and thus has processed all the vertices in the polygon. The VL vertices found by the algorithm are the vertices of the MPP

Polygonal Approximations Example 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors letting WC = BC = V0 = VL = (1.4). The next vertex is V1 = (2, 3). sgn(VL, WC, V1) = 0 and sgn(VL, BC, V1) = 0 condition (b) holds. We let BC = V1 = (2, 3) and WC is at (1, 4), VL is still at (1,4) because no new MPP-vertex was found.

Polygonal Approximations Next, we look at V2 = (3, 3). sgn(VL, WC, V2) = 0, and sgn(VL, BC, V2) = 1 condition (b) of the algorithm holds again. Because V2 is a W (convex) vertex , we let WC = V2 = (3, 3). At this stage, the crawlers are at WC = (3,3) and BC = (2,3); VL remains un- changed. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations If we we examine V5 = (7,1) ,we’ll have: sgn(VL, WC, V5) = 9 so condition (a) holds set VL = WC = (4, 1). Because a new MPP vertex was found, we reinitialize the algorithm by setting WC = BC = VL and start again with the next vertex being the vertex after the newly found VL. The next vertex is V5, so we visit it again. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Polygonal Approximations 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Continuing as above with this and the remaining vertices yields the MPP vertices in Fig below.

Polygonal Approximations 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Representation Chain Codes Polygonal Approximations Signatures 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Chain Codes Polygonal Approximations Signatures Signatures Boundary Segments Skeletons

Signatures represents a shape by a one dimensional function derived from shape boundary point. 1-D function is easier to describe than the original 2-D boundary. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Signatures 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Signatures 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Problem: Signatures generated by this approach are invariant to translation, but they do depend on rotation. Solution for rotation: We can normalize with respect to rotation . find a way to select the same starting point to generate the signature, regardless of the shape’s orientation. One way to do so is to select the starting point as the point farthest from the centroid, assuming that this point is unique for each shape of interest.

Signatures 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Solution for scaling: We can normalize with respect to scaling. Scale all functions so that they always span the same range of values, e.g., [0, 1]. Advantage: simplicity disadvantage :depends on only two values the minimum and maximum.

Representation Chain Codes Polygonal Approximations Signatures 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Chain Codes Polygonal Approximations Signatures Boundary Segments Boundary Segments Skeletons

Boundary Segments 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Decomposition reduces the boundary’s complexity and thus simplifies the description process. In this case, use of the convex hull of the region enclosed by the boundary is a powerful tool for robust decomposition of the boundary.

Boundary Segments convex hull H of an arbitrary set S is the smallest convex set containing S. The set difference H - S is called the convex deficiency D of the set S. The region boundary can be 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. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Convex deficiency

Boundary Segments S = dbabcbabdbabcbab 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors S = dbabcbabdbabcbab

Boundary Segments Another use of convex deficiency is to recognize human actions. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Boundary Segments Problem: digital boundaries tend to be irregular . These effects usually result in convex deficiencies that have small, meaningless components scattered randomly through-out the boundary. Solution: smooth a boundary prior to partitioning. traverse the boundary and replace the coordinates of each pixel by the average coordinates of k of its neighbors along the boundary. works for small irregularities, but it is time-consuming and difficult to control Large values of k  excessive smoothing small values of k insufficient in some segments of the boundary 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Boundary Segments Better solution: use a polygonal approximation prior to finding the convex deficiency of a region. Most digital boundaries of interest are simple .Graham and Yao [1983] give an algorithm for finding the convex hull of such polygons. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Representation Chain Codes Polygonal Approximations Signatures 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons Skeletons

Skeletons Problem of some skeletonizing algorithms: 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Represent structural shape of a plane region reduce it to a graph. Use skeletonizing algorithm via a thinning Problem of some skeletonizing algorithms: the procedure made no provisions for keeping the skeleton connected. The skeleton of a region may be defined via the medial axis transformation (MAT) proposed by Blum [1967].

Skeletons medial axis is the locus of centers of maximal disks that fit within the shape. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Skeletons Difference between MAT and skeleton 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Skeletons 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors The MAT of a region R with border B: For each point p in R, find its closest neighbor in B. If p has more than one such neighbor, it is said to belong to the skeleton of R. Problem: direct implementation of this definition is expensive computationally. Implementation involves calculating the distance from every interior point to every point on the boundary of a region.

Skeletons Solution: thinning algorithms that iteratively delete boundary points of a region subject to the constraints that deletion of these points (1) does not remove end points (2) does not break connectivity (3) does not cause excessive erosion of the region. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Skeletons Algorithm for thinning binary regions Step 1 flags a contour point p, for deletion if the following conditions are satisfied: 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors N(p1) is the number of nonzero neighbors of p1 T(p1) is the number of 0-1 transitions in the ordered sequence p2, p3, ,..., p8. p9, p2.

Skeletons Example 0 0 1 1 p1 0 1 0 1 N(p1) = 4 and T(p1) = 3 0 0 1 1 p1 0 1 0 1 N(p1) = 4 and T(p1) = 3 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Skeletons Step 2 conditions (a) and (b) remain the same, but conditions (c) and (d) are changed to 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Algorithm: 1.Apply Step 1 to every border pixel in the binary region . 2. if all conditions (a)-(d) are satisfied the point is flagged for deletion. 3.Delete flagged points (changed to 0). 4.Apply step 2 to the resulting data

Skeletons 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

11.2 Boundary Descriptors

Boundary Descriptors Some Simple Descriptors Some Simple Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some Simple Descriptors Some Simple Descriptors Shape Numbers Fourier Descriptors Statistical Moments

Some Simple Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors There are several simple geometric measures that can be useful for describing a boundary. The length of a boundary: the number of pixels along a boundary gives a rough approximation of its length. Curvature: the rate of change of slope To measure a curvature accurately at a point in a digital boundary is difficult The difference between the slops of adjacent boundary segments is used as a descriptor of curvature at the point of intersection of segments

Boundary Descriptors Some Simple Descriptors Shape Numbers 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some Simple Descriptors Shape Numbers Shape Numbers Fourier Descriptors Statistical Moments

Shape Numbers First difference The shape number of a boundary is defined as the first difference of smallest magnitude. The order n of a shape number is defined as the number of digits in its representation.

Shape Numbers

Shape Numbers

Boundary Descriptors Some Simple Descriptors Shape Numbers 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some Simple Descriptors Shape Numbers Fourier Descriptors Fourier Descriptors Statistical Moments

Fourier Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors This is a way of using the Fourier transform to analyze the shape of a boundary. The x-y coordinates of the boundary are treated as the real and imaginary parts of a complex number. Then the list of coordinates is Fourier transformed using the DFT (chapter 4). The Fourier coefficients are called the Fourier descriptors. The basic shape of the region is determined by the first several coefficients, which represent lower frequencies. Higher frequency terms provide information on the fine detail of the boundary.

Fourier Descriptors Fourier descriptor: view a coordinate (x,y) as a complex number (x = real part and y = imaginary part) then apply the Fourier transform to a sequence of boundary points. Let s(k) be a coordinate of a boundary point k : Fourier descriptor : Reconstruction formula Boundary points

Fourier Descriptors

Fourier Descriptors Some properties of Fourier descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some properties of Fourier descriptors

Boundary Descriptors Some Simple Descriptors Shape Numbers 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some Simple Descriptors Shape Numbers Fourier Descriptors Statistical Moments Statistical Moments

Statistical Moments Moments are statistical measures of data. 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Moments are statistical measures of data. They come in integer orders. Order 0 is just the number of points in the data. Order 1 is the sum and is used to find the average. Order 2 is related to the variance, and order 3 to the skew of the data. Higher orders can also be used, but don’t have simple meanings.

Statistical Moments Let r be a random variable, and g(ri) be normalized (as the probability of value ri occurring), then the moments are 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

11.3 Regional Descriptors

Regional Descriptors Some Simple Descriptors Some Simple Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some Simple Descriptors Some Simple Descriptors Topological Descriptors Texture Moments of Two-Dimensional Functions

Some Simple Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some simple descriptors The area of a region: the number of pixels in the region The perimeter of a region: the length of its boundary The compactness of a region: (perimeter)2/area The mean and median of the gray levels The minimum and maximum gray-level values The number of pixels with values above and below the mean

Some Simple Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Regional Descriptors Some Simple Descriptors Topological Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some Simple Descriptors Topological Descriptors Topological Descriptors Texture Moments of Two-Dimensional Functions

Topological Descriptors Topological property 1: the number of holes (H) 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Topological property 2: the number of connected components (C)

Topological Descriptors Topological property 3: Euler number: the number of connected components subtract the number of holes E = C - H 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors E=0 E= -1

Topological Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Topological property 4: the largest connected component.

Regional Descriptors Some Simple Descriptors Topological Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some Simple Descriptors Topological Descriptors Texture Texture Moments of Two-Dimensional Functions

Texture 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Texture is usually defined as the smoothness or roughness of a surface. In computer vision, it is the visual appearance of the uniformity or lack of uniformity of brightness and color. There are two types of texture: random and regular. Random texture cannot be exactly described by words or equations; it must be described statistically. The surface of a pile of dirt or rocks of many sizes would be random. Regular texture can be described by words or equations or repeating pattern primitives. Clothes are frequently made with regularly repeating patterns. Random texture is analyzed by statistical methods. Regular texture is analyzed by structural or spectral (Fourier) methods.

Texture 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Statistical Approaches Let z be a random variable denoting gray levels and let p(zi), i=0,1,…,L-1, be the corresponding histogram, where L is the number of distinct gray levels. The nth moment of z: The measure R: The uniformity: The average entropy:

Statistical Approaches Smooth Coarse Regular

Statistical Approaches Structural concepts: Suppose that we have a rule of the form S→aS, which indicates that the symbol S may be rewritten as aS. If a represents a circle [Fig. 11.23(a)] and the meaning of “circle to the right” is assigned to a string of the form aaaa… [Fig. 11.23(b)] .

Spectral Approaches 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Spectral Approaches

Regional Descriptors Some Simple Descriptors Topological Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Some Simple Descriptors Topological Descriptors Texture Moments of Two-Dimensional Functions Moments of Two-Dimensional Functions

Moments of Two-Dimensional Functions For a 2-D continuous function f(x,y), the moment of order (p+q) is defined as The central moments are defined as

Moments of Two-Dimensional Functions If f(x,y) is a digital image, then The central moments of order up to 3 are

Moments of Two-Dimensional Functions The central moments of order up to 3 are

Moments of Two-Dimensional Functions The normalized central moments are defined as 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Moments of Two-Dimensional Functions A seven invariant moments can be derived from the second and third moments:

Moments of Two-Dimensional Functions This set of moments is invariant to translation, rotation, and scale change

Moments of Two-Dimensional Functions

Moments of Two-Dimensional Functions Table 11.3 Moment invariants for the images in Figs. 11.25(a)-(e).

11.4 Use of Principal Components for Description

Principal Components for Description Purpose: to reduce dimensionality of a vector image while maintaining information as much as possible. Let Mean: Covariance matrix

Principal Components for Description Let 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Where A is created from eigenvectors of Cx as follows Row 1 contain the 1st eigenvector with the largest eigenvalue. Row 2 contain the 2nd eigenvector with the 2nd largest eigenvalue. …. and Then elements of are uncorrelated. The component of y with the largest l is called the principal component.

Principal Components for Description 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors Eigenvector and eigenvalue of Matrix C are defined as Let C be a matrix of size NxN and e be a vector of size Nx1. If Where l is a constant We call e as an eigenvector and l as eigenvalue of C

Principal Components for Description 6 spectral images from an airborne Scanner.

Principal Components for Description Component l 3210 931.4 118.5 83.88 64.00 13.40

Original image After Hotelling transform

Principal Components for Description 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

11.5 Relational Descriptors

Relational Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Relational Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Relational Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Relational Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Relational Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors

Relational Descriptors 11.1- Representation 11.2- Boundary Descriptors 11,3- Regional Descriptors 11.4- Use of Principal Components for Description 11.5 - Relational Descriptors