Representation and Description Digital Image Processing Lecture 11,12 – Representation and Description Lecturer: Ha Dai Duong Faculty of Information Technology 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 used When colour or texture is important, an internal representation is used Digital Image Processing Representation and Description I Introduction „ „ The description of a region is based on its representation, for example a boundary can be described by its length The 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 Digital Image Processing I Introduction „ Contents: ‰ ‰ ‰ Representation Boundary Descriptors Region Descriptors Digital Image Processing Representation and Description 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 Digital Image Processing II.1 Chain code „ „ Generation of chain code: follow the boundary in an clockwise direction and assign a direction to the segment between successive pixels Difficulties: ‰ ‰ „ Code generally very long Noise changes the code Solution: Resample the boundary using a larger grid spacing Digital Image Processing Representation and Description II.1 Chain code „ Example Digital Image Processing II.1 Chain code „ „ „ Chain code depends on starting point Normalization: consider the code to be circular and choose the starting point in such a way that the sequence represents the smallest integer Example: Digital Image Processing Representation and Description 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 3 2 first difference: 33133030 „ „ 3 Scale invariance: chance grid size Note: when objects differ in scale and orientation (rotation), they will be sampled differently 2 Digital Image Processing 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 segments Several polygonal approximation techniques of modest complexity and processing requirements are well suited for image processing applications Digital Image Processing 10 Representation and Description II.2 Polygonal Approximations „ 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 ‰ Digital Image Processing 11 II.2 Polygonal Approximations Merging techniques „ Consider an arbitrary point on the boundary Consider the next point and fit a line through these two points: E = (least squares error is zero) Now consider the next point as well, and fit a line through all three these points using a least squares approximation Calculate E Repeat until E > T Store a and b of y = ax + b, and set E = Find the following line and repeat until all the edge pixels were considered Calculate the vertices of the polygon, that is where the lines intersect Digital Image Processing 12 Representation and Description II.2 Polygonal Approximations Merging techniques „ ‰ Least Squares Error „ Calculate E as: where Digital Image Processing 13 II.2 Polygonal Approximations Splitting techniques „ ‰ ‰ ‰ ‰ Joint the two furthest points on the boundary → line ab Obtain 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 possible Now obtain a polygon by joining c and d with a and b Repeat until the perpendicular distance is less than some predefined fraction of ab Digital Image Processing 14 Representation and Description II.2 Polygonal Approximations Splitting techniques - example „ Digital Image Processing 15 II.3 Signatures „ „ ‰ ‰ A signature is a 1-D representation of boundary It might be generated by various ways Simplest approach: plot r(θ) r: distance from centroid of boundary to boundary point θ: angle with the positive x-axis Digital Image Processing 16 Representation and Description II.3 Signatures Translation invariant, but not rotation or scale invariant Normalization 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 and ‰ (2) Divide each sample by its variance - assuming it is not zero ‰ Problem: sensitive to noise Digital Image Processing 17 II.3 Signatures Alternative approach: plot Φ(θ) „ ‰ ‰ ‰ ‰ ‰ ‰ Φ: angle between the line tangent to the boundary and a reference line θ: angle with the positive x-axis Φ(θ) carry information about basic shape characteristics Alternative approach: use the so-called slope density function as a signature, that is a histogram of the tangent-angle values Respond 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) Digital Image Processing 18 Representation and Description II.4 Boundary segments „ „ „ „ Boundary segments are usually easier to describe than the boundary as a whole We need a robust decomposition: convex hull A 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 Digital Image Processing 19 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 Digital Image Processing 20 10 Representation and Description III.3 Fourier Discriptors „ 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 Digital Image Processing 37 III.3 Fourier Discriptors „ Some basic properties of Fourier descriptors Digital Image Processing 38 19 Representation and Description III.4 Statistical moments „ „ Statistical moments can be used to describe the shape of a boundary segment A boundary segment can be represented by a 1-D discrete function g(r) Digital Image Processing 39 III.4 Statistical moments Digital Image Processing 40 20 Representation and Description IV.1 Simple Regional Descriptors Area: Number of pixels in region Perimeter: Length of boundary Compactness: Perimeter2/Area Mean and median gray levels Min and max gray level values Number of pixels with values above or below mean „ „ „ „ „ „ Digital Image Processing 41 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 holes Number of connected conponents Digital Image Processing 42 21 Representation and Description IV.2 Topological Descriptors Example „ Digital Image Processing 43 IV.2 Topological Descriptors Euler number: „ ‰ E denotes Euler number, it is denined as: E=C–H Where C: The number of connected components H: The number of holes ‰ Euler number is also topological property Digital Image Processing 44 22 Representation and Description IV.2 Topological Descriptors Example „ Digital Image Processing 45 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 „ „ ‰ ‰ ‰ Smoothness Coarseness and Regularity Digital Image Processing 46 23 Representation and Description IV.3 Texture Digital Image Processing 47 IV.3 Texture Statistical approaches „ ‰ ‰ Using the statistical moments of gray-level histogram of an image or region Let 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 Digital Image Processing 48 24 Representation and Description IV.3 Texture „ Statistical approaches Digital Image Processing 49 IV.3 Texture „ Statistical approaches Digital Image Processing 50 25 Representation and Description 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 R Third moment: generally is useful for determining the degree of symmetry of histogram: negative - left, positive – right Uniformity: First subimage is smoother (more uniform than the rest) Entropy: same idea with standard deviation Digital Image Processing 51 V Use Principal Components „ „ „ Applicable for boundaries and regions Can 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 Digital Image Processing 52 26 Representation and Description V Use Principal Components „ For K vector samples from a random population, the mean vector can be approximated as „ The covariance matri Cx can be approximated as follows: Digital Image Processing 53 V Use Principal Components „ Note Digital Image Processing 54 27 Representation and Description V Use Principal Components „ Example Digital Image Processing 55 V Use Principal Components Digital Image Processing 56 28 Representation and Description V Use Principal Components Digital Image Processing 57 V Use Principal Components Digital Image Processing 58 29 Representation and Description V Use Principal Components Digital Image Processing 59 V Use Principal Components Digital Image Processing 60 30 Representation and Description V Use Principal Components Digital Image Processing 61 V Use Principal Components Digital Image Processing 62 31 Representation and Description V Use Principal Components Instead of storing all six images, we only store two transformed images, along with mx and the first two rows of A Digital Image Processing 63 V Use Principal Components Digital Image Processing 64 32 Representation and Description VI Relational Descriptors Digital Image Processing 65 VI Relational Descriptors Digital Image Processing 66 33 ... Processing 54 27 Representation and Description V Use Principal Components „ Example Digital Image Processing 55 V Use Principal Components Digital Image Processing 56 28 Representation and Description. .. Processing 58 29 Representation and Description V Use Principal Components Digital Image Processing 59 V Use Principal Components Digital Image Processing 60 30 Representation and Description V... 20 Representation and Description IV.1 Simple Regional Descriptors Area: Number of pixels in region Perimeter: Length of boundary Compactness: Perimeter2/Area Mean and median gray levels Min and