Computer Assisted Image Analysis

Transcription

1 Computer Assisted Image Analysis Lecture 0 - Object Descriptors II Amin Allalou amin@cb.uu.se Centre for Image Analysis Uppsala University A. Allalou (Uppsala University) Object Descriptors II / 26

2 Reading Instructions Chapters for this lecture Chapter.3.4 in Gonzales-Woods. A. Allalou (Uppsala University) Object Descriptors II / 26

3 Today Regional Descriptors Simple Topological Texture Statistical Spectral Moments Other Descriptors Principal components A. Allalou (Uppsala University) Object Descriptors II / 26

4 Simple Descriptors Simple Descriptors Area - number of pixels in region. Perimeter - boundary length. Compactness - P2A = P 2 /A (circularity). Graylevel measures: mean, median, max etc. A. Allalou (Uppsala University) Object Descriptors II / 26

5 Topological Descriptors topology The study of properties of a figure that are unaffected by any deformation. Topological Descriptors Number of holes in region, H. Number of connected components, C. Euler number, E = C H. A. Allalou (Uppsala University) Object Descriptors II / 26

6 Topological Descriptors Using Connected Components in Segmentation IR image Thresholded image Largest connected Skeleton A. Allalou (Uppsala University) Object Descriptors II / 26

7 Texture Figure: Smooth, coarse and regular textures Statistical and spectral approaches. A. Allalou (Uppsala University) Object Descriptors II / 26

8 Statistical Texture Statistical Texture Descriptors Histogram based Statistical moments Uniformity Average entropy Co-occurence matrix based Maximum probability etc. A drawback with using the histogram for texture analysis is that no positional information is used. Only gives information such as smooth or coarse not if patterned. A. Allalou (Uppsala University) Object Descriptors II / 26

9 Histogram Based Descriptors Statistical Moments Properties of the graylevel histogram, of an image or region, used when calculating statistical moments. z - discrete random variable representing discrete graylevels in the range [0, L ]. p(z i ) - normalised histogram component corresponding to the ith value of z. The nth Statistical Moment (about its mean) L L µ n (z) = (z i m) n p(z i ), m = z i p(z i ) i=0 µ 2 (z) - variance of z (i.e., σ 2 (z)). µ 3 (z) - histogram skewness. µ 4 (z) - relative flatness of histogram. i=0 A. Allalou (Uppsala University) Object Descriptors II / 26

10 Histogram Based Descriptors Uniformity and Average Entropy Also use the histogram properties z and p(z i ). Uniformity L U = p 2 (z i ) U is maximum for an image in which all graylevels are equal (maximally uniform). Average Entropy i=0 i=0 L e = p(z i ) log 2 p(z i ) Measure of variability. Is 0 for constant images. A. Allalou (Uppsala University) Object Descriptors II / 26

11 Co-occurence Matrix For an image with N graylevels, and P, a positional operator, generate A, a N N matrix, where a i,j is the number of times a pixel with graylevel value z i is in relative position P to graylevel value z j. Divide all elements in A with the sum of all elements in A (the number of times P was satisfied in the image). This gives a new matrix C, where c i,j is the probability that a pair of pixels fulfilling P has graylevel values z i and z j, which is called the co-occurrence matrix. A. Allalou (Uppsala University) Object Descriptors II / 26

12 Co-occurence Matrix Examples Ex.) P = one pixel down A = , C = Ex.2) A 2 = 4, C 2 = st row: pixel value 0 in pos P to other values 2 nd row: pixel value in pos P to other values 3 rd row: pixel value 2 in pos P to other values A. Allalou (Uppsala University) Object Descriptors II / 26

13 Co-occurence Matrix Descriptors Maximum Probability (strongest response to P) max(c ij ) i,j Uniformity Entropy (randomness) i i j c 2 ij c ij log 2 c ij j Characterize content of C using these descriptors. A. Allalou (Uppsala University) Object Descriptors II / 26

14 Co-occurence Matrix Descriptors Examples Example C = Maximum probability: Uniformity: Entropy: max(c ij ) = i,j 3 c 2 ij i i j c ij log 2 c ij = undef. j A. Allalou (Uppsala University) Object Descriptors II / 26

15 Co-occurence Matrix Descriptors Examples Example 2 C 2 = Maximum probability: Uniformity: Entropy: max(c ij ) = i,j 3 c 2 ij 0.67 i i j c ij log 2 c ij 2.98 j A. Allalou (Uppsala University) Object Descriptors II / 26

16 Co-occurence Matrix Descriptors Examples Figure: Match sample image with correct co-occurrence matrix A. Allalou (Uppsala University) Object Descriptors II / 26

17 Spectral Texture Peaks in the Fourierspectrum give information about direction and spatial period of pattern. Express spectrum in polar coordinates S(r, θ). For each direction θ, S(r, θ) is a D function S θ (r). Similarly, for each frequency r, S r (θ) is a D function. A global description can be obtained by summing S θ (r) and S r (θ): S(r) = π R 0 S θ (r), S(θ) = S r (θ) θ=0 r= Spectral-energy description (entire region). A. Allalou (Uppsala University) Object Descriptors II / 26

18 Spectral Texture Example Image Spectrum S(r) S(θ) A. Allalou (Uppsala University) Object Descriptors II / 26

19 Moments For a 2D continuous function f (x, y), the moment of order (p + q) is defined as m pq = x p y q f (x, y) dx dy for p, q = 0,, 2,.... The moment sequence (m pq ) is uniquely determined by f (x, y), and conversely, (m pq ) uniquely determines f (x, y). The central moments are defined as µ pq = (x x) p (y ȳ) q f (x, y) dx dy where x = m 0 m 00 and ȳ = m 0 m 00. A. Allalou (Uppsala University) Object Descriptors II / 26

20 Moments If f (x, y) is a digital image, then the central moments become µ pq = x (x x) p (y ȳ) q f (x, y). y The normalised central moments, denoted η pq, are defined as η pq = µ pq µ γ 00 where for p + q = 2, 3,.... γ = p + q 2 + A. Allalou (Uppsala University) Object Descriptors II / 26

21 Moments A set of seven invariant moments can be derived from the second and third moments. φ = η 20 + η 02 φ 2 = (η 20 η 02 ) 2 + 4η 2 φ 3 = (η 30 η 2 ) 2 + (3η 2 η 03 ) 2 φ 4 = (η 30 + η 2 ) 2 + (η 2 + η 03 ) 2... (see textbook) The set of moments is invariant to translation, rotation, and scale change. A. Allalou (Uppsala University) Object Descriptors II / 26

22 Moments Invariant Example A. Allalou (Uppsala University) Object Descriptors II / 26

23 Principal Components Image, region or boundary. R G B Each pixel represented as a 3D vector. Pixel vector and mean vector: R(i, j) mean(r) x i,j = G(i, j), m x = E{x} = mean(g) = K x k K B(i, j) mean(b) k= Covariance matrix C x = E{(x m x )(x m x ) T } = K K x k x T k m x m T x k= A. Allalou (Uppsala University) Object Descriptors II / 26

24 Principal Components C x is real and symmetric. e i and λ i are the eigenvectors and corresponding eigenvalues. (C x e i = λ i e i ) A is a matrix with the eigenvectors as rows, ordered corresponding to decreasing eigenvalue. Use A to transform x to y: y = A(x m x ). This is called the Hotelling transform or principal component transform. Covariance matrix C y = AC x A T = diagonal matrix with eigenvalues on the diagonal. Any vector x can be recovered from y by: x = A T y + m x and approximated by only using some (say k) of the eigenvalues and an A k matrix constructed from the k eigenvectors. A. Allalou (Uppsala University) Object Descriptors II / 26

25 Principal Components This gives a possibility to store the data more efficiently. For a 2D image the Hotelling or principal component transform is used to align an object or boundary according to its eigenaxes. This removes rotational effects, and the eigenvalues can be used for size normalization. Translational effects are also removed since the object is centered around its mean. A. Allalou Figure: (Uppsala University) Match sample image Objectwith Descriptors correct II co-occurrence matrix / 26

26 Principal Components Example A. Allalou (Uppsala University) Object Descriptors II / 26