Digital Image Processing: Image Restoration Matrix Formulation - Duong Anh Duc

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Digital Image Processing: Image Restoration Matrix Formulation - Duong Anh Duc provides about matrix Formulation of Image Restoration Problem; constrained least squares filtering (restoration); a brief review of matrix differentiation; Pseudo-inverse Filtering; Minimum Mean Square Error (Wiener) Filter; Parametric Wiener Filter.

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Nội dung Text: Digital Image Processing: Image Restoration Matrix Formulation - Duong Anh Duc

Digital Image Processing Image Restoration Matrix Formulation 21/11/15 Duong Anh Duc - Digital Image Processing 1
Matrix Formulation of Image Restoration Problem 1-D Case:  We will consider the 1-D version first, for simplicity: g(m) = f(m)*h(m) + (m)  We will assume that the arrays f and h have been zero- padded to be of size M, where M length(f) + length(h) - 1.  Henceforth, we will not explicitly mention the zero-padding.  The degradation equation: can be written in matrix-vector form as follows: g = Hf + n, where 21/11/15 Duong Anh Duc - Digital Image Processing 2
Matrix Formulation of Image Restoration Problem h0 h 1 h 2  h M 1 h1 h0 h 1  h M 2 H H1 h2 h1 h0  h M 3      hM 1 hM 2 hM 3  h0 h0 0 0  0 h1 h0 0  0 h2 h1 h0  0      hM 1 hM 2 hM 3  h0 21/11/15 Duong Anh Duc - Digital Image Processing 3
Matrix Formulation of Image Restoration Problem  However, since the arrays f and h are zero- padded, we can equivalently set: h0 hM 1 hM 2  h1 h1 h0 hM 1  h2 H H2 h2 h1 h0  h3      hM 1 hM 2 hM 3  h0  Notice that the (second) matrix H is circulant; i.e., each row of H is a circular shift of the previous row. 21/11/15 Duong Anh Duc - Digital Image Processing 4
Matrix Formulation of Image Restoration Problem  Example: A = length of array f = 3  B = length of array h = 2 M A + B – 1 = 4, say M = 4. f 0 h0 f 1 h1 f h f 2 0 0 0 21/11/15 Duong Anh Duc - Digital Image Processing 5
Matrix Formulation of Image Restoration Problem h0 h 1 h 2 h 3 h1 h0 h 1 h 2 H1 h2 h1 h0 h 1 h3 h2 h1 h 0 h0 0 0 0 h0 0 0 0 h1 h0 0 0 h1 h0 0 0 h2 h1 h0 0 0 h1 h0 0 h3 h2 h1 h0 0 0 h1 h0 21/11/15 Duong Anh Duc - Digital Image Processing 6
Matrix Formulation of Image Restoration Problem h0 hM 1 hM 2 hM 3 h1 h0 hM 1 hM 2 H2 h2 h1 h0 hM 1 h3 h2 h1 h0 h0 h3 h2 h1 h0 0 0 h1 h1 h0 h3 h2 h1 h0 0 0 h2 h1 h0 h3 0 h1 h0 0 h3 h2 h1 h0 0 0 h1 h0 Notice that H1f = H2f. Indeed 21/11/15 Duong Anh Duc - Digital Image Processing 7
Matrix Formulation of Image Restoration Problem h0 0 0 0 f 0 h1 h0 0 0 f 1 H1f 0 h1 h0 0 f 2 0 0 h1 h0 0 h0 0 0 h1 f 0 h1 h 0 0 0 f 1 H 2f 0 h1 h 0 0 f 2 0 0 h1 h 0 0 Henceforth, we will use H = H2 , so that we can apply properties of circulant matrices to H. 21/11/15 Duong Anh Duc - Digital Image Processing 8
Matrix Formulation of Image Restoration Problem 2-D Case:  Suppose g, f, h, are M N arrays (after zero- padding). The degradation equation can be written in matrix-vector format as follows: g = Hf + n, where 21/11/15 Duong Anh Duc - Digital Image Processing 9
Matrix Formulation of Image Restoration Problem g 0,0 f 0,0 0,0    g 0, N 1 f 0, N 1 0, N 1 g 1,0 f 1,0 1,0    g f n g 1, N 1 f 1, N 1 1, N 1    g M 1,0 f M 1,0 M 1,0    g M 1, N 1 MN 1 f M 1, N 1 MN 1 M 1, N 1 MN 1 21/11/15 Duong Anh Duc - Digital Image Processing 10
Matrix Formulation of Image Restoration Problem H0 H M1 H M 2  H1 H1 H0 HM 1  H2 H H2 H1 H0  H3      HM 1 HM 2 HM 3  H0 MN MN  Note that H is a MN MN block-circulant matrix with M M blocks. 21/11/15 Duong Anh Duc - Digital Image Processing 11
Matrix Formulation of Image Restoration Problem  Each block Hj is itself an N N circulant matrix. Indeed, the matrix Hj is a circulant matrix formed from the j-th row of array h(m,n): h j ,0 h j, N 1 h j, N 2  h j ,1 h j ,1 h j ,0 h j, N 1  h j ,2 Hj h j ,2 h j ,1 h j ,0  h j ,3      h j, N 1 h j, N 2 h j, N 3  h j ,0 21/11/15 Duong Anh Duc - Digital Image Processing 12
Matrix Formulation of Image Restoration Problem  Given the degradation equation: g = Hf + n, our objective is to recover f from observation g.  We will assume that the array h(m,n) (usually referred to as the blurring function) and statistics of the noise (m,n) are known. The problem becomes very complicated if array h(m,n) is unknown and this case is usually referred to as blind restoration or blind deconvolution. 21/11/15 Duong Anh Duc - Digital Image Processing 13
Matrix Formulation of Image Restoration Problem  Notice that, even when there is no noise; i.e. (m,n) = 0, or the values of (m,n) were exactly known, and matrix H is invertible, computing ^f = H-1(g-n) directly would not be practical.  Example: Suppose M = N = 256. Therefore MN = 65536 and H would be a 65536 by 65536 matrix to be inverted! 21/11/15 Duong Anh Duc - Digital Image Processing 14
Matrix Formulation of Image Restoration Problem  Naturally, direct inversion of H would not be feasible.  But H has several useful properties; in particular:  H is block circulant.  H is usually sparse (has very few non-zero entries). We will exploit these properties to obtain ˆf more efficiently.  In particular, we will derive the theoretical solutions to the restoration problem using matrix algebra. However, when it comes to implementing the solution, we can resort to the Fourier domain, thanks to the properties of circulant matrices. 21/11/15 Duong Anh Duc - Digital Image Processing 15
Constrained least squares filtering (restoration)  Recall that the knowledge of blur function h(m, n) is essential to obtain a meaningful solution to the restoration problem.  Often, knowledge of h(m, n) is not perfect and subject to errors.  One way to alleviate sensitivity of the result to errors in h(m, n) is to base optimality of restoration on a measure of smoothness, such as the second derivative of the image. 21/11/15 Duong Anh Duc - Digital Image Processing 16
Constrained least squares filtering (restoration)  We will approximate the second derivative (Laplacian) by a matrix Q. Indeed, we will first formulate the constrained restoration problem and obtain its solution in terms of a general matrix Q. 21/11/15 Duong Anh Duc - Digital Image Processing 17
Constrained least squares filtering (restoration)  Later different choices of matrix Q will be considered, each giving rise to a different restoration filter.  Suppose Q is any matrix (of appropriate dimension). In constrained image restoration, we choose ^f to minimize ||Qˆf||2, subject to the constraint, ||g-Hˆf||2= ||n||2. (Recall the degradation equation g=Hˆf +n g-Hˆf = n.) 21/11/15 Duong Anh Duc - Digital Image Processing 18
Constrained least squares filtering (restoration)  Introduction of matrix Q allows considerable flexibility in the design of appropriate restoration filters (we will discuss specific choices of Q later). So our problem is formulated as follows: min ||Qˆf||2 subject to ||g-Hˆf||2= ||n||2 or ||g-Hˆf||2- ||n||2=0 21/11/15 Duong Anh Duc - Digital Image Processing 19
A brief review of matrix differentiation  Suppose x1 x x2 and f(x1,x2) is a function of two variables. Then f x1 , x2 f x1 , x2 x1 x f x1 , x2 x2 21/11/15 Duong Anh Duc - Digital Image Processing 20