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. | 5/14/2020 4:35:04 AM Duong Anh Duc - Digital Image Processing Digital Image Processing Image Restoration Matrix Formulation 5/14/2020 4:35:04 AM Duong Anh Duc - Digital Image Processing 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 5/14/2020 4:35:04 AM Duong Anh Duc - Digital Image Processing Matrix Formulation of Image Restoration Problem 5/14/2020 4:35:04 AM Duong Anh Duc - Digital Image Processing Matrix Formulation of Image Restoration Problem However, since the arrays f and h are zero-padded, we can equivalently set: Notice that the (second) matrix H is circulant; ., each row of H is a circular shift of | 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Digital Image Processing Image Restoration Matrix Formulation 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing 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 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Matrix Formulation of Image Restoration Problem 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Matrix Formulation of Image Restoration Problem However, since the arrays f and h are zero-padded, we can equivalently set: Notice that the (second) matrix H is circulant; ., each row of H is a circular shift of the previous row. 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Matrix Formulation of Image Restoration Problem Example: A = length of array f = 3 B = length of array h = 2 M ³ A + B – 1 = 4, say M = 4. 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Matrix Formulation of Image Restoration Problem 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Matrix Formulation of Image Restoration Problem Notice that H1f = H2f. Indeed 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Matrix Formulation of Image Restoration Problem Henceforth, we will use H = H2 , so that we can apply properties of circulant matrices to H. 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing 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 5/14/2020 5:31:15 AM Duong Anh Duc
