Blind Deconvolution Sample Clauses

Blind Deconvolution. Another example of a separable nonlinear least squares problem (3.1) arises in image deblurring, when the blurring operator is not known exactly. This problem is often referred to as blind deconvolution in the image processing literature [18, 71, 76, 120, 130]. It is assumed that the observed image b is data measured by an imaging device (such as a camera, telescope, micro- scope, or medical imaging scanner), and A(y) is an operator that models how the image is captured. If the true parameters y true were available, the problem reduces to the linear image deblurring problem discussed in Chap- ter 2. However, realistically the vector y is obtained through a calibration process, for example, by collecting images of known objects. Thus, it is only an approximation of the true parameters y true. Matrix A(y) models the blurring operation and can be written as A(y) = A(P(y)) , where P(y) is a PSF. In many applications the blur is assumed to be spa- tially invariant, which means P(y) is an image of a point source object and A(P(y)) is structured. The precise structure depends on the imposed bound- ary conditions, but it is usually a combination of Toeplitz and Xxxxxx ma- trices; see [69] for more details. 1

Blind Deconvolution. The Jacobian of the reduced cost functional for blind deconvolution can also be computed using the chain rule in the following way: ∂ [ A( P(y) ) x ] Jϕ = = ∂y ∂ [ A( P(y) ) x ] ∂P · ∂ [ P(y) ] ∂ [ P(y) ] ∂y = A(X) · , (3.18) ∂y where x = vec(X), i.e. x is the vector obtained by stacking columns of matrix ∂y