Convolution Model of Image Formation Sample Clauses

Convolution Model of Image Formation. ‌ Each pixel in a blurred image can be represented as a weighted average of pixels in the true image scene. The point spread function (PSF) defines these weights. The PSF can sometimes be obtained by calibration of the optical instrument, or expressed by a mathematical formula. If the PSF is assumed to be spatially invariant (as is often the case), then mathematically this image formation process is known as convolution, and can be written as Yk,l Hi,j Xk i,x x Nk,l , (1.1) i,j where Yk,l is a pixel of the observed image at xxx xx, xx xxxxxxxx, Xx,x is the pi, jq entry of the PSF, Xk i,x x is a pixel of the exact original image at the pk i, l jq position, and Nk,l is additive noise. The additive noise may come from a combination of background noise, electronic sensor noise, modeling errors, measurement errors, etc. Since the focus of this disserta- tion is deblurring rather than denoising, without loss of generality, we can usually assume Nk,L is zero when describing much of the mathematical and computational approaches proposed in this thesis. However, as we discuss later in this thesis, regularization must be incorporated into the algorithms to provide stability in the presence of noise. Estimating the original image from the blurred image is called the de- blurring problem. We can further classify deblurring problems into two sub- classes: when the PSF is known, the deblurring problem is called decon- volution, and when the PSF is unknown, it is called blind deconvolution. We consider the deconvolution problem in Chapters 2 and 3, and the blind deconvolution problem in Chapter 4.