These results illustrate the process of denoising on several real-life (slightly) blurred images. The process is more correctly viewed as deconvolution, as the blur kernel plays an important role in the quality of the recovered image. In all these examples, the kernel is estimated from the input image using a Fourier domain computation which assumes that image gradients are correlated (according to the AR model described below). In some cases, the kernel is estimated from a cropped subregion of the image (see code for precise details). This method assumes that the kernel is symmetric, which is often not true for blur due to camera shake, and is thus of limited use.
The estimated kernels are cropped to a size of 11 x 11 pixels, and entries less than 1% of the maximum are set to 0.
λ is the regularization parameter; larger values give more importance to the prior and less to image fidelity. The sparse prior refers to the hyper-Laplacian distribution for image gradients with parameter α = 0.8. The AR priors assume that the image gradients are correlated (according to a simple independent 2-D auto-regressive model with correlation parameters 0.3 and 0.6). The IID priors assume that the gradients are independent.
For most of these examples, the AR Sparse prior with λ = 0.001 gives the best visual result, especially compared to the IID Sparse prior which tends to oversmooth. The Gaussian prior with λ = 0.01 also gives reasonable results, depending on the image, and has the advantage of being a lot faster. The even faster Richardson-Lucy method also performs quite well.
Image details
1, 2. From personal collection.
3. From mathematician and amateur astronomer Sunil Chebolu. Due to considerable variation in distance of different parts of the moon from the camera, the blur is not the same in different parts of the input image. The kernel used here is estimated from a cropped subregion. See also the results of denoising with locally estimated kernel.
4. A frame from the Buster Keaton film The Electric House, obtained from The Internet Archive. Also used to illustrate super-resolution.