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. Further, for the direct method, entries less than 1% of the maximum are set to 0. For the conjugate gradient method, the solver runs for 100 iterations for each invocation.
λ is the regularization parameter; larger values give more importance to the prior and less to image fidelity. 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, 3. From personal collection.
4. Old family photograph from personal collection.
5. 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.
6, 7. From Fergus et al (2006), Removing camera shake from a single image. The recovered images from the original work (using Richardson-Lucy) are available here and here. These are substantially better than the results we obtain assuming a symmetric kernel, showing the importance of estimating the kernel well. See this demo for results of deconvolution using the kernels estimated by Fergus et al instead of our simpler method.
8, 9. Frames from the Satyajit Ray film Nayak. The original in the first frame has good focus, but is still visibly improved by denoising. The focus is slighly off in the second frame, which is considerably improved by denoising. See also the results of denoising with locally estimated kernel.
10. A frame from the Buster Keaton film The Electric House, obtained from The Internet Archive. Also used to illustrate super-resolution.