Chapter 3

Discrete Gaussian filters Discuss the following issues with implementing a discrete Gaussian filter: \- If you just sample the cquation of a continuous Gaussian filter at discrete locations, will you get the desired properties, e.g., will the coefficients sum up to 0? Similarly, if you sample a derivative of a Gaussian, do the samples sum up to 0 or have vanishing higher-order moments? \- Would it be preferable to take the original signal, interpolate it with a sinc, blur with a continuous Gaussian, then pre-filter with a sinc before re- sampling? Is there a simpler way to do this in the frequency domain? \- Would it make more sense to produce a Gaussian frequency response in the Fourier domain and to then take an inverse FFT to obtain a discrete filter? \- How does truncation of the filter change its frequency response? Does it introduce any additional artifacts? \- Are the resulting two-dimensional filters as rotationally invariant as their continuous analogs? Is there some way to improwe this? In fact, can any $2\mathrm{D}$ discrete (separable or non-separable) filter be truly rotationally invariant?

Steerable filters Implement Freeman and Adelson's (1991) stecrable filter algorithm. The input should be a grayscale or color image and the output should be a multi-banded image consisting of $G_1^{0^{\circ }}$ and $G_1^{{90}^{\circ }}$. The coefficients for the filters can be found in the paper by Freeman and Adelson (1991). Test the various order filters on a number of images of your choice and see if you can reliably find corner and intersection features. These filters will be quite useful later to detect elongated structures, such as lines (Section 4.3).

Wiener filtering Estimate the frequency spectrum of your personal photo collection and use it to perform Wiener filtering on a few images with varying degrees of noise. 1\. Collect a few hundred of your images by re-scaling them to fit within a $512\times 512$ window and cropping them. 2\. Take their Fourier transforms, throw away the phase information, and average together all of the spectra. 3\. Pick two of your favorite images and add varying amounts of Gaussian noise, ${\sigma }_{\text{ri }}\in$ $1,2,5,10,20$ gray levels. 4\. For each combination of image and noise, determine by cye which width of a Gaussian blurring filter ${\sigma }_s$ gives the best denoised result. You will have to make a subjective decision between sharpness and noise. 5\. Compute the Wiener filtered version of all the noised images and compare them against your hand-tuned Gaussian-smoothed images. 6\. (Optional) Do your image spectra have a lot of energy concentrated along the horizontal and vertical axes $(f_x=0$ and $f_y=0)$ ? Can you think of an explanation for this? Does rotating your image samples by ${45}^{\circ }$ move this energy to the diagonals? If not, could it be due to edge effects in the Fourier transform? Can you suggest some techniques for reducing such effects? Ex 3.17: Deblurring using Wiener filtering Use Wiener filtering to deblur some images. 1\. Modify the Wiener filter derivation $(3.66-3.74)$ to incorporate blur $(3.75)$. 2\. Discuss the resulting Wiener filter in terms of its noise suppression and frequency boosting characteristics. 3\. Assuming that the blur kernel is Gaussian and the image spectrum follows an inverse frequency law, compute the frequency response of the Wiener filter, and compare it to the unsharp mask. 4\. Synthetically blur two of your sample images with Gaussian blur kernels of different radii, add noise, and then perform Wiener filtering. 5\. Repeat the above experiment with a "pillbox" (disc) blurring kernel, which is characteristic of a finite aperture lens (Section 2.2.3). Compare these results to Gaussian blur kernels (be sure to inspect your frequency plots). 6\. It has been suggested that regular apertures are anathema to de-blurring because they introduce zeros in the sensed frequency spectrum (Veeraraghavan, Raskar, Agrawal et al. 2007). Show that this is indeed an issue if no prior model is assumed for the signal, i.e., $P_{\ast }^{-1}l1$. If a reasonable power spectrum is assumed, is this still a problem (do we still get banding or ringing artifacts)?