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3.9 Exercises 175

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 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 charac-
teristic 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 −1 l1. If a reasonable power spectrum is assumed, is this still a problem (do we
s
still get banding or ringing artifacts)?

Ex 3.18: High-quality image resampling Implement several of the low-pass filters pre-
sented in Section 3.5.2 and also the discussion of the windowed sinc shown in Table 3.2 and
Figure 3.29. Feel free to implement other filters (Wolberg 1990; Unser 1999).
Apply your filters to continuously resize an image, both magnifying (interpolating) and
minifying (decimating) it; compare the resulting animations for several filters. Use both a
synthetic chirp image (Figure 3.65a) and natural images with lots of high-frequency detail
(Figure 3.65b-c). 27
You may find it helpful to write a simple visualization program that continuously plays the
animations for two or more filters at once and that let you “blink” between different results.
Discuss the merits and deficiencies of each filter, as well as its tradeoff between speed and
quality.

Ex 3.19: Pyramids Construct an image pyramid. The inputs should be a grayscale or color
image, a separable filter kernel, and the number of desired levels. Implement at least the
following kernels:

27 These particular images are available on the book’s Web site.