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3.9 Exercises 173

5. Interpolate adjacent CDFs for final lookup.

Ex 3.8: Padding for neighborhood operations Write down the formulas for computing
˜
the padded pixel values f(i, j) as a function of the original pixel values f(k, l) and the image
width and height (M, N) for each of the padding modes shown in Figure 3.13. For example,
for replication (clamping),

k = max(0, min(M − 1,i)),
˜
f(i, j)= f(k, l),
l = max(0, min(N − 1,j)),
(Hint: you may want to use the min, max, mod, and absolute value operators in addition to
the regular arithmetic operators.)
• Describe in more detail the advantages and disadvantages of these various modes.

• (Optional) Check what your graphics card does by drawing a texture-mapped rectangle
where the texture coordinates lie beyond the [0.0, 1.0] range and using different texture
clamping modes.

Ex 3.9: Separable filters Implement convolution with a separable kernel. The input should
be a grayscale or color image along with the horizontal and vertical kernels. Make sure
you support the padding mechanisms developed in the previous exercise. You will need this
functionality for some of the later exercises. If you already have access to separable filtering
in an image processing package you are using (such as IPL), skip this exercise.
• (Optional) Use Pietro Perona’s (1995) technique to approximate convolution as a sum
of a number of separable kernels. Let the user specify the number of kernels and report
back some sensible metric of the approximation fidelity.

Ex 3.10: Discrete Gaussian filters Discuss the following issues with implementing a dis-
crete Gaussian filter:

• If you just sample the equation 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 improve this? In fact, can any 2D discrete (separable or
non-separable) filter be truly rotationally invariant?