Written Assignment 2

Due in class on October 3, 9:30 AM.

Late assignments will be accepted by the TAs until the answer key is posted.

1. Convolution Boundaries

Convolve the following 1D signal:

  0 0 0 6 6 6 0 0 0 4 4 4 0 0 0 2 2 2 0 0 0 4 4 4

with the kernel:

  1/4 [1 2 1]

in the following 2 ways:

1. using zero padding
2. using kernel renormalization

2. Seperable and Squareable Kernels

Some 2D filter kernels have a special property that they are separable. This means that they can be divided up into two 1D convolutions (one in each direction). Gaussians and binomials have this property.

As it turns out, using the separated form is preferable (when it is possible) because it is much more efficient. Not only are there fewer operations, but the memory accesses are all in a "row" (often, it is best to implement things by grabbing a column out of the image into a row, computing the convolution, and then putting the row back into the column.

1. Compute the width 5 1D binomial filter. Then compute the 5x5 binomial filter, both by repeatedly "squaring" the unit 2D box, and by applying the 1D kernel in both directions.
2. This property of squaring a filter to get a "bigger" kernel, that effectively has a lower pass band, does not work with ideal low-pass filters (filters with gain 1 inside their pass band, and zero outside). Why?
3. Give an explanation of why no non-zero filter with a finite kernel size (in the time domain) can be a true low-pass filter. (NOTE: there is a confusing double negative/existential quantifier in that sentence - a simpler way to say it: a finite kernel cannot be a true LPF) HINT: consider what 2B says about the "square" of the kernel, and consider the edges of the kernel when it is convolved.

3. Downsampling Filters

When we downsample an image, we must be careful to do proper filtering to avoid aliasing. Too much, or too little filtering, and you'll get a bad result.

Consider reducing an image by a factor of 3.

1. Describe/sketch an "image" (a discrete 1D signal) that would be good for testing whether or not the sampling is done correctly. Describe/sketch the results if the frequency limit was too high or too low.
2. Give an example of a 1D binomial kernel that has too low a frequency cutoff and too high a frequency cutoff, and show what each one does.

4. Reconstruction kernels

Consider the following sampled signal (you can assume it has zeros before and after):

  0 0 5 5 4 4 3 2 1 5 1 0 

Sketch the continuous reconstruction of this signal with:

• A unit box reconstruction kernel of "radius" 1/2 (p 89 of the book)
• A tent reconstruction kernel of "radius" 1 (p 89 of the book)
• A tent reconstruction kernel of "radius" 3/2 (p 89 of the book)

5. Reconstruction kernels

Consider the following sampled signal (again, assume it has zeros before and after):

  0 1 1 1 0 0 1 1 0 1 0

And resampling this to double the number of samples.

What would the resulting sampled signal (this will be the 11 original samples and the 10 samples "in between") be if:

• you use the "interpolating tent" filter (r=1)?
• you use the interpolating catmul-rom filter (p 91 of the book)

Why might you prefer one filter over the other?