CSCI 497P/597P - Homework 1

Complete the following problems and submit your solutions to the HW1 assignment on Canvas. For all questions, justify your answers either or showing your work or giving a brief explanation. Please typeset your solutions using latex or similar¹; if you feel that typesetting your answers is a major burden, please email me I will try to help or make alternative arrangements. You may work with your classmates on these problems, but you must write up your own solutions individually without using notes or photos made during your collaborative discussions. This is largely new material, so please let me know if you think anything is unclear or ambiguous and I’ll make corrections or clarifications as needed.

1. Consider the following 3x3 image: \[ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} \]

   1. (4 points) Use this image and an identity filter to show that (for this image and filter) commutativity holds when performing convolution, but not cross-correlation.

   2. (Optional, for 2 points of extra credit) Recall that the definition of convolution is: \[ g(x,y) = \sum_{i=-k}^{k}\sum_{j=-k}^{k} w(i,j) f(x-i, y-j) \] Prove that convolution is commutative in general. It may be helpful to first modify the above definition so that all functions are defined everywhere but zero outside their bounds, and both summations run from \(-\infty\) to \(\infty\).

2. (4 points) As we’ve seen in class, a common approach to noise reduction is to blur the image using a Gaussian filter. Explain the tradeoffs involved when choosing a filter size.

3. 1. (2 points) Consider the following 1D “tent” filter, which also serves as a very rough size-3 approximation of a 1D Gaussian filter:

      \(G = \frac{1}{4} \begin{bmatrix}1 & 2 & 1\end{bmatrix}\)

      We can transpose the filter to create an equivalent 1D filter that applies a blur in the vertical direction:

      \(G^T = \frac{1}{4}\begin{bmatrix} 1\\2\\1 \end{bmatrix}\)

      Using zero-padding and full output size, compute \(H = G * G^T\).

   2. (3 points) \(H\) is an instance of a special kind of filter called a separable filter, which has the property that the result of the 2D filtering operation can be computed by doing two 1D filtering operations in series. Prove that convolving an image \(F\) with \(H\) gives the same result as convolving it with \(G\), then convolving the result with \(G^T\). Feel free to use any properties presented in class in your proof.

   3. (3 points) Does the above proof hold true for this particular filter if you use cross-correlation instead of convolution? Does it hold true for all filters?

   4. (4 points) Suppose you wish to filter a 64x64 image \(F\) with \(H\), using valid output size. The image and filter(s) are both represented using floating-point numbers. How many floating-point multiplications between pixel and filter values are required to convolve with \(H\)? How many are required to convolve with \(G\) then \(G^T\). Which is more efficient?