## Description

1. Consider a 1D convolution mask given as (w0, w1, …, w6). Express the convolution of the mask with a 1D

image f as the multiplication of a suitable matrix with the image vector f. What are the properties of this

matrix? What could be a potential application of such a matrix-based construction? [10 points]

2. In bicubic interpolation, the image intensity value is expressed in the form v(x, y) = P3

i=0

P3

j=0 aijx

iy

j

where aij are the coefficients of interpolation and (x, y) are spatial coordinates. This uses sixteen nearest

neighbors of a point (x, y). Given the intensity values of these 16 neighbors, explain with the help of matrixbased equations, how one can determine the coefficients aij that determine the function v(x, y)? Why do

you require 16 neighbors for determining the coefficients? [10 points]

3. Consider a clean image I(x, y) which gets corrupted by additive noise randomly and independently from

a zero mean Gaussian distribution with standard deviation σ. Derive an expression for the PDF of the

resulting noisy image. Assume continuous-valued intensities. [10 points]

4. Prove or disprove: (a) The Laplacian mask with a −4 in the center (see class slides) is a separable filter.

(b) The Laplacian mask with a −4 in the center (see class slides) can be implemented entirely using 1D

convolutions. [5+5=10 points]

5. Suppose I convolve an image f with a mean-filter of size (2a + 1) × (2a + 1) where a > 0 is an integer to

produce a result f1. Suppose I convolve the resultant image f1 with the same mean filter once again to

produce an image f2, and so on until you get image fK in the Kth iteration. Can you express fK as a

convolution of f with some kernel. If not, why not? If yes, with what kernel? Justify. [10 points]

6. Consider a 1D ramp image of the form I(x) = cx + d where c, d are scalar coefficients. Derive an expression

for the image J which results when I is filtered by a zero-mean Gaussian with standard deviation σ.

Derive

an expression for the image that results when I is treated with a bilateral filter of parameters σs, σr. (Hint:

in both cases, you get back the same image.) Ignore any border issues, i.e. assume the image had infinite

extent. [10 points]

7. Prove that the Laplacian operator is rotationally invariant. For this consider a rotation of the coordinate

system from (x, y) to u = x cos θ − y sin θ, v = x sin θ + y cos θ, and show that fxx + fyy = fuu + fvv for any

image f. [10 points]

8. Consider the two images in the homework folder ‘barbara256.png’ and ‘kodak24.png’. Add zero-mean Gaussian noise with standard deviation σ = 5 to both of them. Implement a bilateral filter and show the outputs

of the bilateral filter on both images for the following parameter configurations: (σs = 2, σr = 2); (σs =

0.1, σr = 0.1); (σs = 3, σr = 15). Comment on your results in your report. Repeat when the image is

corrupted with zero-mean Gaussian noise of σ = 10 (with the same bilaterial filter parameters).

Comment on your results in your report. For the bilateral filter implementation, write a MATLAB function

mybilateralfilter.m which takes as input an image and parameters σr, σs. Implement your filter using

at the most two nested for-loops for traversing the image indices.

For creating the filter, use functions like

meshgrid and vectorization for more efficient implementation. Include all image ouputs as well as noisy

images in the report. [15 points]

9. Implement local histogram equalization of sizes 7 × 7, 31 × 31, 51 × 51, 71 × 71 on the images ‘LC1.jpg’

and ‘LC2.jpg’ from the homework folder. Comment on your results in your report and compare it to global

histogram equalization, which you can use from the image processing toolbox of MATLAB. Point out regions

where the local method produces better local contrast than the global histogram equalization. [15 points]