– Computer Vision

Assignment  No. 3

1. (a) [6 points] Let f(x,y) be an image. Let h(x,y) be the image obtained by applying the a 3 by 3 spatial low pass mask (averaging filter) to f(x,y).  Similarly let g(x,y) be the image obtained by applying a 3 by 3 spatial high pass mask to f(x,y).  Prove that

g(x,y) =  f(x,y) – h(x,y)

Note: one or more examples do not constitute a proof.

(b) [5 points] Is the high pass mask separable? What is the implication of separability on computations?

2. [4+5 points] Suppose that the image gray level values under amask are

3     2  1

7     8     4

3     6     5

Determine the value of the corresponding pixel in the output image for:

(a)  Median,   (b) Harmonic mean.

In each case comment on the suitability of the filter for reducing Gaussian noise, and provide reasoning for your comments.

3. [8 points] Find the output images if Sobel edge operators are applied to the following 8 by 8 input image.  Note that you will have three gradient images, one in x-direction, one in y-direction and one gradient magnitude. Ignore the border effects, and produce only 6 by 6 output images.

2   2   2   2   2   2   2   2

           2   2   2   2   2   2   2   7

           2   2   2   2   2   2   7   7

           2   2   2   2   2   7   7   7

           2   2   2   2   7   7   7   7

           2   2   2   7   7   7   7   7

          2   2   7   7   7   7   7   7

           2   7   7   7   7   7   7   7

4. (a) [6 points] Use the definitions of the derivatives as , and similarly for to obtain the Laplacian mask.  Why is Laplacian is rarely used alone.

(b) [4 points] What will be the Laplacian mask if the derivative is defined as , and similarly for ?  Why is this definition is not suitable for obtaining the Laplacian?

5. [7 points] Compute the Fourier transform of the one-dimensional image f(0) = 8,  f(1) = 4,     f(2) = 2, f(3) = 1.  Find Fourier spectrum |F(u)|.  Comment on your results.

6. [4+4+4+4 points] Answer the following questions and support your answers with reasoning and analysis

• Why is it necessary to move the origin of the Fourier transformed image to the

center (i.e. to u = n/2, v = n/2).  How is this shifting implemented?

(b) Why is bit reversal needed in FFT? Explain.

(c) The Fourier spectrum |F(u,v)| of an image f(x,y) is known, but f(x,y) is not known.

Can f(x,y) be computed? Explain.

(d) Prove that the two-dimensional Fourier transform of an image f(x,y) can be

achieved using two one-dimensional transforms.  What is the significance of this?