– Computer Vision
Assignment No. 3
- (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?
- [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.
- [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
- (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?
- [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.
- [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?