1) Discussions about KLT. (3/10 points)
In Experimental Assignment #2, we are learning eignen faces for K=80 images of N=25x30=750pixels. Observe how many eigen faces you need accounting for 50%, 90% and 100% of the covariance. What is the maximum number of eigenfaces we can get no matter how the K faces look like?
If you were to design the eigenface experiment end-to-end, i.e. experimental setup, data capture, pre-processing, learning and face reconstruction, what would you do differently to learn eigen faces more reliably? What if the system is meant for gender- and age- discrimination using faces? A brief summary of the idea should suffice.
2) G&W book problem 5.19 (2/10 points)
Write or sketch the filter causing the rotational blur in an appropriate coordinate space, explain how you would "undo" the rotational motion blur of a space craft as described in the problem.
3) Inverse Filtering and Pseudo-inverse Filtering. (5/10 points)
a) (1pt) An observed image is affected by convolution with a Point Spread Function h(x, y), plus random additive noise n(x, y), so that g(x, y) = f(x, y) ∗ h(x, y) + n(x, y). Describe how the image is restored using the Inverse Filtering method. Give a mathematical expression for the noise n'(x,y) in the restored image f'(x,y). i.e. f'(x,y) = f(x,y) + n'(x,y).
The distorted image g(x, y) shown below is affected by constant speed motion plus noise. Hence h(x, y) is a square impulse function in x. Below right is shown the transfer function H(u, v) which is the Fourier transform of h(x, y), also shown is a slice through this transfer function the first local minimum of this function is -0.217 and the first local maximum has value 0.128).
To restore the image the Pseudo-Inverse method is used with two different values of the minimum amplitude of H, e= 0.25 ande= 0.15. Copy theH(u, 0) slice and draw on the restoring filter slices R(u, 0) for both values of e.
b) (1 pts) After applying the Pseudo-Inverse filter for e=0.25 the restored image equals the true image convolved by a residual PSF plus noise. Give a drawing of the residual PSF.
d) (1 pts) After applying the Pseudo-Inverse filter for e=0.15 what is the shape of the residual transfer function? Give a drawing of the residual PSF with which the restored image is convolved.
e) (2 pts) Compare the redidual random noise in the three restorations in (a-c) above, rank them in terms of total noise power and justify your ranking.
Prepared by Lexing Xie < xlx at ee dot columbia dot edu >, 2007-03-18