# Problem Set I Solution

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## Description

1. (20%) Prove the properties of convolution. For all continuous function f, g, and h, the following axioms hold:

Associativity: (f g) h = f (g h)

Distributivity: f (g + h) = f g + f h

Di erentiation rule: (f g)0 = f0 g = f g0

Convolution theorem: F(g h) = F(g)F(h), where F denotes Fourier transform

1. (25%) Frequency smoothing:

1. Compute Fourier transform of the given image lenaNoise.PNG by using fft2 function in Matlab and then center the low frequencies (fftshift).

1. Keep di erent number of low frequencies (e.g., 102; 202; 402 and up to the full dimension), but set all other high frequencies to 0.

1. Reconstruct the original image (ifft2) by using the new generated frequencies in step

(b).

Submit the code and include the restored images with di erent number of low frequencies in your report.

1. (55%) Implement gradient decent algorithm for ROF model with total variation minimiza-tion. All codes and a two-page report including problem description, your solution, and experimental results (denoised image, convergence graph, etc.) with discussions should be submitted.

NOTE that

The forward / backward di erence for computing image gradient is given in Dx.m / Dxt.m. Feel free to use it or write your own.