This problem is about graph signal inpainting via total variation regularization. Signal in- painting is the process of estimating missing signal values from available noisy measurements. Assume that the observed graph signal is y = [ y T1 y T2 ] T , where y 1 ? R M × 1 is known and y 2 = 0 ? R ( N ? M ) × 1 is missing or unknown. Now two cases are possible: the known part of the signal y 1 may be noisy or noiseless.

(a)





(b)


Hint: Use the method of Lagrange multipliers.


Note: The problem in the book has a small error. Above is the corrected version.

(a) Weight matrix–based framework














(b) Weight matrix–based framework









To satisfy the constraint, we can put




f1=y1 in the above expression and then take derivative




with respect to f2