+ w Observations Operator Unknown signal Noise Several problems Inpaiting Super-resolution Regularization x? 2 argmin x 2RN 1 2 || y x ||2 2 + J ( x ) Tuesday, November 8, 11
+ w Observations Operator Unknown signal Noise x? 2 argmin x = y J ( x ) Noiseless 0 Several problems Inpaiting Super-resolution Regularization x? 2 argmin x 2RN 1 2 || y x ||2 2 + J ( x ) Tuesday, November 8, 11
of natural images in frequency-space domain J(x) = min ↵ || ↵ ||1 subject to x = D↵ Synthesis Sparsity of Natural Images frequency-space domain Orthogonal Wavelets Tuesday, November 8, 11
6= 0 D x ↵ = D⇤ x ↵ Synthesis argmin ↵2RQ 1 2 ||y ↵||2 2 + ||↵||1 = D x = D↵ Analysis argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
x ? in D Sparse Regularizations = 6= 0 D x ↵ = D⇤ x ↵ Synthesis argmin ↵2RQ 1 2 ||y ↵||2 2 + ||↵||1 = D x = D↵ Analysis argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 Tuesday, November 8, 11
Solutions x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 P depends on sign(D ⇤ x ? (y)) and AP is a linear operator If [y, ¯ y] ⇢ P x ?(¯ y ) = AP x ?( y ) u y ¯ y P Tuesday, November 8, 11
? piecewise a ne Scaling Behaviour of Solutions x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 P depends on sign(D ⇤ x ? (y)) and AP is a linear operator If [y, ¯ y] ⇢ P x ?(¯ y ) = AP x ?( y ) u y ¯ y P Tuesday, November 8, 11
? piecewise a ne Scaling D 7! x ? open question Dictionary Behaviour of Solutions x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 P depends on sign(D ⇤ x ? (y)) and AP is a linear operator If [y, ¯ y] ⇢ P x ?(¯ y ) = AP x ?( y ) u y ¯ y P Tuesday, November 8, 11
x0 ||2 = O(||w||2) supp( D ⇤ x ?) ✓ supp( D ⇤ x0) then, for every x0 such that supp(D ⇤ x0) = I, If RC(I) < 1 and > ||w||2CI, RC ( I ) : function of the support Robustness of Estimators D ⇤ x0 D ⇤ x ? x?( , , D ) = argmin x 2RN 1 2 || y x ||2 2 + || D ⇤ x ||1 y = x0 + w Tuesday, November 8, 11