Total Variation Blind Deconvolution: The Devil is in the Details Technical Report

First, notice that the smallest value1 of r is minx∈[−L,L−1] r[x] = r[−1] = δ0 + δ1 − UL and occurs at x = −1. Next, consider solving the taut string problem by enforcing only the constraint |s[−1] − r[−1]| ≤ λ. The cost of the taut string problem is minimum for the shortest path s through a point at x = −1. We can decompose such path into the concatenation of the shortest path from x = −L− 1 to x = −1 and the shortest path from x = −1 to x = L − 1. Given that each of these paths are only constrained at the end points, a direct solution will give a line segment between the end points, i.e.,