An illustration of de-scratching. (a) The original signal with obvious catastrophe points. (b, e, h) The residual signal in each iteration. (c, f, i) After subtracting the residual signals in (b, e, h) during each iteration, we get restored signal marked by blue dots. (d, g) The initial signal of each iteration is the restoration result from the former iteration. (j) Compared with original signal represented by red points, after 3 times iteration, our algorithm produces a smooth result without ringing effect. (k) The final results of joint de-noising with scratch restoration (blue dots) and the original signal (red points).