On the effective measure of dimension in total variation minimization

Total variation (TV) is a widely used technique in many signal and image processing applications. One of the famous TV based algorithms is TV denoising that performs well with piecewise constant images. The same prior has been used also in the context of compressed sensing for recovering a signal from a small number of measurements. Recently, it has been shown that the number of measurements needed for such a recovery is proportional to the size of the edges in the sampled image and not the number of connected components in the image. In this work we show that this is not a coincidence and that the number of connected components in a piecewise constant image cannot serve alone as a measure for the complexity of the image. Our result is not limited only to images but holds also for higher dimensional signals. We believe that the results in this work provide a better insight into the TV prior.