Points on zero-level set of bandlimited function

Outline

Aim

We consider the model where points lie on the zero-level set curve of a band-limited function. We study the problems of:

• Recovery of the curve from few points
• Recovering these points from corrupted or undersampled measurements

Method

We define a feature matrix where each column is a high dimensional mapping of a particular point. We show that the null-space vectors of this matrix can be used to recover the curve uniquely, considering that we have a sufficient number of points. The rank-deficient nature of the feature matrix can also be exploited to solve inverse problems. We propose an optimization problem using the nuclear norm of the feature matrix as a regularizer. This problem can be solved efficiently using an IRLS formulation.

Result

The proposed scheme can successfully recover the curve, given sufficient number of points, as predicted by our theory. The proposed optimization problem can recover the points lying on such curves from corrupted measurements.

References

• Recovery of Noisy points on Bandlimited Surfaces: Kernel Methods Re-explained

S. Poddar, M. Jacob. ICASSP 2018 (Accepted)