Level Set Technique for Image Reconstruction

Abstract

Level Set (LS) technique is shown to be an ef- fective regularization technique in non-linear inverse prob- lems because of its topological based representation of un- known structures using level zero (a front) of a higher di- mension function (level set function). The representation of structures using the level set function is of great use to ad- dress the need of image reconstruction from ill-posed inverse problem containing limited set of available data and prior information. The cost functional of the classical LSRM is based on the quadratic formulations (least squares fitting or L2 norms) of data mismatch and regularization terms. How- ever, the L2 norm optimization problems are not robust to outliers and measurement noise. To achieve a high robust- ness against outliers and noise, a general inverse problem can be formulated in terms of the L1 norms or a combina- tion of the L1 norms and the L2 norms, instead of merely usage of the L2 norms. In this paper, we derive a novel LS based regularization method (LSRM) which allows any pos- sible combinations of the L1 and the L2 norms on the inverse problem terms. To show the implementation of the derived LSRM, we use an ill-posed inverse problem called Electrical Impedance Tomography (EIT). The image reconstruction re- sults of the proposed LSRM are compared to those of four state of the art regularization methods: Gauss-Newton (GN) with Tikhonov regularization term, GN with NOSER algo- rithm, Total Variation (TV), and the PDIPM. According to our results, the proposed LSRM produces more robust re- sults in the presence of high level of noise (additive 60dB Gaussian noise) and strong outliers (loss of measurement data) when compared with the competing methods.