Reducing the Effects of Noise in Image Reconstruction

Fourier spectral methods have proven to be powerful tools that are frequently employed in image reconstruction. However, since images can be typically viewed as piecewise smooth functions, the Gibbs phenomenon often hinders accurate reconstruction. Recently, numerical edge detection and reconstruction methods have been developed that effectively reduce the Gibbs oscillations while maintaining high resolution accuracy at the edges. While the Gibbs phenomenon is a standard obstacle for the recovery of all piecewise smooth functions, in many image reconstruction problems there is the additional impediment of random noise existing within the spectral data. This paper addresses the issue of noise in image reconstruction and its effects on the ability to locate the edges and recover the image. The resulting numerical method not only recovers piecewise smooth functions with very high accuracy, but it is also robust in the presence of noise.