ComForWaRD: Complex Fourier-Wavelet Regularized Deconvolution in Computerized Tomography

The inversion of the Radon transform in the presence of noise is numerically unstable in tomographic image reconstruction and is said to be ill conditioned. We propose an efficient hybrid Complex Fourier-Wavelet Regularization (ComForWaRD) that comprises blurring operator inversion followed by noise suppression via scalar shrinkage in both Fourier and Wavelet domains. The Fourier shrinkage exploits the structure of the noise inherent in deconvolution, while the Wavelet shrinkage exploits the piecewise smooth structure of real world signals and images. Moreover the Complex Discrete Wavelet Transform (CWT) has an additional properties of nearly shift invariant and good directional selective. The CWT is particularly suitable for representation of images and other multi-dimensional signals. The proposed hybrid regularization yields state of the art mean squared error (MSE) performance in practice. Further for certain problems the hybrid algorithm guarantees an optimal rate of MSE decay with increasing resolution. All the algorithms are implemented in Matlab.