A Direct Comparison of SAR Processing as Non-Orthogonal Transform to both Fourier and Wavelet Transform

Abstract

The term 'SAR processing' is commonly used to denote the process of producing fully focused SAR images from SAR raw data. Beside its main task of applying a two-dimensional range-dependent convolution it simultaneously performs many other incorporated processing steps such as motion compensation, interference filtering, calibration, coregistration, geometry conversions, apodization techniques or speckle filtering. Above all, SAR processing can roughly be described as applying a two-dimensional convolution designed in order to reverse a two-dimensional convolution performed during the imaging process. This forward-backward process can be modelled as non-orthogonal transform and there are great similarities to both Fourier and Wavelet transform. Most modern fast SAR processing techniques rely on the Fourier transform as it is known to be the state-of-the-art technique for performing fast convolutions. However, in recent years, Wavelet transforms outperform the Fourier transform in terms of computational burden. It only requires O(n) or even O(log n) operations instead of O(n log n) operations. In a previous work, we derived the matrix form of SAR raw and image data from the continuous case and defined the SAR transform which maps the two matrices onto each other. We concluded that the SAR transform is a Wavelet transform. In this paper, we would like to do a direct comparison of Fourier, Wavelet and SAR transform. We analyse the way Fourier and Wavelet transform gain computational speed and show a perspective how this can be done for SAR processing.