Super-resolution Fourier transforms by optimisation, and ISAR imaging

5A method for super-resolution ISAR imaging, denoising, and error estimation is developed using a novel Fourier transform that exploits the a priori information that the image is sparse, i.e. contains relatively few bright points. The method applies nonlinear optimisation to the complex-valued pixels to estimate the image by minimising its l¹-norm. Noiseless images require linear programming, while quadratic programming with logarithmic barriers is necessary when complex-valued Gaussian noise is present. The novel Fourier transform, which is referred to as the l¹-FFT, works with ´missing´ data points, making ´jackknife´ estimates of the mean and variance of each pixel value possible. These estimates should aid in image classification. This work extends earlier work of Chen, Donoho and Saunders on basis pursuit and denoising to complex signals, by formulating and solving the corresponding complex-valued nonlinear optimisation problems