A new method for D-dimensional exact deconvolution

Deconvolution is an important problem of signal processing, and conventional approaches, including Fourier methods, have stability problems due to the zeros of the convolution kernel. We present a new method of multidimensional exact deconvolution. This method is always stable, even when the convolution kernel h(n) has zeros on the unit circle, and there exist closed-form solutions for the one-dimensional (1-D) case (D=1). For the multidimensional case (D>1), the proposed method yields stable solutions when det(h)=D. This solution set covers a portion of all possible convolution kernels, including the ones that have zeros on the multidimensional unit circle. This novel time-domain method is based on the fact that the convolution inverse of a first-order kernel can be found exactly in multidimensional space. Convolution inverses for higher order kernels are obtained using this fact and the zeros of the convolution kernel. The presented method is exact, stable, and computationally efficient. Several examples are given in order to show the performance of this method in 1-D and multidimensional cases.