The removal of unknown image blurs by homomorphic filtering

This report describes a homomorphic method for the estimation and removal of unknown image blurs, which are presumed to have been caused by a linear stationary system. Previous methods have required a more or less perfect a priori knowledge of the identity of the blur. the method described here requires no information about the original scene which led to the blurred image, but does presently require that the blur be phaseless for best results. The estimation method uses a homomorphic mapping to map sections of the blurred image, which is a convolutional combination of the undistorted of the undistorted image and the blur, into a domain (the log spectrum) in which the components corresponding to the undistorted image and blur are added. An averaging method allows the component corresponding to the undistorted image to be estimated and subtracted, leaving an estimate of the component corresponding to the blur. This estimate is then used to generate a homomorphically estimated restoration filter. Properties of the homomorphically estimated restoration filter in the presence of additive noise are derived, and it is shown that the homomorphic filter has noise and resolution properties intermediate between those of the Weiner filter and those of the simple inverse filter. The concept of a family of filters with variable noise and resolution properties is discussed. This family is bounded by the simple inverse and Weiner filters and includes the homomorphic filter. It is shown that any member of the family of filters can be generated from the homomorphic filter and an estimate of the noise level. A homomorphic method for the treatment of multiplicative noise is presented. Finally, the limitations of the method with respect to phase are discussed and some results using real-world blurred images are presented.