Digital Image Restoration by Constrained Deconvolution

The fundamental identity between the operations of vector convolution and polynomial multiplication is exploited to provide a general-purpose alternative to the method of spatial filtering for digitally deconvolving noisy, degraded images of incoherently illuminated objects. The method is remotely related to those of linear programming, but differs significantly from them in its exploitation of the special properties of convolution. Sampled image arrays are treated as points in euclidean n space. The convolution relation, together with bounds on individual recorded and point-spread image irradiance values, defines a set of linear constraints on the restored image-irradiance values. These constraints define a convex region of possible restorations in n space. A method is described for selecting a point (i.e., an estimate of the restored image) from near the center of this region. The human viewer may then readjust the original constraints to reflect the new information revealed by his interpretation of the restored-image estimate. The deconvolution calculations can then be repeated with the readjusted constraints, to yield a possibly better estimate. The method is applicable to restoration problems in which both the recorded image and the point-spread image contain noise. Furthermore, it is applicable to any problem requiring the numerical solution of a convolution equation involving measured data. The connection with Fourier-transform theory and comparison with spatial-filtering methods are touched upon briefly. A few computer restorations are shown, to illustrate the practicality and potential of the method.