A deblurring filter is designed adaptively to reduce adverse effects of noise.

An algorithm for adaptive deblurring of images has been designed to be less adversely affected by image noise than are prior deblurring algorithms. The need for this or another noise-tolerant deblurring algorithm arises as follows: For a typical imaging instrument, in which the blurring function (also known as the point-spread function) approximates a Gaussian function in the spatial-frequency domain, a simplistic spatial-frequency-domain deblurring function equal to the inverse of the blurring function magnifies the noise at high spatial frequencies. In the present adaptive deblurring algorithm, the spatial-frequency-domain deblurring function is the product of (1) the inverse of spatial-frequency-domain blurring function and (2) a smoothing or low-pass filter function denoted variously as a power window or a P-deblurring filter function (wherein "P" signifies "power"). The term "adaptive" in the name of the algorithm characterizes the process for choosing the parameters of the power window.

Figure 1. This Block Diagram and the spatial-frequency spectral plots represent selected aspects of the image-acquisition process (including blurring and addition of noise) and the deblurring process.
Figure 1 schematically depicts the image-acquisition and deblurring processes as modelled by the following equations. The observed or measured blurred image r(x,y) is given by


where x and y are Cartesian position coordinates in the image plane, f(x,y) is the non-blurred input image that one seeks to recover, g(x,y) is the blurring function in position space, ⊗ is the convolution operator, and n(x,y) is the additive image noise. In the spatial-frequency domain, the observed image is given by


where kx and ky are spatial frequencies corresponding to the Cartesian coordinates and R(kx,ky), F(kx,ky), G(kx,ky), and N(kx,ky) are the Fourier transforms of r(x,y), f(x,y), g(x,y), and n(x,y), respectively. For the purpose of the algorithm, it is assumed that, as in most practical imaging systems, the blurring and noise functions are radially symmetric, so that their Fourier transforms can be expressed as G(ρ) and N(ρ), respectively, where ρ≡(kx2 + ky2)(1⁄2). It is further assumed that, as in most practical imaging systems, the noise is white in that its power spectral density, ⏐N(ρ)⏐2 can be considered approximately constant over the entire spatial-frequency range of interest.