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Digital Image Processing (2000)

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8.7 Image Restoration

The image restoration problem (also known as image deconvolution or image deblurring) is one of reconstruction or estimation of an undegraded image from a blurred and noisy one [Gon02, Bov00]. In effect, the task is to undo the degradation imposed on the original image. The methods of image restoration described here all assume that the degradation is spatially invariant and that it can be modeled by linear filters and additive noise. Additionally, we assume that the degradation is known and that some characteristics of the noise are also known or can be estimated. Following a common convention, in the remainder of this section we refer to the blurring function or filter as the point spread function (psf). Given an original image f[n1,n2], a point spread function h[n1,n2] and a noise signal w[n1,n2] we obtain the following formulation for the observed, or degraded, image g[n1,n2]:
where symbol * represents convolution (see Section 5.2). Due to the assumption of linearity the degradation process has the following Fourier domain formulation:
where G[k1,k2] , F[k1,k2] , H[k1,k2] , and W[k1,k2] are the discrete Fourier transforms of the 2D sequences g[n1,n2], f[n1,n2], h[n1,n2], and w[n1,n2], respectively. In what follows, to simplify the formulas, the spatial and frequency variables will be dropped.

Image restoration functions.

The simplest approach to restoration is to use the inverse filter. However, this approach also suffers from problems that in most cases produce unacceptable results, for example, the inverse filter may not exist or the degradation may include additive noise. The preferred approach is therefore to use methods based on least squares. The so-called Wiener filter is the classic solution to the problem of minimizing the mean squared restoration error, the difference between the original and restored images. The Wiener filter has the form
where SN and SF are the power spectra of the noise and original signal, respectively, and H* denotes the complex conjugate of H. An approach used frequently when the spectra SN and SF are not known or cannot be estimated is to approximate the filter with the following expression.
where K is a user specified constant. Using either form of the Wiener filter the restored image becomes
This loads the necessary packages.
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The following calculations demonstrate image restoration using the Wiener filter.
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This shows the original, the distorted, and two restoration results.
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As can be observed, the Wiener filter does a good job of suppressing noise, but is only marginally effective in deblurring the image. Two generalizations of the classic Wiener filter are available. The geometric mean filter uses two parameters to vary the sensitivity of the filter to noise and blurring while the constrained least-squares filter is another optimal filter that minimizes the difference between the observed and a blurred version of the restored image. The geometric mean filter has the form:
Here are two examples of geometric mean restoration.
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The next restoration examples show the constrained least-squares method. The computed value of the parameter used in constrained restoration was 0.0568.
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Options for LucyRichardsonRestoration.

The Lucy-Richardson algorithm is an iterative technique used heavily for the restoration of atronomical imagery. The algorithm effectively computes
where symbol denotes correlation, symbol denotes convolution, and is the estimate of the original image f at step k of the iteration. The iteration starts with f0=g h. First, here is a 1D example of Lucy-Richardson restoration.
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This returns 50 iterations of the accelerated algorithm.
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This shows the restored values at each of the 50 steps of the restoration process.
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Here are the results of Lucy-Richardson restoration of the example image.
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