DSP Methods for Blur Reduction

Inverse Filtering

The "classical" approach to digital image restoration is the class of algorithms which implement, in one way or another, inverse filtering. The most prevalent of these (and the one which we implemented) is the Wiener filter. Other types of inverse filtering include Power Spectrum Equalization and the Geometric Mean Filter.

Inverse filters in general have quite good performance, but are limited in two key areas: an inability to handle certain types of filters, including lens aberrations[1], and the need to know the response of the filter which caused the degradation.

The Wiener Filter

If the response of the degrading filter is known, the ideal correction would be to invert that response and apply it to the degraded image. However, most degrading filters will reach zero at some point, and therefore cannot be inverted. The trick is to find the range over which the filter should be inverted. This is what the Wiener filter does.

The Wiener filter operates in the Fourier domain, and both (approximately) inverts a degrading filter and removes noise. Given the original image I, the degrading filter H, and a noise constant n, it produces the corrected image J (H, I, and J are all in Fourier domain), by the following equation: 

            conj(H(u,v)) I(u,v)
G(u,v) = _________________________

              |H(u,v)|^2 + n
This provides the "optimal method for rolling off the deconvolution transfer function"[1].

Matlab implementation
Performance analysis

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References