User Manual - pancroma

IMPORTANT NOTE: In reality, the convolution process may not be easilyreversible. Inverting convolution operations has many potential issues• The inversion process is usually very sensitive to small perturbations.• The inversion process is always corrupted by rounding errors (noise).• Instead of the desired solution we often see the amplified noise only.• If the transformed convolution matrix is close to singular, we can easilylose important information.In addition, the FFT does not necessarily transform an image from the spatialdomain to the frequency domain with complete fidelity. In particular, certainimage features, combined with certain convolution kernel characteristics cancause the reverse FFT of the product array to "ring". If one attempts to reversethe convolution process (deconvolution), the convoluted array does notcorrespond to the input array, and the deconvoluted array will be corrupted. Ifsigma for example is too large, the deconvolution may "blow up" into anincomprehensible noisy mess. However, if sigma is not too large, the inversiongenerally works for most areas of the image. Image discontinuities likeshorelines for example may still exhibit ringing effects.Taking the deconvolution idea a step further, what happens if we attempt todeconvolve an image that has not been blurred (by us)? Images can be blurryfor a variety of reasons, including lens distortion, CCD sensor distortion,atmospheric turbulence, and many other reasons. If we knew the nature of thedistortion we might recover the image using the same technique describedabove.The distortion is described by the Point Spread Function (PSF), which as itsname suggests describes how a point source of light actually ends up gettingdisplayed in our image. Rather than a point mapping to a single pixel, it isactually spread among many pixels. Deconvolving with the Point SpreadFunction can recover the “true” image.292