Quarktet Blind Deconvolution

From home photographers to world-class astronomers, anyone who has taken a photograph, has taken a blurry picture, including NASA. At these times, it would be nice if there was a process that could magically remove the blur from the photograph. In certain situations, there is a method., which is called "Deconvolution".

One way to grasp this idea is to think of an image of a star. With the star being so distant from our viewing point, the star would only affect a single point on the image (barring atmospheric effects). The star is an example of what we call a "point source". Now if your camera is not in focus, the light from that star will not be focused to a single point, but will instead fall on a collection of points surrounding the intended point. This distribution of a point source in an image is called a "point spreaed function" or PSF. If the image is of an extended scene, a house for example, each point in the image has the spread of the PSF associated with it.

All images have a characteristic PSF. If the PSF is large enough, the image appears blurry. Common sources of blur include out-of-focus, object motion, camera motion, atmospheric effects, and optical defects. The physical term for the influence of the PSF on the image is called a "convolution". To perform a "deconvolution," you must have some method of finding the PSF. There are three basic approaches for doing so:

The PSF can be calculated by accurately knowing the optical system and the cause of the blur. There exists software packages that perform this operation. However, one must have exact knowledge of the optics and their positions, and the actual optics must closely match the specifications.
The PSF can be measured by imaging a point source. The point source must be small enough compared to the camera system's resolution to act as a point source. The frequency (color) of the point source should also closely match light created or reflected by the object.
The PSF can be estimated using information about either the object, optical system, or educated guesses. Since this is an indirect approach, this method is what we typically call "Blind Deconvolution".

Researchers have been studying blind deconvolution methods for several decades, and have approached the problem from different directions. Here, we summarize the more common methods.

APEX and BEAK---Alfred Carasso developed two non-iterative approaches that work well for Gaussian and Lorentzian-distributed PSFs. By choosing a specific type of blur function, the authors narrow the scope in which the blind deconvolution can be applied. The algorithm fits parameters to match the logarithm of the modulation transfer function with a modeled PSF. These methods, called APEX and BEAK, are not expected to be effective for out-of-focus, and motion blurs.
Caron Filter---This filter is an approximation of the SeDDaRA process. The truth image is approximated by a user-defined constant. A little bit of effort is needed to choose the right constant. Once chosen, this process operates faster than the SeDDaRA process. However, the SeDDaRA process generally produces a better result.

Cepstrum Transform---This filter is similar in approach to the Caron Filter. A frequency function, derived from a Cepstrum transform, is imposed on the blurry image to produce a PSF, which is then used to clean up the blurred image. This approach is more restrictive than the Caron filter, is about as fast, but will operate successfully on a smaller variety of images.

Maximum Entropy Method---This appears to be the most popular form of deconvolution. It stems from research in information theory. MEM functions by minimizing a smoothness function, called "entropy" in an image. Adding constraints such as positivity improves the chances that this iterative approach converges to a solution.
Nearest Neighbor Technique---Nearest neighbor is most closely associated with microscopy whereby a series of images can be created using different focus positions. The middle position is at the optimum focus position whereas the other two are out-of-focus positions on either side. Using information from the two 'neighbors', the PSF is calculated.

SeDDaRA---The SeDDaRA approach calculates the PSF by comparing the blurred image to an in-focus image, or "reference image", that contains the desired spatial frequency content. Now this sounds like it is a difficult task. But the Quarktet technique greatly relaxes this condition to enable easy implementation. The process is non-iterative, requires only a couple of parameters, and takes seconds to perform (depending on image size and computational speed.)

Support Constraints---This technique is applied to bright objects on a dark background, and makes two assumptions. The truth image is assumed to be positive and comprised of an object with known support against a uniform background. A 'support' is defined to the smallest rectangle that can be drawn around the object. This technique can be further improved by controlling the noise parameters using 'regularization'. Regularization methods try to alleviate the method's sensitivity to noise by eliminating eigen-components of the solution belonging to noise subspace.