# Online Exam а¤•а¤ѕ а¤Ўа¤° аҐ¤ Online Exam Demo |

The task of restoration of a blurred image consists in finding the best approximation f'(x, y) to the sourceimage. Let's consider each component in a more detailed way. As for functions f(x, y) and g(x, y),everything is quite clear with them. But as for h(x, y) I need to say a couple of words - what is it? In theprocess of blurring the each pixel of a source image turns into a spot in case of defocusing and into a line segment (or some path) in caseof a usual blurring due to movement. Or we can say otherwise, that each pixel of a blurred image is "assembled" frompixels of some nearby area of a source image. All those overlap each other, which fact results in a blurred image. Theprinciple, according to which one pixel becomes spread, is called the blurring function. Other synonyms -PSF (Point spread function), kernel and other. The size of this function is lower than the size of the imageitself - for example, when we were considering the first "demonstrational" example the size of the function was 2,because each result pixel consisted of two pixels.

## Online exam а¤•а¤ѕ а¤Ўа¤° аҐ¤ online exam Demo |

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Let us see what typical blurring functions look like. Hereinafter we will use the tool which has already become standardfor such purposes - Matlab, it contains everything required for the most diverse experiments with image processing(among other things) and allows to concentrate on algorithms, shifting all the routine work to function libraries.However, this is only possible at the cost of performance. So, let's get back to PSF, here are their examples:

This is called inverse filtering, but in practice it almost never works. Why so? In order to answer this question, letus see the last summand in the formula (5) - if the function H(u, v) gives values, which are close to zero orequal to it, then the input of this summand will be dominating. This can be almost always seen in real examples - toexplain this let's remember what a spectrum looks like after the Fourier transform. So, we take the source image,

Another interesting approach was offered by Richardson (1972 year) and Lucy independently (1974 year), so this approachis called as method Lucy-Richardson. Its distinctive feature consists in the fact that it is nonlinear, unlike the firstthree - potentially this can give a better result. The second feature - this method is iterative, so there arisedifficulties with the criterion of iterations stop. The main idea consists in using the maximum likelihood method forwhich it is supposed that an image is subjected to Poisson distribution. Calculation formulas are quite simple, withoutthe use of Fourier transform - everything is done in the spatial domain: (8)

And at the end of the first part we will consider examples of real images. Before that all blurs were artificial, whichis quite good for practice and learning, but it is very interesting to see how all this will work with real photos. Hereis one example of such image, shot with the Canon 500D camera using manual focus (to get blur):

And an example with a real blur due to movement - in order to make this the camera was fixed on a tripod, there was setquite long exposure value and with an even movement at the moment of exposure the following blur was obtained:

High speed. Processing of an image with the size of 2048*1500 pixels takes about 300ms in the Preview mode (when adjustment sliders can move). But high-quality processing may take a few minutes

Real-time parameters changes applying (without any preview button)

Full resolution processing (without small preview window)

Deep tuning of kernel parameters

Easy and friendly user interface

Help screen with image example

Deconvolution methods: Wiener, Tikhonov, Total Variation prior

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