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There are especial cases: blurring due to incorrect focus andblurring due to movement - and these very defects (which each of you knows very well, and which are very difficult torepair) were selected as the subject of this article. As for other image defects (noise, incorrect exposure,distortion), the humanity has learned how to correct them, any good photo editor has that tools.

Why is there almost no means for correction of blurring and defocusing (except unsharp mask) - maybe it is impossible todo this at all? In fact, it is possible - development of a respective mathematical theory started approximately 70 yearsago, but like other algorithms of image processing, deblurring algorithms became wide-used just recently. So, below is acouple of pictures to demonstrate the WOW-effect:

I decided not to use well-known Lena, but just found my own picture ofVenice. The right image was really made from the left one, moreover, no optimization like 48-bit format (in this casethere will be almost 100% restoration of the source image) were used - there is just a regular PNG with syntetic blur onthe left side. The result is impressive... but in practice not everything is so good.

Let's start from afar. Many people think that blurring is an irreversible operation and the information in this case islost for good, because each pixel turns into a spot, everything mixes up, and in case of a big blur radius we will get aflat color all over the image. But it is not quite true - all the information just becomes redistributed in accordancewith some rules and can be definitely restored with certain assumptions. An exception is only borders of the image, thewidth of which is equal to the blur radius - no complete restoration is possible here.

After blurring the value of each pixel is added to the value of the left one: x'i = xi +xi-1. Normally, it is also required to divide it by 2, but we will drop it out for simplicity. As a result wehave a blurred image with the following pixel values:x1 + x0 x2 + x1 x3 + x2 x4 +x3... - Blurred image

Now we will try to restore it, we will consequentially subtract values according to the following scheme - the first pixelfrom the second one, the result of the second pixel from the third one, the result of the third pixel from the fourthone and so on, and we will get the following:x1 + x0 x2 - x0 x3 + x0 x4 -x0... - Restored image

As a result, instead of a blurred image, we got the source image with added to the each pixel an unknownconstant x0 with the alternate sign. This is much better - we can choose this constant visually, we cansuppose that it is approximately equal to the value x1, we can automatically choose it with such a criteriathat values of neighboring pixels were changing as little as possible, etc. But everything changes as soon as we addnoise (which is always present in real images). In case of the described scheme on each stage there will accumulate thenoise value into the total value, which fact eventually can produce an absolutely unacceptable result, but as wealready know, restoration is quite possible even using such a primitive method.

Now let's pass on to more formal and scientific description of these blurring and restoration processes. We willconsider only grayscale images, supposing that for processing of a full-color image it is enough torepeat all required steps for each of the RGB color channels. Let's introduce the following definitions:f(x, y) - source image (non-blurred)h(x, y) - blurring functionn(x, y) - additive noiseg(x, y) - blurring result image

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.

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:

The process of applying of the blurring function to another function (in his case, to an image) is called convolution,i.e. some area of the source image convolves into one pixel of the blurred image. It is denoted through the operator"*", but do not confuse it with a simple multiplication! Mathematically, for an image f with dimensions M x N andthe blurring function h with dimensions m x n it can be written down as follows:

It is only left to consider the last summand, which is responsible for noise, n(x, y) in the formula (1). Therecan be many reasons for noise in digital sensors, but the basic ones are - thermal vibrations (Brownian motion) and dark current.The noise volume also depends on a number of factors, such as ISO value, sensor type, pixel size, temperature, magneticfield value, etc. In most cases there is Gaussian noise (which is set by two parameters - the average and dispersion), andit is also additive, does not correlate with an image and does not depend on pixel coordinates. The last threeassumptions are very important for further work.

Let us get back to the initial task of restoration - we need to somehow reverse the convolution, bearing in mind thenoise. From the formula (2) we can see that it is not so easy to get f(x, y) from g(x, y) - if we calculateit straightforward, then we will get a huge set of equations. But the Fourier transform comes to the rescue, we will notview it in details, because a lot has been said about this topic already. So, there is the so called convolutiontheorem, according to which the operation of convolution in the spatial domain is equivalent to regular multiplicationin the frequency domain (where the multiplication - element-by-element, not matrix one). Correspondingly, the operationwhich is opposite to convolution is equivalent to division in the frequency domain, i.e. this can be expressed asfollows: (3)

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, 041b061a72