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# Documentation

## Understanding Deblurring

### Causes of Blurring

The blurring, or degradation, of an image can be caused by many factors:

• Movement during the image capture process, by the camera or, when long exposure times are used, by the subject

• Out-of-focus optics, use of a wide-angle lens, atmospheric turbulence, or a short exposure time, which reduces the number of photons captured

• Scattered light distortion in confocal microscopy

### Deblurring Model

A blurred or degraded image can be approximately described by this equation g = Hf + n, where

 g The blurred image H The distortion operator, also called the point spread function (PSF). In the spatial domain, the PSF describes the degree to which an optical system blurs (spreads) a point of light. The PSF is the inverse Fourier transform of the optical transfer function (OTF). In the frequency domain, the OTF describes the response of a linear, position-invariant system to an impulse. The OTF is the Fourier transform of the point spread function (PSF). The distortion operator, when convolved with the image, creates the distortion. Distortion caused by a point spread function is just one type of distortion. f The original true image n Additive noise, introduced during image acquisition, that corrupts the image

 Note   The image f really doesn't exist. This image represents what you would have if you had perfect image acquisition conditions.

#### Importance of the PSF

Based on this model, the fundamental task of deblurring is to deconvolve the blurred image with the PSF that exactly describes the distortion. Deconvolution is the process of reversing the effect of convolution.

 Note   The quality of the deblurred image is mainly determined by knowledge of the PSF.

To illustrate, this example takes a clear image and deliberately blurs it by convolving it with a PSF. The example uses the fspecial function to create a PSF that simulates a motion blur, specifying the length of the blur in pixels, (LEN=31), and the angle of the blur in degrees (THETA=11). Once the PSF is created, the example uses the imfilter function to convolve the PSF with the original image, I, to create the blurred image, Blurred. (To see how deblurring is the reverse of this process, using the same images, see Deblurring with the Wiener Filter.)

```I = imread('peppers.png');
I = I(60+[1:256],222+[1:256],:); % crop the image
figure; imshow(I); title('Original Image');```

```LEN = 31;
THETA = 11;
PSF = fspecial('motion',LEN,THETA); % create PSF
Blurred = imfilter(I,PSF,'circular','conv');
figure; imshow(Blurred); title('Blurred Image');```

### Deblurring Functions

The toolbox includes four deblurring functions, listed here in order of complexity. All the functions accept a PSF and the blurred image as their primary arguments.