09-08-2012, 04:56 PM
The Image Deblurring Problem
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How Images Become Arrays of Numbers
Having a way to represent images as arrays of numbers is crucial to processing images using
mathematical techniques. Consider the following 9 × 16 array:
If we enter this into a MATLAB variable X and display the array with the commands
imagesc(X), axis image, colormap(gray), then we obtain the picture shown at
the left of Figure 1.1. The entries with value 8 are displayed as white, entries equal to zero
are black, and values in between are shades of gray.
Color images can be represented using various formats; the RGB format stores images
as three components, which represent their intensities on the red, green, and blue scales. A
pure red color is represented by the intensity values (1, 0, 0) while, for example, the values
(1, 1, 0) represent yellow and (0, 0, 1) represent blue; other colors can be obtained with
different choices of intensities.
A Blurred Picture and a Simple Linear Model
Before we can deblur an image, we must have a mathematical model that relates the given
blurred image to the unknown true image. Consider the example shown in Figure 1.2. The
left is the “true” scene, and the right is a blurred version of the same image. The blurred
image is precisely what would be recorded in the camera if the photographer forgot to focus
the lens.
Deblurring Using a General Linear Model
Underlying all material in this book is the assumption that the blurring, i.e., the operation
of going from the sharp image to the blurred image, is linear. As usual in the physical
sciences, this assumption is made because in many situations the blur is indeed linear, or at
least well approximated by a linear model.