A-level Physics (Advancing Physics)/Digital Processing

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As we have already seen, a digital image consists of pixels, with each pixel having a value which represents its colour. For the purposes of understanding how digital images are manipulated, we are going to consider an 8-bit grayscale image, with pixel values ranging from 0 to 255, giving us 256 (28) levels of grey. 0 represents white, and 255 represents black. This is the image we are going to consider:

000 000 000 000 000 150 150 150 050 150
000 000 000 000 000 150 150 150 150 150
000 000 235 000 000 150 150 150 150 150
000 000 000 000 000 150 205 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
255 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 095
000 000 000 000 000 150 150 150 150 150
000 000 000 185 000 150 150 150 150 150

The image consists of an edge, and some random noise. There are two methods of smoothing this image (i.e. removing noise) that you need to know about:

Mean Smoothing[edit]

In order to attempt to remove noise, we can take the mean average of all the pixels surrounding each pixel (and the pixel itself) as the value of the pixel in the smoothed image, as follows:

000 000 000 000 050 100 150 133 133 133
000 026 026 026 050 100 150 139 139 139
000 026 026 026 050 106 173 173 150 150
000 026 026 026 050 106 173 173 150 150
000 000 000 000 050 106 173 173 150 150
043 028 000 000 050 100 150 150 150 150
043 028 000 000 050 100 150 150 150 150
043 028 000 000 050 100 150 150 144 141
000 000 000 000 050 100 150 150 144 141
000 000 021 021 071 100 150 150 144 141
000 000 31 31 081 100 150 150 150 150

This does remove the noise, but it blurs the image.

Median Smoothing[edit]

A far better method is, instead of taking the mean, to take the median, as follows:

000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150
000 000 000 000 000 150 150 150 150 150

For this image, this gives a perfect result. In more complicated images, however, data will still be lost, although, in general, less data will be lost by taking the median than by taking the mean.

Edge Detection[edit]

We can detect the positioning of edges in an image using the 'Laplace rule', or 'Laplace kernel'. For each pixel in the image, we multiply its value by 4, and then subtract the values of the pixels above and below it, and on either side of it. If the result is negative, we treat it as 0. So, taking the median-smoothed image above, edge detection gives the following result:

000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000
000 000 000 000 000 150 000 000 000 000

Questions[edit]

1. How could the above methods be applied to a digital sound sample?

2. Which of the above methods would be suitable for smoothing sharp edges? Why?

3. Use median smoothing to remove noise from the following image of a white cat in a snowstorm (the black pixels have a value of 255):

000 255 000 000
000 000 000 255
255 000 000 000
000 000 255 000

4. Why would mean sampling not be appropriate for smoothing the image given in question 3?

5. Use mean smoothing to remove noise from the following image of a black cat in a coal cellar:

255 255 255 255
255 255 000 255
255 255 255 255

Worked Solutions