When we introduce the mean and the median in basic courses on
statistics my feeling is that the median seems a little unnatural for
the students. On the other hand they are very familiar with the mean
because it is the useful operation to compute the final grade. If the
mean is the "natural average" why should they bother about the median? The
importance of the median relies mainly on its stability when we have
outliers. There is a simple way of "seeing" the difference with a
well-known application to digital image filtering.
Let us start with an image. Beauty is not mandatory but it can capture the audience.
Now we add "salt and pepper noise". For a grayscale image this means that with a certain probability p
a pixel is set to on/off and the result is that the image is
contaminated with black and white dots. For color images the situation
is a little more complicate because we have three channels (RGB) and
then with the same parameter the probability of a pixel of being
changed is 1-(1-p)3. In our case we have take p=0.15 and consequently something less than the 40% of the pixels are corrupted.
This 40% implies that still the majority of the pixels are OK. With
this idea in mind one can take the average of the values of the pixels
in each 3x3 block (a central pixel and the 8 surrounding pixels) hoping
that majority wins. The result is shown below and it is quite
disappointing, the image has just blurred. If we force to team a single very bad student with
groups of good students, we can feel the
presence of the bad student in the average result of a contest.
The situation is completely different with the median because it
is unresponsive to extreme values if the rest of the values are alike.
Taking the median in each 3x3 block has a magical effect. The image is
almost completely clean.
The code
The matlab code that produces the images from the initial one is:
imag = imread('niki.jpg');
%imag = rgb2gray(imag);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NOISY IMAGE
imagn = imnoise(imag,'salt & pepper',0.15);
imwrite(imagn, 'niki_n.bmp')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MEAN FILTER
h2 = ones(3)/9;
imwrite(imfilter( imagn, h2, 'conv'), 'niki_c.bmp')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MEDIAN FILTER
imagm = imagn;
% simply imagm = medfilt2(imagn,[3,3]); for grayscale images
for ii = 1:3
imagm(:,:,ii) = medfilt2(imagn(:,:,ii),[3,3]);
end
imwrite(imagm, 'niki_m.bmp')
Quite simple!
