Error of Averaging a Large Vector in MATLAB

How do you accurately take the average of large sets of integers in MATLAB?
I have two large vectors (2672x4008 in dimensions) I am dealing with, each the result of pixels in an image. Hence, the resulting vector is filled with values 0 to 256, all integers. My problem is that I want an accurate value of the average intensity of these grey-scale images. To do this, I used the line

meanvalue = mean(I(:))


This yielded a value of meanvalue = 155.9335 in the output line of MATLAB.

Next, I added 20 to each value of the vector, as below (this should raise the intensity of the overall image, if I am understanding correctly).

Ipt = I + 20;


I then took the mean value of this new vector, Ipt

meanvaluept = mean(Ipt(:))


and matlab spat out a value of meanvaluept = 175.8916. I'm no math wizard, but I know enough to know that 175.8916 - 20 ≠ 155.9335.

Any help would be appreciated, either mathematically (how to increase the precision of MATLAB), or procedurally (there is some built-in function of MATLAB which will find the intensity).

3 Answers
3

Since you are referring to "grey-scale images", and you have integers in the range 0-255 (the 256 you mention must be a typo), my guess is that your I is of type uint8.

I

uint8

In this case, MATLAB uses saturated addition, in which results larger than 255 are clamped to 255. The effect you describe is caused by this saturated addition.

Here is an example:

>> I = uint8(randi(255,1000,1000));
>> mean( I(:)+20 )
ans =
147.1954
>> mean(I(:)) + 20
ans =
148.0151


The solution is to convert to doubles first:

>> mean( double(I(:)) + 20 )
ans =
148.0151


Have you checked the image datatype?

It's true that if your the mean of image I is

meanvalue = mean(I(:)) = 155.9335


and you added 20 to each pixels

Ipt = I + 20


you supposed to have

meanept = mean(Ipt(:)) = meanvalue + 20 = 175.9335


But, don't forget that image's datatype is uint8, which limit the pixels value to 0-255. It means if you added 20 to a pixel and its value is greater than 255, its value will set to 255, and the same if you substract some value and it's lower than 0.

Maybe, some of your pixels restricted to 255 when normally you'll have more than 255.

I have vector X in double

X = [1 1 1; ...
1 1 1; ...
1 1 240];


The mean of X is

mean(X(:)) = 27.5556


since

( 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 240)/9 = 27.5556


If I added 20 to each pixels

X20 = X + 20
= [(1 + 20) (1 + 20) (1 + 20); ...
(1 + 20) (1 + 20) (1 + 20); ...
(1 + 20) (1 + 20) (240 + 20)];
= [21 21 21; ...
21 21 21; ...
21 21 255];


Notice that X20(3,3) is 255, not 260. It cause

meanX20 = mean(X20(:)) = 47


but if I change X's datatype to double

X_double = double(X)


and added 20 to each pixels

X20_double = X_double + 20
= [(1 + 20) (1 + 20) (1 + 20); ...
(1 + 20) (1 + 20) (1 + 20); ...
(1 + 20) (1 + 20) (240 + 20)];
= [21 21 21; ...
21 21 21; ...
21 21 260];


and the average of X20_double is

X20_double_mean = mean(X20_double(:)) = 47.5556


See the difference?
The double X20's mean is 47.5556 and the uint8 X20's mean is 47.
I hope this will help :)

There is a very important note in your question :

Suppose I = [2 3;4 9]

I = [2 3;4 9]

meanvalue = mean(I(:)) = 4.5


When you add 20 with I , You will have :

Ipt = I + 20;


Ipt = [22 23;24 29]

so you add 20 to all elements in I, therefore your mean will increase 20 value.

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