how to calculate variance of image histogram?

[Mohd Farhan]

HELLO;
I AM NEW IN MATLAB IMAGE PROCESSING. I AM STUCK IN PROGRAMMING OF WITH-IN CLASS VARIANCE OF HISTOGRAM!!

THE HISTOGRAM OF AN IMAGE IS GIVEN AS:
I = imread('camera.tif');
histogram = imhist(I);
bar(histogram,'BarWidth',1)
figure;
J = histeq(I);
imshow(J);
figure;
histogramEq = imhist(J);
bar(histogramEq,'BarWidth',1)

NOW HOW TO CALCULATE THE "WITH-IN VARIANCE" OR "VARIANCE" OF GIVEN HISTOGRAMS?
ITS VARIANCE IS GIVEN AS:
square of (s) = summation of {square of (graylevel - mean) x no. of pixels in its perticular greylevel} / (total pixels in an image).

refer: http://itl.nist.gov/div898/handbook/eda/section3/eda356.htm

PLEASE HELP ME IN THIS ISSUE. THANKS IN ADVANCE.

[Mohd Farhan]

"Mohd Farhan " <farha...@gmail.com> wrote in message <j8djl2$704$1...@newscl01ah.mathworks.com>...

Please anyone help me..

[Mohd Farhan]

"Mohd Farhan " <farha...@gmail.com> wrote in message <j8djl2$704$1...@newscl01ah.mathworks.com>...

Please anyone help me..

[ImageAnalyst]

That's just the regular histogram. And I think you want the variance
of the image, not the variance of the histogram. So just do this
varianceOfImage = var(imageArray(:));

[Mohd Farhan]

ImageAnalyst <imagea...@mailinator.com> wrote in message <43d64d1a-dcc1-4121...@g21g2000yqc.googlegroups.com>...

> That's just the regular histogram. And I think you want the variance
> of the image, not the variance of the histogram. So just do this
> varianceOfImage = var(imageArray(:));

THANKS FOR YOUR REPLY.
var(imageArray(:)) GIVING GOOD RESULT WHEN IMAGE IS CONSIDERED AS WHOLE i.e. WHEN (THRESHOLD) T=0, BUT NOT FOR OTHER THRESHOLD VALUES.
LET SUPPOSE I HAD THIS 6x6 MATRIX:
refer: http://www.labbookpages.co.uk/software/imgProc/otsuThreshold.html
R =
0 0 1 4 4 5
0 1 3 4 3 4
1 3 4 2 1 3
4 4 3 1 0 0
5 4 2 1 0 0
5 5 4 3 1 0

NOW HOW TO CALCULATE THE VARIANCE OF BACKGROUND & FOREGROUND IMAGE?
For Different Threshold, Background & Foreground variance is coming out to be:
Threshold Background variance Foreground variance
T=0 0 3.1196
T=1 0 1.9639
T=2 0.2489 0.7755
T=3 0.4637 0.5152
T=4 1.4102 0.2130
T=5 2.5303 0

HOW TO CALCULATE THESE VARIANCES?

[ImageAnalyst]

The web page you gave talked about the variance of the image and
showed a way to get it via the histogram. This, as I hope you know,
is entirely different than the variance of the histogram. Now, to get
the foreground and background you can do this:
foregroundMask = imageArray >= thresholdValue;
foregroundPixelsValues = imageArray(foregroundMask);
backgroundPixelsValues = imageArray(~foregroundMask);
then pass each of those into imhist() to get the histograms, and use
the formulas on your web page.

[Mohd Farhan]

ImageAnalyst <imagea...@mailinator.com> wrote in message <a384766e-4c26-4a7b...@ek5g2000vbb.googlegroups.com>...

THANKS FOR YOUR REPLY.

I DID WHAT YOU QUOTED. BUT ONE PROBLEM ARRIVES:
The O/P result is:
backgroundPixelsValues =
0
0
1
0
1
1
2
2
1
1
1
0
0
1
0
0
0

foregroundPixelsValues =
4
5
5
3
4
4
5
3
4
3
4
4
4
3
4
3
5
4
3

Histogram {hist} of this backgroundPixelsValues gives range from 1-3 & also same for foregroundPixelsValues!!
I used this program:
A = backgroundPixelsValues(:);
H= hist(A,2);
bar(H,'BarWidth',1)

& SAME FOR foregroundPixelsValues.

NOW WHEN I DID HISTOGRAM EQUALIZATION:
histeq(backgroundPixelsValues); it gives wrong result as compared to theoretical calculation!
===================================
As theoretical histogram-equalization calculation gives:
backgroundPixelsValues:
for 0 --> 0.9412 quantized to 1.
for 1 --> 1.7648 quantized to 2.
for 2 --> quantized to 2.
===================================
but the above MATLAB program gives:
for 0 --> 0.4762 quantized to 1.
for 1 --> quantized to 1.
for 2 --> quantized to 1. !!

& same problem for foregroundPixelsValues.

[Mohd Farhan]

my above problem is almost solved. thanks for ur replies.

[Mohd Farhan]

ImageAnalyst <imagea...@mailinator.com> wrote in message <a384766e-4c26-4a7b...@ek5g2000vbb.googlegroups.com>...

WHAT WE HAVE TO DO IF WE WANT TO DIVIDE HISTOGRAM IN MORE THAN 2 PARTS? LIKE 4,8 etc.

I TOOK THIS MATRIX:
R = [6 0 1 7 4 5; 0 7 3 4 3 4; 6 6 4 2 1 3; 4 4 3 1 6 0; 5 4 7 1 0 0; 5 7 4 3 1 0];

& APPLIED THIS FOR DIVIDING INTO THREE PARTS FROM 0-2, 3-4 & 5-7:
foregroundMask = R >= 5;
midgroundMask = 5 > R >= 2;
backgroundMask = 2 > R >= 0;
foregroundPixelsValues = R(foregroundMask);
midgroundPixelsValues = R(midgroundMask);
backgroundPixelsValues = R(backgroundMask);

BUT IT IS NOT WORKING.. PLEASE HELP ME..

[ImageAnalyst]

On Nov 1, 8:01 am, "Mohd Farhan " <farhan7...@gmail.com> wrote:
> foregroundMask = R >= 5;
> midgroundMask = 5 > R >= 2;
> backgroundMask = 2 > R >= 0;
> foregroundPixelsValues = R(foregroundMask);
> midgroundPixelsValues = R(midgroundMask);
> backgroundPixelsValues = R(backgroundMask);
>
> BUT IT IS NOT WORKING.. PLEASE HELP ME..

--------------------------------------------------------------
You can't do stuff like 5 > R >= 2.
You need to do it this way instead.

foregroundMask = R >= 5

midgroundMask = (5 > R) & (R >= 2)
backgroundMask = R < 2

[Mohd Farhan]

ImageAnalyst <imagea...@mailinator.com> wrote in message <ac859295-c385-4cd6...@o19g2000vbk.googlegroups.com>...

REALLY REALLY THANKS.. IT SOLVED MY PROBLEM..

[Mohd Farhan]

refer: http://www.labbookpages.co.uk/software/imgProc/otsuThreshold.html

in above web-page, there is the calculation of within-class variance when Threshold is considered.
At T=0, ?2W = 3.1196
At T=1, ?2W = 1.5268
At T=2, ?2W = 0.5561
At T=3, ?2W = 0.4909
At T=4, ?2W = 0.9779
At T=5, ?2W = 2.2491

HOW CAN I DO THE PROCESS FOR AN 256X256 GREY IMAGE ? i.e. I WANT TO PRINT:: {ALL THE THRESHOLD VALUES OF AN IMAGE, WITH ALL CALCULATED WITHIN-CLASS VARIANCE} SO THAT I CAN COMPARE THE RSULT.

my program are showing errors.. may be i am doing something wrong somewhere!

[Mohd Farhan]

"Mohd Farhan " <farha...@gmail.com> wrote in message <j8tf3t$ddl$1...@newscl01ah.mathworks.com>...

Please help me. How can i calculate & print all the with-in class variances? so that i can see & compare it.

[Mohd Farhan]

let a image of 256x256.
its complete histogram had 65536 pixels.
Now i divide its histogram into two parts. how to calculate the total pixels of two sub-histograms?

pixelCount only showing total pixels in an image! but i want to know the pixel count of sub-histograms. so that weight can be calculated.

[ImageAnalyst]

----------------------------------------------------------
So you divided the histogram somehow into two sub-histograms? OK,
then to get the pixel count in each subhistogram you simply sum up the
counts in each subhistogram array:
pixelsInSubHistogram = sum(subHistPixelCounts(:))

[Mohd Farhan]

ImageAnalyst <imagea...@mailinator.com> wrote in message <a932bd34-c6b7-4e0d...@9g2000yqv.googlegroups.com>...

thanks i works.. i was doing one error before.

can you help me calculating minimum & maximum [& other if possible] within-class histogram variance of an 256x256 image.
here: http://www.labbookpages.co.uk/software/imgProc/otsuThreshold.html : 6 values are calculated but as image size increases its no. increases.
So please help me how can i calculate with-in class histogram variance of larger images.

please.

[ImageAnalyst]

Well the page tells you so just follow the example:

% Calculate the histogram:
[pixelCounts GLs] = imhist(pixelValues);

% Calculate the mean:
meanGL = sum(pixelCounts .* GLs) / sum(pixelCounts);

% Subtract the mean from the gray levels:
m0 = pixelCounts - meanGL;

% Square that difference:
diffSquared = m0 .^ 2

% Multiply by the count in that histogram bin:
numerator = diffSquare .* pixelCounts;
denominator = sum(pixelCounts);

% Finally:
histVariance = numerator / denominator;

Of course you could make that more compact. I just expanded it out
and make it real explicit so that you'd be able to follow along
easily. In fact you could just do
histVariance = var(pixelValues(:));
and get the same answer, but you wanted the way to do it from a
histogram.

[Mohd Farhan]

ImageAnalyst <imagea...@mailinator.com> wrote in message <1c31528d-b211-4be9...@u28g2000yqb.googlegroups.com>...

This formula is for variance only.
But i want to calculate with-in class histogram variance.

With-in class histogram variance is calculated by dividing histogram in two grounds: foreground & background for every grey-level. i.e. for grey level image, we have 256 with-in class variances.

1st when threshold is took at 0 grey level (i.e. T=0) --> so only ground is generated.
2nd when threshold is took at 1 grey level (i.e. T=1) --> so two grounds are generated, foreground is the from (2 to 256) & back ground is from (0 to 1).
.
.
.
.
150th when threshold is took at 149th grey level (i.e. T=149) --> so two grounds are generated, foreground is the from (149 to 256) & back ground is from (0 to 148).
.
.
.
till 255th..

THIS IS WHAT I MEAN.
I WANT TO CALCULATE, ALL THE WITH-IN CLASS VARIANCES. SO THAT I CAN SEE & COMPARE WHICH THRESHOLD LEVEL GIVES MINIMUM & MAXIMUM WITH-IN CLASS VARIANCE.

i think now it is clear.

[ImageAnalyst]

On Nov 5, 9:09 am, "Mohd Farhan " <farhan7...@gmail.com> wrote:
> THIS IS WHAT I MEAN.
> I WANT TO CALCULATE, ALL THE WITH-IN CLASS VARIANCES. SO THAT I CAN SEE & COMPARE WHICH THRESHOLD LEVEL GIVES MINIMUM & MAXIMUM WITH-IN CLASS VARIANCE.
>
> i think now it is clear.

------------------------------------------------------------------
I know. It was clear before already. That's why I gave you the
code. pixelValues is either the foreground or background pixels
depending on which class you're calculating the variance for. I think
it's clear with me (what you want), I'm not so sure about you. If you
don't understand why my code won't work, then write back and say why.

[Mohd Farhan]

ImageAnalyst <imagea...@mailinator.com> wrote in message <349469fe-5090-4eae...@o19g2000vbk.googlegroups.com>...

Your formula works.. no doubt.. but that is for variance. not for WITHIN CLASS VARIANCE.

WITHIN CLASS VARIANCE have 256 different values for one gray-image,
but variance has only 1 value for one gray-image.

I am looking for that 256 values.
WITHIN CLASS VARIANCE = (weight of foreground histogram * variance of foreground histogram) + (weight of background histogram * variance of background histogram).

& these are 256 different values for grey-scale image.

refer: http://www.labbookpages.co.uk/software/imgProc/otsuThreshold.html

[Mohd Farhan]

actually what i am in need is that, a loop type of thing which automatically sets a threshold value (from 0 to 255) [& therefore divides histogram in two grounds], & give us output value as WITH-IN CLASS VARIANCE. So that i can compare the result of minimum & maximum values.

[Mohd Farhan]

please help me.