1. science
2. mathematics
3. statistics

Histogram Matching (Specification)
OBJECTIVES
1. Define histogram matching
(histogram specification)
2. Sample application
3. Derive formula in continuous
domain
4. Convert to digital domain +
example
1
Definition of Histogram Matching (Specification)
A(x, y) - Input
C(x, y) - Output
Hc(DC)
HA(DA)
0
2
255
DA
0
255
DC
Histogram Matching (Specification)
3
PRIMARY USES
• Improve display (contrast &
brightness)
• Preprocessing step for comparison of
images
COMPARISON TO EQUALIZATION
• More general than histogram
equalization since histogram of output
image need not be flat only
• Interactive enhancement technique user can draw desired histogram
Example - Application
We wish to check if a circuit board (image 1)
matches the template (image 2) from which it
was manufactured. Any defects?
1 - Manufactured
4
2 - Template
Compute difference image (defined later),
threshold by setting pixels with non-zero
absolute difference to 1 and all other pixels to 0:
Example - Application
What if the overall brightness of image 1 is
different from that of image 2?
1 - Manufactured
5
2 - Template
Difference image is white everywhere because
there is a difference in the brightness of all
pixels.
One solution would be to match the histograms
of the two images and then do the subtraction.
255
DA
0
HB(DB)
B(x, y)
255
DB
0
HC(DC)
DB=f 2(DC)
C(x, y)
255
DC
(see match.doc)
0
HA(DA)
A(x, y)
DB=f1 (DA)
DC=f (DA)
Histogram Matching (Specification)
6
DERIVATION - CONTINUOUS
Example – Continuous-tone
Suppose you are given two X-ray films taken immediately
before and after injection of a contrast medium (dye) into the
arteries of a patient’s heart. Radiologists are studying the films
to determine whether coronary bypass surgery or heart valve
replacement is required. Normally they use digital image
subtraction to visualize the dye as it fills the arterial
passageways. In this case, however, problems in exposure and
development of the two films make a direct comparison
inconclusive. The patient is too weak to undergo the
angiography procedure again. Only digital subtraction (Chapter
7) of the two images will reveal the extent of coronary disease.
To a good approximation, the histograms of the two images are
given by the Rayleigh distribution where Dm = 63, α = 16 for the
first image, and α = 24 for the second image. What GST would:
(a) flatten the histogram of the preinjection image? (b) flatten
the histogram of the postinjection image? (c) make the
histogram of the postinjection image match that of the
preinjection image? You can assume continuous variables.
[Question 8, Chapter 6 of textbook.]
H ( D) =
DDm
α2
 D2 
exp − 2 
 2α 
7
Example (cont’d)
8
Example (cont’d)
9
Example – Digital
10
Given a 3-bit gray-level image A(i, j) with the
following histogram:
DA
0
1
2
3
4
5
6
7
HA(DA)
1028
3544
5023
3201
1867
734
604
383
Design point operation to match it to the following
histogram:
DC
0
1
2
3
4
5
6
7
HC(DC)
0
0
1638
4096
4916
4096
1638
0
Example - Digital
11
1. Equalize the input image to get the intermediate
image B(i, j):
DA
0
1
2
3
4
5
6
7
HA(DA)
1028
3544
5023
3201
1867
734
604
383
DB
0
2
4
5
6
7
7
7
Can now find B(i, j) from A(i, j).
2. Find GST that would equalize the desired image
C(i, j) if it were known:
DC
0
1
2
3
4
5
6
7
HC(DC)
0
0
1638
4096
4916
4096
1638
0
DB
0
0
1
2
5
6
7
7
Example - Digital
12
Therefore, the GST DB = f(DC) that maps gray levels
DC into DB is:
DC
0
1
2
3
4
5
6
7
D B = f2 (D C )
3. Calculate the inverse function DC = f -1(DB) that
maps gray levels DB into DC:
DB
DC = f 2−1 ( DB )
0
1
2
3
4
5
6
7
4. Examine image B(i, j) pixel-by-pixel, replacing
each gray level DB by DC using the table in 3 above.
Rules for Calculating Inverse
• For unique pairs (DB, DC), it is
easy to fill in a row of the table
• If a DB value has more than one
potential DC value associated
with it, use the lower DC value
• If a DB value does not appear in
the table, use the closest DC
value; if two or more DC values
are equally close, use the
smaller DC value
13
Example
14
>> D = 0:255;
>> h = exp(-(D-50).^2/(2*16^2)) + exp(-(D-200).^2/(2*16^2));
>> im2 = histeq(im, h);
original
Desired histogram
After
Summary
• Definition of histogram
matching (specification)
• Describe applications
• Derivation and application of
formula in continuous domain
• Conversion to digital domain
Should be able to do Homework 2.
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