Example of Histogram Equalization

http://en.wikipedia.org/wiki/Histogram_equalization

Examples

Small image

The following is the same 8x8 subimage as used in JPEG. The 8-bit greyscale image shown has the following values:

The histogram for this image is shown in the following table. Pixel values that have a zero count are excluded for the sake of brevity.

Value	Count	Value	Count	Value	Count	Value	Count	Value	Count
52	1	64	2	72	1	85	2	113	1
55	3	65	3	73	2	87	1	122	1
58	2	66	2	75	1	88	1	126	1
59	3	67	1	76	1	90	1	144	1
60	1	68	5	77	1	94	1	154	1
61	4	69	3	78	1	104	2
62	1	70	4	79	2	106	1
63	2	71	2	83	1	109	1

The cumulative distribution function (cdf) is shown below. Again, pixel values that do not contribute to an increase in the cdf are excluded for brevity.

Value	cdf	cdf, scaled
52	1	0
55	4	12
58	6	20
59	9	32
60	10	36
61	14	53
62	15	57
63	17	65
64	19	73
65	22	85
66	24	93
67	25	97
68	30	117
69	33	130
70	37	146
71	39	154
72	40	158
73	42	166
75	43	170
76	44	174
77	45	178
78	46	182
79	48	190
83	49	194
85	51	202
87	52	206
88	53	210
90	54	215
94	55	219
104	57	227
106	58	231
109	59	235
113	60	239
122	61	243
126	62	247
144	63	251
154	64	255

This cdf shows that the minimum value in the subimage is 52 and the maximum value is 154. The cdf of 64 for value 154 coincides with the number of pixels in the image. The cdf must be normalized to. The general histogram equalization formula is:

Where cdf_min is the minimum non-zero value of the cumulative distribution function (in this case 1), M × N gives the image's number of pixels (for the example above 64, where M is width and N the height) and L is the number of grey levels used (in most cases, like this one, 256). The equalization formula for this particular example is:

For example, the cdf of 78 is 46. (The value of 78 is used in the bottom row of the 7th column.) The normalized value becomes

Once this is done then the values of the equalized image are directly taken from the normalized cdf to yield the equalized values:

Notice that the minimum value (52) is now 0 and the maximum value (154) is now 255.

Original	Equalized

Full-sized image

An unequalized image	Corresponding histogram (red) and cumulative histogram (black)
The same image after histogram equalization	Corresponding histogram (red) and cumulative histogram (black)

Notes

Jump up^ Laughlin, S.B (1981). "A simple coding procedure enhances a neuron’s information capacity".Z. Naturforsch. 9–10(36):910–2.