Image analysis via the general theory of moments*

Michael Reed Teague

doi:10.1364/JOSA.70.000920

Journal of the Optical Society of America
Vol. 70,
Issue 8,
pp. 920-930
(1980)
•https://doi.org/10.1364/JOSA.70.000920

Image analysis via the general theory of moments

Michael Reed Teague

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Michael Reed Teague, "Image analysis via the general theory of moments*," J. Opt. Soc. Am. 70, 920-930 (1980)

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Abstract

Two-dimensional image moments with respect to Zernike polynomials are defined, and it is shown how to construct an arbitrarily large number of independent, algebraic combinations of Zernike moments that are invariant to image translation, orientation, and size. This approach is contrasted with the usual method of moments. The general problem of two-dimensional pattern recognition and three-dimensional object recognition is discussed within this framework. A unique reconstruction of an image in either real space or Fourier space is given in terms of a finite number of moments. Examples of applications of the method are given. A coding scheme for image storage and retrieval is discussed.

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Equations (96)

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μ₂₀ – μ₀₂	μ ₁₁	ϕ
Zero	Zero	0	$ξ \equiv \frac{2 μ_{11}}{μ_{20} - μ_{02}}$
Zero	Positive	+45°
Zero	Negative	−45°
Positive	Zero	0
Negative	Zero	−90°
Positive	Positive	(1/2)tah⁻¹ξ	(0 < ϕ < 45°)
Positive	Negative	(1/2)tan⁻¹ξ	(−45° < ϕ < 0)
Negative	Positive	(1/2)tan⁻¹ξ + 90°	(45° < ϕ < 90°)
Negative	Negative	(1/2) tan⁻¹ξ − 90°	(−90° < ϕ < −45°)

Maximum order of moments included	E	F
2	133	149
3	132	149
4	103	118
5	102	119
6	90	102
7	91	101
8	87	98
9	83	93
10	81	92
11	74	81
12	67	75
13	61	65
14	54	60
15	47	56
16	46	51
17	40	48
18	36	43
19	33	39
20	27	33

μ₂₀ – μ₀₂	μ ₁₁	ϕ
Zero	Zero	0	$ξ \equiv \frac{2 μ_{11}}{μ_{20} - μ_{02}}$
Zero	Positive	+45°
Zero	Negative	−45°
Positive	Zero	0
Negative	Zero	−90°
Positive	Positive	(1/2)tah⁻¹ξ	(0 < ϕ < 45°)
Positive	Negative	(1/2)tan⁻¹ξ	(−45° < ϕ < 0)
Negative	Positive	(1/2)tan⁻¹ξ + 90°	(45° < ϕ < 90°)
Negative	Negative	(1/2) tan⁻¹ξ − 90°	(−90° < ϕ < −45°)

Maximum order of moments included	E	F
2	133	149
3	132	149
4	103	118
5	102	119
6	90	102
7	91	101
8	87	98
9	83	93
10	81	92
11	74	81
12	67	75
13	61	65
14	54	60
15	47	56
16	46	51
17	40	48
18	36	43
19	33	39
20	27	33

Maximum order of moments included	E	F
2	133	149
3	132	149
4	103	118
5	102	119
6	90	102
7	91	101
8	87	98
9	83	93
10	81	92
11	74	81
12	67	75
13	61	65
14	54	60
15	47	56
16	46	51
17	40	48
18	36	43
19	33	39
20	27	33

Maximum order of moments included	E	F
2	133	149
3	132	149
4	103	118
5	102	119
6	90	102
7	91	101
8	87	98
9	83	93
10	81	92
11	74	81
12	67	75
13	61	65
14	54	60
15	47	56
16	46	51
17	40	48
18	36	43
19	33	39
20	27	33

Maximum order of moments included	E	F
2	133	149
3	132	149
4	103	118
5	102	119
6	90	102
7	91	101
8	87	98
9	83	93
10	81	92
11	74	81
12	67	75
13	61	65
14	54	60
15	47	56
16	46	51
17	40	48
18	36	43
19	33	39
20	27	33

Maximum order of moments included	E	F
2	133	149
3	132	149
4	103	118
5	102	119
6	90	102
7	91	101
8	87	98
9	83	93
10	81	92
11	74	81
12	67	75
13	61	65
14	54	60
15	47	56
16	46	51
17	40	48
18	36	43
19	33	39
20	27	33

Image analysis via the general theory of moments

Author Affiliations

Abstract

Cited By

Figures (4)

Tables (2)

Equations (96)

Journal of the Optical Society of America

Maximum order of moments included	E	F
2	133	149
3	132	149
4	103	118
5	102	119
6	90	102
7	91	101
8	87	98
9	83	93
10	81	92
11	74	81
12	67	75
13	61	65
14	54	60
15	47	56
16	46	51
17	40	48
18	36	43
19	33	39
20	27	33

Maximum order of moments included	E	F
2	133	149
3	132	149
4	103	118
5	102	119
6	90	102
7	91	101
8	87	98
9	83	93
10	81	92
11	74	81
12	67	75
13	61	65
14	54	60
15	47	56
16	46	51
17	40	48
18	36	43
19	33	39
20	27	33