## Affine moment invariants

The affine transform is general linear transformation of space coordinates of the image:

u = a0 + a1x + a2y
v = b0 + b1x + b2y
.

The affine moment invariants are features for pattern recognition computed from moments of objects on images that do not change their value in affine transformation. The geometric moment of order p + q of the image f is defined

.

We can normalize them with respect to translation, then we obtain central moments

,

where xt=m10/m00 and yt=m01/m00 are coordinates of the centroid. The ratios

are then invariant to translation and scaling. There is a few approaches to the further procedure, we can use an area of a triangle and integrate a product of a few triangle areas over the object in a graph method [7,9]. Another approach is based on solution of Cayley - Aronhold differential equation [1,2,11]

.

We can also continue in the normalization and normalize subsequently to first rotation, stretching and second rotation [8], then we obtain normalized moments.

If we use one of the first two methods, we obtain many dependent invariants that must be eliminated. The invariants can be zero, identical, an invariant can be a product of other invariants or their linear combination. If all these dependent invariants are eliminated, we obtain a set of irreducible invariants. The invariants can be also polynomially dependent and their elimination has not been automated yet [10].

An example shows using affine moment invariants for registration of two satellite images:

 LANDSAT SPOT

The satellite images show the same area in Bohemia, north-west from Prague. One of them is Landsat from August 28, 1990, the other is Spot from May 13, 1986, the images were scaled twice down. The Landsat image has resolution 30 m and 7 spectral bands. We computed principal component analysis and used 3 most important components (upper left image) as input for color segmentation by means of region growing. The Spot image has resolution 20 m and its 3 spectral bands were used directly for the segmentation (upper right image). The Spot image was used without correction to skew caused by Earth's rotation to test the affine invariance of the features. A noise was reduced in both images by minimization of the Mumford - Shah functional (middle row of images). Segmented regions are in the bottom row of images, that with a counterpart in the other image are numbered, the matched ones have numbers in a circle. All three methods yielded similar results.

Contact person: Tomáš Suk

Publications:

1. Flusser Jan, Suk Tomáš, Zitová Barbara: Moments and Moment Invariants in Pattern Recognition, Wiley & Sons Ltd., 2009 (in print).
2. Suk Tomáš, Flusser Jan: Affine Moment Invariants Generated by Automated Solution of the Equations, Proceedings of the 19th International Conference on Pattern Recognition, ICPR 2008, (Tampa, US, 8.12.2008-11.12.2008), Download PDF
3. Suk Tomáš, Flusser Jan: The Independence of the Affine Moment Invariants, Information Optics, p. 387-396, Eds: Cristobal G., International Workshop on Information Optics. WIO'06 /5./, (Toledo, ES, 05.06.2006-07.06.2006), Download PDF
4. Suk Tomáš, Flusser Jan: Tables of Affine Moment Invariants Generated by the Graph Method, ÚTIA AV ČR, (Praha 2005)Research Report 2156, Download PDF
5. Suk Tomáš, Flusser Jan: Affine normalization of symmetric objects, Proceedings of the 7th International Conference on Advanced Concepts for Intelligent Vision Systems vol.3708, - (2005), p. 100-107, Advanced Concepts for Intelligent Vision Systems 2005 /7./, (Antwerp, BE, 20.09.2005-23.09.2005), Download PDF
6. Suk Tomáš, Flusser Jan: Graph method for generating affine moment invariants, Proceedings of the 17th International Conference on Pattern Recognition. ICPR 2004, p. 192-195, International Conference on Pattern Recognition. ICPR 2004 /17./, (Cambridge, GB, 23.08.2004-26.08.2004), Download PDF
7. Flusser Jan, Suk Tomáš: Affine moment invariants: a new tool for character recognition, Pattern Recognition Letters vol.15, p. 433-436, Download PS
8. Flusser Jan, Suk Tomáš:A moment-based approach to registration of images with affine geometric distortion, IEEE Transactions on Geoscience and Remote Sensing and IEEE Transactions on Geoscience Electronics vol.32, 2 (1994), p. 382-387, Download PDF
9. Flusser Jan, Saic Stanislav, Suk Tomáš:Registration of images with affine geometric distortion by means of moment invariants, Image and Signal Processing for Remote Sensing, p. 843-852, Eds: Desachy J., SPIE, (Bellingham 1994), European Symposium on Satellite Remote Sensing, (Roma, IT, 26.09.1994-30.09.1994)
10. Flusser Jan, Suk Tomáš, Šimberová Stanislava: An Experimental Study on Uniqueness of the Affine Moment Invariants, Czech Pattern Recognition Workshop '93, p. 119-125, Eds: Hlaváč V., Pajdla T., Czechoslovak Pattern Recognition Society, (Praha 1993), Czech Pattern Recognition Workshop '93, (Temešvár, CZ, 04.11.1993-06.11.1993)
11. Flusser Jan, Suk Tomáš:Pattern Recognition by Affine Moment Invariants, Pattern Recognition vol.26, 1 (1993), p. 167-174, Download PDF
12. Flusser Jan, Suk Tomáš:Rozpoznávání objektů pomocí afinních momentových invariantů, ÚTIA ČSAV, (Praha 1991)Research Report 1726

Relevant publications by other authors:

1. D. Shen and H. H. S. Ip, Generalized affine invariant image normalization, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 5, pp. 431–440, 1997.
2. D. Hilbert, Theory of Algebraic Invariants. Cambridge University Press, 1993.
3. T. H. Reiss, Recognizing Planar Objects using Invariant Image Features, vol. 676 of LNCS. Berlin: Springer, 1993.