Image analysis by moment invariants using a set of step-like basis functions
Introduction
Moment invariants have become a major topic in image description research from their initial proposal by Hu (1962). From then on, many attention has been paid to improve the theoretical basis of their definition and to generate new and better ways of building such descriptors (Reiss, 1991, Flusser and Suk, 2009). One of the main streams in this research has been to define moment invariants using orthonormal basis, given the links between them and Fourier decompositions postulated by Hilbert algebra; according this theoretical corpus, in addition to the advantages of using invariants, orthonormal basis offer optimal image description and reconstruction, in terms of computational effort (Rudin et al., 1991, Teague, 1980). In this way, many works have been devoted to explore the available orthonormal basis and their performance as image descriptors, setting comparative tests among them (Teh and Chin, 1988), or exploring their specific capabilities (Khotanzad and Hong, 1990). Given the description power that this kind of descriptors have shown for 2D images, some efforts have been devoted even to extend descriptions to 3D objects (Xu and Li, 2008). Recent efforts have been routed to mix the power of moment invariants with the analysis capabilities of waveletes, like (Chen et al., 2011).
In this paper, a set of orthonormal functions for moment invariants analysis is defined, being its main distinctive feature that they are discontinuous in a finite set of points, contrary to other well known moment sets, as Zernike or Legendre. This feature has been introduced in order to get a set that is well suited to analyse signals with discontinuities as well, like images have (Lee and Tarng, 1999), that originate undesired effects like Gibbs phenomenon when continuous functions are used to analyse them (Hewitt and Hewitt, 1979); it has been reported to generate visible artifacts in image reconstruction after filtering, as well (Bovik et al., 2009).
According to Hilbert Algebra, orthonormal basis have the property that descriptors obtained by projecting an image over each basis function contains exclusive information, i.e. it is not contained in any other descriptor of the series. This fact allows, in one hand, having the more compact representation in terms of descriptor series length, and on the other that images can be easily reconstructed just by combining their moment values, providing that the subspace spanned by the basis elements is dense (Rudin et al., 1991).
The goal of this work is, therefore, to generate a set of orthonormal functions that yields an image decomposition gathering the following properties:
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is well suited for analyzing 2D functions that may include step discontinuities, images in this case. It has been proved that discontinuous basis, like Haar, are well suited to analyze discontinuous signals.
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it allows the definition of rotation invariants, in order to generate the same descriptors for rotated instances of the same object.
In order to test the achievements of this work, results have been compared to those reached using Zernike moments; there are many works supporting the idea that Zernike moments are among the best suited sets of moments for image analysis (Teh and Chin, 1988, Chen et al., 2011).
Section snippets
Derivation of the basis functions
The general definition of moments is the following: given an image , its moment is computed as (Flusser and Suk, 2009):where is a function which belongs to the moment basis, having orders n for the x component and m for the y, and and are the limits of the region where both the image and are defined.
Note that according to Hilbert Algebra, Eq. (1) can be interpreted as the inner product between both functions,
Noise sensitivity
In real world applications images are seldom completely clean, having arbitrary amounts of noise. In this section, the proposed family of invariants is tested against images corrupted with different amounts of additive Gaussian noise, in order to check their ability to preserve image information from noise, following the experiment described in Teh and Chin (1988). For the sake of providing a baseline, results are compared to those obtained using Zernike moments under the same conditions.
Results in image retrieval
The proposed family of invariants has been used as image descriptor in two image retrieval tasks in order to test it. Given the results shown in Section 3, their order has been kept very low, using the ranks , which sum up a total of 56 descriptors per image. Zernike moments have been extracted up to 17th order, that according with (Teh and Chin, 1988, Mukundan, 2004) represents a good tradeoff between quality of the representation and noise sensibility, giving a total of 90
Conclusions
In this paper, a set of invariants has been introduced, having the same properties and description power as Zernike moments but requiring much shorter descriptions to achieve these results. These new invariants have been derived from the concepts of Hilbert space and Hilbert basis, and share with Zernike invariants the properties of rotation invariance and orthonormality, that yields optimal descriptions of the images to be analyzed.
The new set of moments has been tested against additive
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