EigenMoments

EigenMoments is a set of orthogonal, noise robust, invariant to rotation, scaling and translation and distribution sensitive moments. Their application can be found in signal processing and computer vision as descriptors of the signal or image. The descriptors can later be used for classification purposes.
It is obtained by performing orthogonalization, via eigen analysis on geometric moments.
Framework summary.
EigenMoments are computed by performing eigen analysis on the moment space of an image by maximizing Signal to Noise Ratio in the feature space in form of Rayleigh quotient.
This approach has several benefits in Image processing applications:
Problem formulation.
Assume that a signal vector formula_1 is taken from a certain distribution having coorelation formula_2, i.e. formula_3 where E[.] denotes expected value.
Dimension of signal space, n, is often too large to be useful for practical application such as pattern classification, we need to transform the signal space into a space with lower dimensionality.
This is performed by a two-step linear transformation:
formula_4
where formula_5 is the transformed signal, formula_6 a fixed transformation matrix which transforms the signal into the moment space, and formula_7 the transformation matrix which we are going to determine by maximizing the SNR of the feature space resided by formula_8. For the case of Geometric Moments, X would be the monomials. If formula_9, a full rank transformation would result, however usually we have formula_10 and formula_11. This is specially the case when formula_12 is of high dimensions.
Finding formula_13 that maximizes the SNR of the feature space:
formula_14
where N is the correlation matrix of the noise signal. The problem can thus be formulated as
formula_15
subject to constraints:
formula_16 where formula_17 is the Kronecker delta.
It can be observed that this maximization is Rayleigh quotient by letting formula_18 and formula_19 and therefore can be written as:
formula_20, formula_21
Rayleigh quotient.
Optimization of Rayleigh quotient has the form:
formula_22
and formula_23 and formula_24, both are symmetric and formula_24 is positive definite and therefore invertible.
Scaling formula_26 does not change the value of the object function and hence and additional scalar constraint formula_27 can be imposed on formula_26 and no solution would be lost when the objective function is optimized.
This constraint optimization problem can be solved using Lagrangian multiplier:
formula_29 subject to formula_30
formula_31
equating first derivative to zero and we will have:
formula_32
which is an instance of Generalized Eigenvalue Problem (GEP).
The GEP has the form:
formula_33
for any pair formula_34 that is a solution to above equation, formula_26 is called a generalized eigenvector and formula_36 is called a generalized eigenvalue.
Finding formula_26 and formula_38 that satisfies this equations would produce the result which optimizes Rayleigh quotient.
One way of maximizing Rayleigh quotient is through solving the Generalized Eigen Problem. Dimension reduction can be performed by simply choosing the first components formula_39, formula_40, with the highest values for formula_41 out of the formula_42 components, and discard the rest. Interpretation of this transformation is rotating and scaling the moment space, transforming it into a feature space with maximized SNR and therefore, the first formula_43 components are the components with highest formula_43 SNR values.
The other method to look at this solution is to use the concept of simultaneous diagonalization instead of Generalized Eigen Problem.
Simultaneous diagonalization.
formula_48
formula_50.
Where formula_51 is a diagonal matrix sorted in increasing order. Since formula_24 is positive definite, thus formula_53. We can discard those eigenvalues that large and retain those close to 0, since this means the energy of the noise is close to 0 in this space, at this stage it is also possible to discard those eigenvectors that have large eigenvalues.
Let formula_54 be the first formula_43 columns of formula_56, now formula_57 where formula_58 is the formula_59 principal submatrix of formula_51.
formula_61
and hence:
formula_62.
formula_49 whiten formula_24 and reduces the dimensionality from formula_42 to formula_43. The transformed space resided by formula_67 is called the noise space.
formula_69,
where formula_70. formula_71 is the matrix with eigenvalues of formula_68 on its diagonal. We may retain all the eigenvalues and their corresponding eigenvectors since the most of the noise are already discarded in previous step.
formula_73
where formula_13 diagonalizes both the numerator and denominator of the SNR,
formula_75, formula_76 and the transformation of signal formula_77 is defined as formula_78.
Information loss.
To find the information loss when we discard some of the eigenvalues and eigenvectors we can perform following analysis:
formula_79
Eigenmoments.
Eigenmoments are derived by applying the above framework on Geometric Moments. They can be derived for both 1D and 2D signals.
1D signal.
If we let formula_80, i.e. the monomials, after the transformation formula_81 we obtain Geometric Moments, denoted by vector formula_82, of signal formula_83,i.e. formula_84.
In practice it is difficult to estimate the correlation signal due to insufficient number of samples, therefore parametric approaches are utilized.
One such model can be defined as:
formula_85,
where formula_86. This model of correlation can be replaced by other models however this model covers general natural images.
Since formula_87 does not affect the maximization it can be dropped.
formula_88
The correlation of noise can be modelled as formula_89, where formula_90 is the energy of noise. Again formula_90 can be dropped because the constant does not have any effect on the maximization problem.
formula_92
formula_93
Using the computed A and B and applying the algorithm discussed in previous section we find formula_13 and set of transformed monomials formula_95 which produces the moment kernels of EM. The moment kernels of EM decorrelate the correlation in the image.
formula_96,
and are orthogonal:
formula_97
Example computation.
Taking formula_98, the dimension of moment space as formula_99 and the dimension of feature space as formula_100, we will have:
formula_101
and
formula_102
2D signal.
The derivation for 2D signal is the same as 1D signal except that conventional Geometric Moments are directly employed to obtain the set of 2D EigenMoments.
The definition of Geometric Moments of order formula_103 for 2D image signal is:
formula_104.
which can be denoted as formula_105. Then the set of 2D EigenMoments are:
formula_106,
where formula_107 is a matrix that contains the set of EigenMoments.
formula_108.
EigenMoment invariants (EMI).
In order to obtain a set of moment invariants we can use normalized Geometric Moments formula_109 instead of formula_82.
Normalized Geometric Moments are invariant to Rotation, Scaling and Transformation and defined by:
formula_111
where:formula_112 is the centroid of the image formula_113 and
formula_114.
formula_115 in this equation is a scaling factor depending on the image. formula_115 is usually set to 1 for binary images.