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ZERNIKE MOMENTS FOR IMAGE

CLASSIFICATION

Sandeep kumar (001)

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The use of moments for image analysis and pattern recognition was inspired by Hu[4].

Hu[4] stated that if f(x, y) is piecewise continuous and has nonzero values only in a finite region of the (x, y) plane, then the moment sequence is uniquely determined by f(x, y) and conversely f(x,y) is uniquely determined by

Hu’s[4] Uniqueness Theorem states that if f(x,y) is piecewise continuous and has nonzero values only in the finite part of the (x,y) plane. then geometric moments of all orders exist.

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REGULAR (CARTESIAN) MOMENTS

A regular moment has the form of projection of onto the monomial function

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PROBLEMS OF REGULAR MOMENTS

The basis set is not orthogonal The moments contain redundant information.

As increases rapidly as order increases, high computational precision is needed.

Image reconstruction is very difficult.

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BENEFITS OF REGULAR MOMENTS

Simple translational and scale invariant properties

By preprocessing an image using the regular moments we can get an image to be translational and scale invariant before running Zernike moments

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ORTHOGONAL FUNCTIONS

• A set of polynomials orthogonal with respect to integration are also orthogonal with respect to summation.

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ORTHOGONAL MOMENTS Moments produced using orthogonal basis sets.

Require lower computational precision to represent images to the same accuracy as regular moments.

Requirement for Zernike Zernike is defined on a unit disk. So, for an

image, the disk takes its center at centroid of the image with radius being minimal , that the support is enclosed[6].

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ZERNIKE POLYNOMIALS

• Set of orthogonal polynomials defined on the unit disk(orthogonal complex rotational moment).

• Constraints : m <= n & n-m = even

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ZERNIKE MOMENTS

Simply the projection of the image function onto these orthogonal basis functions.

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ADVANTAGES OF ZERNIKE MOMENTS

Simple rotation invariance Higher accuracy for detailed shapes Orthogonal

Less information redundancy (moments are uncorrelated).

Much better at image reconstruction (vs. normal moments).

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SCALE AND TRANSLATIONAL INVARIANCE

Scale:

• Image normalization by dividing with center of mass m0 (or m00).� This allows one to compute moments which depend only on the shape and not the magnitude of f(x)

Translational: Shift image’s origin to centroid (computed from normal first order moments)

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ROTATIONAL INVARIANCE

The magnitude of each Zernike moment is invariant under rotation.

Coefficient of variance income example by chocobar vs airplane

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IMAGE RECONSTRUCTION

Orthogonality enables us to determine the individual contribution of each order moment.

Simple addition of these individual contributions reconstructs the image.

Nmax highest order

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IMAGE RECONSTRUCTIONReconstruction of a crane shape via Zernike moments up to order 10k-5, k = {1,2,3,4,5}.

(a) (b)

(c) (d)

(e) (f)

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DETERMINING MIN. ORDER

After reconstructing image up to moment i1. Calculate the Hamming distance, H( fe, f )

which is the number of pixels that aredifferent between fe and f .

2. Since, in general, H( fe, f ) decreases as i increases, finding the first for which

H( fe, f ) < ϵ will determine the minimum order to reach a predetermined accuracy.

fe estimated image,f original image

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Zernike suffer from high computation cost and numerical instability at high order of moments.

There are three recursive methods which are normally used in ZMs calculation-Prata's, Kintner's and q-recursive methods.

The earlier studies have found the q-recursive method outperforming the two other methods.

The modified Prata's method for low orders (<90) and Kintner's fast method for high orders (>90) of ZMs. Are recommende & for further description refer paper by Chandan singh & Ekta Walia[2]

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CONCLUSION

Zernike moments have rotational invariance, and can be made scale and translational invariant, making them suitable for many applications.

Zernike moments are accurate descriptors even with relatively few data points.

Reconstruction of Zernike moments can be used to determine the amount of moments necessary to make an accurate descriptor.

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FUTURE RESEARCH

Apply Zernike’s to supervised classification problem

Make hybrid descriptor which combines Zernike’s and contour curvature to capture shape and boundary information

But, it has been shown that high order ZM's are not useful for clustering purposes because they capture too much noise as published in a paper for binary leaf classification as published by Michael Vorobyov[5].

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REFERENCES 

1. Image Analysis by Moments by Simon Xinmeng Liao

A Thesis Submitted to the Faculty of Graduate Studies

in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy The Department of Electrical and Computer Engineering The University of Manitoba

 2. Fast and numerically stable methods for the computation of Zernike moments by Chandan singh & Ekta Walia

 3. F_L_ Alt_ Digital pattern recognition by moments.J.Assoc. Computing Machinery. Vol 9 

4. M_K_ Hu_ Visual problem recognition by moment invariant.IRE Trans. Inform.Theory, Vol.IT 8 

5. Shape Classification Using Zernike Moments

Michael Vorobyov iCamp at University of California Irvine August 5, 2011

6. Zernike moments by Patrick C Hew university of west Australia 1996