A Tutorial on Image Moments

8/12/2019 A Tutorial on Image Moments

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A TUTORIAL

ON

IMAGE MOMENTS

BY

ANIL SUDHAKAR KURHEKAR


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1. To Compute Centroidof an imageThe centroid is based on pixel intensity. The centroid, located at pixel

coordinates (x0,y0) is calculated using the following equation:

Where mpqis given by

N is the horizontal dimensionality of the image and M is the vertical

dimensionality of the image.

Is the pixel intensity at the location (x,y)

Center is at 270.046 248.303


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2. COMPUTE MAXIMUM RADIUSCompute maximum distance of all pixels from the centroid of the image.Alternatively, it is the radius of the the smallest disk that covers the entire

image with its origin at the centroid. The radius is determined by:

Where mpqis given by

MAXIMUM RADIUS =376.7252

3.Compute Image TransformThis transformation is performed by finding the Euclidean coordinates of

points which are contained within the smallest circle with origin at the

centroid or center of image which can be contained within the image

boundaries. The pixels whose coordinates most closely match these

Euclidean coordinates are plotted at the corresponding polar coordinates.

At each radius, the perimeter is divided into radius * resolution equally

spaced point. Then, the Euclidean coordinates of the points will be used in

the transformation. This results in a upper-triangular matrix. This

transform was created to approximate rotational invariance for

orthogonal moments. The inverse transform may be calculated.


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4. Find the maximum order of finite momentsFunction to find the maximum order (and repetition for complex

moments) of moments such that all moment values up to that order (in

both the x and y directions) are finite and not NaN values. The maximum

order will be at most the maximum order represented in the matrix of

moment values. This function looks for the largest size rectangular block

in the moment matrix such that no values are infinity or NaN.

Matrix containing image moments from order 0 up to some order p in the

x- direction and q in the y-direction, or in the case of complex moments, up

to order p and repetition q.

The maximum order of finite moments for Lena=300,200

5. Complex Image ClassThis class contains an image and all information about the image required

for computing the moments of the image. The object has methods for

computing moments, moment invariants, and reconstructing the image

from moments. The types of moments that can be calculated using thisobject are: generalized Pseudo-Zernike, Fourier Mellin, Fourier

Chebyshev, and Radial Harmonic Fourier.

1. Moments


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2. Invariant


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Example-II

The zerothMoment


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The First Moment

The Second Moment

The 14thmoment


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7. The Chebyshev Moments of an Image

8. Bivariate Legendre/Gegenbauer moments up to orders 50and 100 with parameter 1 for Gegenbauer moments


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9.Radial Harmonic-Fourier moment invariants of Image usingup to order 10 and repetition 10


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10. Continuous Chebyshev MomentsExample-I


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Example-II


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11. Generalize Pseudo Zernike PolynomialFor generalized Pseudo-Zernike moments, a single parameter is required,

. A larger value of

decreases the range of the polynomial values used for moment

computation.

For Gegenbauer moments, a single parameter is required,

with the constraints

> 1

and

~ = 0

.A larger value of

increases the values of the polynomials used for moment calculation. If

= 1

, the moments are equivalent to continuous Chebyshev moments. For

Hahn moments, two parameters are required,

and

c

with the constraints

> -1/2

and> abs(c) -1

.

specifies where the polynomials are centered and the difference between

and c is positively correlated to the range of the polynomial values used

to calculate moments.

For Krawtchouk moments, a single parameter is required,

with the constraint

0 < < 1.

specifies where the polynomials are centered, relative to the image

centroid.


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Documents

A Tutorial on Image Moments