polynomial approximation of 2d image patch -part 2

8/13/2019 Polynomial Approximation of 2D image patch -Part 2

1/14

PolynomialApproximation of

2D image -Part 2Pi19404

February 18, 2014


2/14

Contents

Contents

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.1.1 Poynomial Interpolation . . . . . . . . . . . . . . . . . . . . . 0.1.2 Code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.2 code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


3/14

Polynomial Approximation of 2D image -Part 2

Polynomial Approximation of 2Dimage -Part 2

0.1 Introduction

In this article we will look at polynomial approximation of image neigh-borhood and efficient implementation in C/C++ OpenCV platform.

In the earlier articles we looked at how to represent a imageregion using quadratic polynomials

Here we will look at efficient way to compute the polynomialbasis representation of an image.

The concept of seperable symmetric/anit symmetric convolutionfilters described in the earlier articles will be also useful.

The first step is the computation of seperable filter coeffi-cients

The basis functions are as shown in figure 1a

0.1.1 Poynomial Interpolation

Let us assume that the neighborhood is approximated by a quadraticpolynomial

We known the value of ie the image values at all pointsin the image

Let us consider a rectangular neighborhood of size

abouta point.

The aim is to find the coefficients

for this neigh-borhood.

The x and y value ranges from


4/14


(a) 2D basis function


5/14


For each eg we can equate it with gen-eral form of

to get a equation.

These set of equations can be written in form of the matrix

In general it can be written as

This can be expressed as

The matrix can be expressed as

This matrix simply compute the projection of basis onto image


6/14


neighborhood being considered

This can be efficiently computed by performining 6 convolutionoperations on the image

The basis matrix for convolution are

are

The matries are displayed below of a neighhborhood of 7 which


7/14


are matrices.

Each of these filters can be expressed as product of 2 matrices


8/14


R,C where R is row vector and C is a column vector

All the filters are weighted by gaussian kernel that gives moreimportance to pixels near the origin,and can be considered asweighted interpolation.

0.1.2 Code

The source code can be found at git repository

To realize the basis function ,seperable filter for a size 7 areshown below
http://localhost/var/www/apps/conversion/tmp/scratch_3/


9/14


a dot product of

kernels specified above will provided

a 2D kernel corresponding to basis function specfied in the

figure 1a based on the desired size of the neighborhood beingapproximated.

It can be noted that the filter coefficients are symmetric andare either symmetic or anti symmetric and are all of the samesize.

The function to generate the filter coeffients is implementedby the function


10/14


The result of the various filters can be computed by making asingle pass over the entire image

The class is used for achieving this.

The details of seperable convolution using symmetric/anti-symmetricfilters can be found in the article

The polyBasis class implements the polynomial basis computation

The method implements the com-

putation of seperable filter coefficients.

The last step in estimation required is the matrix

The matrix is 6xN matrix
http://localhost/var/www/apps/conversion/tmp/scratch_3/


11/14


12/14


this implies

for a 7x7 neighborhood size we get

This computation is performed by the function

The matrix is called as Vandermonde matrix .

The last step remaining is normalization of the filter coef-ficients as a result of Least sequares estimation process thatrequire application of the matrix on each pixel of multi chan-

nel image.

It can be noted that result of applying the kernel on vectorformed by the channel of length N results in a matrix of length

N.

It can also be noted that the transformation is linear.

This is achieved using the class that can be foundin the article


13/14


Thus a modular and efficient method to compulte the local poly-nomial basis of image is provided in the article.

0.2 code

The code for the same can be found in the git repository

in files ImgBasis/polybasis.cpp andImgBasis/polybasis.hpp files.
https://github.com/pi19404/OpenVisionhttps://github.com/pi19404/OpenVision


14/14

Bibliography

Bibliography
http://dblp.uni-trier.de/db/journals/sigpro/sigpro87.html#AnderssonWK07http://dblp.uni-trier.de/db/journals/sigpro/sigpro87.html#AnderssonWK07http://dblp.uni-trier.de/db/conf/icmcs/icme2002-1.html#AnderssonK02http://dblp.uni-trier.de/db/conf/icmcs/icme2002-1.html#AnderssonK02

polynomial approximation of 2d image patch -part 2

Documents