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 Polynomial approximation of 

2D signal Pi19404

February 6, 2014

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Contents

Contents

 

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

 0.2 Polynomial Approximation of 2D signal   . . . . . . . . . . . . . .  

 0.3 2D projection   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

 References   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  

 

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 Polynomial approximation of 2D signal

 Polynomial approximation of 2Dsignal

 0.1 Introduction

In this article we will look at the concept for polynomial expansion to approximate a neighborhood of a pixel with a polynomial.

 0.2 Polynomial Approximation of 2D signal

  The idea of polynomial expansion is to approximate a neighbor- hood of a point in a 2D function with a polynomial.

  Considering only a quadratic polynomial ,pixel values in a neigh-borhood is given by

         

 

     

 

     

where A is a symmetric matrix,b is a vector and c is a scalar

  The coefficients can be estimated by weighted least square es- timate of pixel values about neighborhood.

  Let us consider a polynomial of order 2.The basis functions of the subspace where the local signal is being approximated are

   

 

 

 

 

  The basis function are defined on the discrete grid of -N to N considering a window/neighborhood of  

 

     ,for the

present example let N=7;

 0.3 2D projection

   We need to project a 2D function onto these basis

   As in the case on 1D projection we simply,take the innerproductof the given signal with the basis

   

 

 

 

 

 

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 Polynomial approximation of 2D signal

 (a) 2D basis function

 (b) 2D gaussian function

 

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 Polynomial approximation of 2D signal

  To compute the projection parameters ,one can directly take the inner production of the signal with each of the basis.

   We also need to consider a weighing function ,in the present

 example we consider a 2D gaussian with variance of 1 along the x and y direction.

   We get the projection matrix as follows.

  

 

 

 

 

 

 

  

           

     

     

           

           

    

 

 

 

 

 

 

 

 

 

   Again we use the same method used in the 1D case

     

 

  

 

 

 

 

 

    

   

where B is the basis vector and W is gaussian weights

 

 

 

    

 will give us a scalar for each of the basis function

  This gives us the projection of the signal onto the basis func- tion

  The inverse transformation gives us the mean square estimateof the polynomial function in the local neighborhood about thepoint.

  In the below function we are obtaining the polynomial estimateof the function

   

 

     

  The below matlab sript computes the 2D polynomial approxima- tion

   At        of result array we get the coefficients                    

This corresonds to result translated by 3.

   Replacing x by   

   we obtain the desired coefficients                    

 

 

 

 

 

 

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 Polynomial approximation of 2D signal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 Bibliography

 Bibliography