polynomial approximation of a 2d signal
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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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Polynomial approximation of 2D signal
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Bibliography
Bibliography