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### Highlights from Regular Control Point Interpolation Matrix with Boundary Conditions

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# Regular Control Point Interpolation Matrix with Boundary Conditions

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### Matt J (view profile)

06 Jan 2010 (Updated )

Creates Toeplitz-like matrices representing interpolation operations with edge conditions.

File Information
Description

The main file interpMatrix.m in this package creates a sparse Toeplitz-like matrix representing a regularly-spaced interpolation operation between a set of control points. The user can specify the interpolation kernel, the number of control points, the spacing between the control points, and certain boundary conditions governing the behavior at the first and last control point.

The tool has obvious applications to interpolation, curve fitting, and signal reconstruction. More generally, the ability to represent interpolation as a matrix is useful for minimizing cost functions involving interpolation operations. For such functions, the interpolation matrix and its transpose inevitably arise in the gradient.

The file Example1D.m in the package gives an example application of the tool to upsampling/signal reconstruction using cubic B-splines with different possible boundary conditions. The screenshot above shows the output of this example, and illustrates how improved signal reconstruction is obtained using boundary extrapolation by mirroring.

Although the matrix generated by interpMatrix() is for 1D interpolation, it can be generalized to n-dimensional tensorial interpolation using kron(). However, a more efficient alternative to kron() is this tool,

http://www.mathworks.com/matlabcentral/fileexchange/25969-efficient-object-oriented-kronecker-product-manipulation

whose usage in conjunction with interpMatrix() is illustrated in the file Example2D.m, a generalization of Example1D.m to two dimensions.

USAGE:

T=interpMatrix(kernel, origin, numCtrlPoints, CtrlPointSep, extraprule)

out:

T: sparse output matrix. The columns of T are copies of a common interpolation
kernel (with adjustments for boundary conditions), but shifted in
increments of the CtrlPointSep parameter (see below) to different control
point locations. The result is that if x is a vector of coefficients,
then T*x is the interpolation of these coefficients on the interval
enclosed by the control points.

in:

kernel: vector containing samples of an interpolation function, shifted copies
of which will be used to create the columns of T. This vector never
the other input arguments below.

origin: Index i such that kernel(i) is located at the first control point.
It is also possible to specify the origin using the following
string options:

'max': origin i will be selected where kernel(i) is maximized.
'ctr': origin i will be selected as ceil((length(kernel)+1)/2).

numCtrlPoints: number of control points in the system.

CtrlPointSep: a stricly positive integer indicating the number of samples between
control points.

extraprule: Initially, the shifted copies of "kernel" form the columns of T. The columns are then modified to satisfy edge conditions indicated by the "extraprules" parameter. Options for this parameter are the strings 'zero', 'mirror','rep', 'circ', or 'allcontrib'. These are explained in the help doc.

MATLAB release MATLAB 7.9 (R2009b)
13 Apr 2015 Michal Kvasnicka

### Michal Kvasnicka (view profile)

Thanks Matt ... nice code (effective, well documented)

07 Jan 2010 1.1

Minor clarifications to summary and description section.

11 Jan 2010 1.2

Small bug fix. The 'allcontrib' option for the "extraprules" input parameter led to extra columns of zeros in output matrix.

14 Jan 2010 1.3

Small edit to Description section

06 Feb 2010 1.4

Removed for-loops from the implementation of interpMatrix.

12 Sep 2010 1.5

Added new options for the "origin" input argument. One can now specify it using a string 'max', which will set the origin automatically to where the kernel is maximum. The string 'ctr' will set it to the center of the kernel profile.

20 Sep 2010 1.6

Edited description only. No new code.

08 Nov 2010 1.7

Small update of Example1D.m and Example2D.m to use the new origin='max' feature. No change to interpMatrix.m