This demo computes a spline-based radon transform.
First, the input image is approximated in a spline space, that means, as a weighted sum of shifted 2D splines of degree DegOnImage. The Radon transform of a 2D tensor-product spline is a 1D convolution of two scaled splines. The approximation of this continuous 1D function in the Dual-spline space of degree DegOnSino of the projection is obtained by a further convolution with a B-spline. Therfore, this spline-based Radon transform involves the compution of so called trikernels (convolution of three B-spline of different widths and degrees).
A similar algorithm (which also involves computation of tri-kernels) is demonstrated for the filtered back-projection.
On the other hand, a more conventional filtered back-projection algorithm is shown, with the standard filters (RamLak, Shepp-Logan, Cosine, Hamming, etc...). There are five modified filters, (B-Spline, Oblique, Haar, DualSpline, CardinalSpline) which all act optimally in the least-square sense, respectively to their underlying approximation space. |
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Description |
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Animation |
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Example |
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Instruction to use the applet
First, choose an input image, then you can set the Spline-based Radon Transform parameters:
Number of angles at which the projections are performed (angle are equally spaced between 0 and pi), the degree of the spline space of the image, the degree of the spline-space on the sinogram.
By Pressing the "Radon Transform" button you launch the computation of the Radon Transform and the resulting sinogram comes up to the right. (A column of the sinogram represents the Radon transform for a given angle)
By Pressing the Button "Add Noise" you can add noise to the Sinogram.
Then you can set the reconstruction parameters: choose the method (spline-based or standard), the filter, the respective spline-space degree on sinogram and image (spline-based algorithm), the filter, the Interpolation degree for the
backprojection, (standard algorithm). For the Dual and the cardinal
filters one can also define the spline degree on the image.
By Pressing the Button "Back Projection" the reconstruction is performed, and the difference image between reconstructed image and the original is computed.s.
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