harmonic_interpolation — Perform a harmonic interpolation on an image region.
harmonic_interpolation(Image, Region : InpaintedImage : Precision : )
The operator harmonic_interpolation reconstructs the
destroyed image data of Image inside the region
Region by solving the discrete Laplace equation
for the corresponding
gray value function u. The unique solution, which exists under
Dirichlet boundary conditions given by Image outside of
Region, is returned in InpaintedImage.
This technique is called harmonic interpolation since in function theory the solutions of the Laplace equation are referred to as harmonic functions.
If Region touches the border of the gray value matrix of
Image and thus some Dirichlet boundary values do not exist,
von Neumann boundary conditions are used instead. This means that
the gray values are mirrored at the border of Image. If no
Dirichlet boundary values exist at all, a constant image with gray
value 0 is returned.
The spatial derivatives are discretized as
and
.
The equation is solved by an
iterative conjugate gradient solver, which iteratively improves the
computational error until the maximum norm of its update step
becomes a smaller fraction than Precision of the norm of
the input data or a maximum of 1000 iterations is
reached. Precision = 0.01 thus means a relative
computational accuracy of 1%.
Note that filter operators may return unexpected results if an image with a reduced domain is used as input. Please refer to the chapter Filters.
Image (input_object) (multichannel-)image(-array) → object (byte / uint2 / real)
Input image.
Region (input_object) region → object
Inpainting region.
InpaintedImage (output_object) image(-array) → object (byte / uint2 / real)
Output image.
Precision (input_control) real → (real)
Computational accuracy.
Default value: 0.001
Suggested values: 0.0, 0.0001, 0.001, 0.01
Restriction: Precision >= 0.0
inpainting_ct,
inpainting_aniso,
inpainting_mcf,
inpainting_texture,
inpainting_ced
L.C. Evans; “Partial Differential Equations”; AMS, Providence;
1998.
W. Hackbusch; “Iterative Lösung großer schwachbesetzter
Gleichungssysteme”; Teubner, Stuttgart;1991.
Foundation