Auswertung Experiment

Database tüddelei
DBHanlder läuft gut für DIESE Datanbank struktur...
App begonnen aber weit entfernt von gutem Stand

Libs/boundedlines/Inpaint_nans/doc/Nomination comments.rtf

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+
+\f0\fs24 \cf0 Nomination comments:\
+\
+Inpaint_nans fills a hole in matlab. (Yes, the pun was intentional.) But there\
+is indeed a niche that inpaint_nans falls into.\
+\
+The alternative to inpaint_nans is griddata (interp1 can be used for the 1-d \
+problems) but griddata fails to extrapolate well. Griddata also has serious\
+problems when its data already lies on a grid, due to its use of a Delaunay \
+triangulation. The other serious problem with the use of griddata is the\
+triangulation itself. The shape of the hole to be filled can sometimes result\
+in triangles with a poor aspect ratio (long, thin triangles) which are in turn\
+poor for interpolation. In fact, Griddata can even leave interior points\
+uninterpolated (see the tests.)\
+\
+A future plan for inpaint_nans is to add an option that will use a locally\
+anisotropic membrane model. This will allow better modeling for certain\
+classes of wavy surfaces. I'm also highly tempted to remove method 5.\
+I've never really liked it, having put it in at the request of one user. It has\
+no valid theory behind it in the context of inpaint_nans.\
+\
+In the interest of openness, I'll also say what inpaint_nans does not do. It\
+does not handle non-uniform grids. It is limited by the amount of memory \
+in the size of the arrays it can handle, although some of the methods were\
+explicitly provided to be more memory efficient than others. Inpaint_nans\
+also makes heavy use of sparse matrices, so surprisingly large problems\
+are accessible.\
+\
+Finally, while inpaint_nans does work for 1-d problems, they are not my\
+target. Interp1 (with 'spline' as the method) is as accurate, and should be\
+faster in general.\
+\
+John\
+}

Libs/boundedlines/Inpaint_nans/doc/methods_of_inpaint_nans.m

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+%{
+
+The methods of inpaint_nans
+
+Digital inpainting is the craft of replacing missing elements in an
+"image" array. A Google search on the words "digita inpainting" will turn 
+up many hits. I just tried this search and found 18300 hits.
+
+If you wish to do inpainting in matlab, one place to start is with my
+inpaint_nans code. Inpaint_nans is on the file exchange:
+
+http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=4551&objectType=file
+
+It looks for NaN elements in an array (or vector) and attempts to interpolate
+(or extrapolate) smoothly to replace those elements.
+
+The name "inpainting" itself comes from the world of art restoration.
+Damaged paintings are restored by an artist/craftsman skilled in matching
+the style of the original artist to fill in any holes in the painting.
+
+In digital inpainting, the goal is to interpolate in from the boundaries
+of a hole to smoothly replace an artifact. Obviously, where the hole is
+large the digitally inpainted repair may not be an accurate approximation
+to the original.
+
+Inpaint_nans itself is really only a boundary value solver. The basic idea
+is to formulate a partial differential equation (PDE) that is assumed to
+apply in the domain of the artifact to be inpainted. The perimeter of the
+hole supplies boundary values for the PDE. Then the PDE is approximated
+using finite difference methods (the array elements are assumed to be
+equally spaced in each dimension) and then a large (and very sparse) linear
+system of equations is solved for the NaN elements in the array.
+
+I've chosen a variety of simple differental equation models the user can
+specify to be solved. All the methods current use a basically elliptic
+PDE. This means that the resulting linear system will generally be well
+conditioned. It does mean that the solution will generally be fairly smooth,
+and over large holes, it will tend towards an average of the boundary
+elements. These are characteristics of the elliptic PDEs chosen. (My hope
+is to expand these options in the future.)
+
+%}
+
+%%
+
+% Lets formulate a simple problem, and see how we could solve it using
+% some of these ideas.
+A = [0 0 0 0;1 NaN NaN 4;2 3 5 8];
+
+% Although we can't plot this matrix using the functions surf or mesh,
+% surely we can visualize what the fudamental shape is.
+
+% There are only two unknown elements, the artifacts that inpaint_nans
+% would fill in: A(2,2) and A(2,3).
+
+% For an equally spaced grid, the Laplacian equation (or Poisson's equation
+% of heat conduction at steady state if you prefer. Or, for the fickle,
+% Ficke's law of diffusion would apply.) All of these result in the PDE
+%
+%   u_xx + u_yy = 0
+%
+% where u_xx is the second partial derivative of u with respect to x,
+% and u_yy is the second partial with respect to y. 
+%
+% Approximating this PDE using finite differences for the partial
+% derivatives, implies that at any node in the grid, we could replace
+% it by the average of its 4 neighbors. Thus the two NaN elements
+% generate two linear equations:
+%
+%  A(2,2) = (A(1,2) + A(3,2) + A(2,1) + A(2,3)) / 4
+%  A(2,3) = (A(1,3) + A(3,3) + A(2,2) + A(2,4)) / 4
+%
+% Since we know all the parameters but A(2,2) and A(2,3), substitute their
+% known values.
+%
+%  A(2,2) = (0 + 3 + 1 + A(2,3)) / 4
+%  A(2,3) = (0 + 5 + A(2,2) + 4) / 4
+%
+% Or,
+%
+%  4*A(2,2)  - A(2,3) = 4
+%  -A(2,2) + 4*A(2,3) = 9
+%
+% We can solve for the unkowns now using 
+u = [4 -1;-1 4]\[4;9]
+
+A(2,2) = u(1);
+A(2,3) = u(2);
+
+% and finally plot the surface
+close
+surf(A)
+title 'A simply inpainted surface'
+
+% Neat huh? For an arbitrary number of NaN elements in an array,
+% the above scheme is all there is to method 2 of inpaint_nans,
+% together with a very slick application of sparse linear algebra
+% in Matlab.
+
+% Method 0 is very similar, but I've optimized it to build as
+% small a linear system as possible for those cases where an array
+% has only a few NaN elements.
+
+% Method 1 is another subtle variation on this scheme, but it
+% tries to be slightly smoother at some cost of efficiency, while
+% still not modifying the known (non-NaN) elements of the array.
+
+% Method 5 of inpaint_nans is also very similar to method 2, except
+% that it uses a simple average of all 8 neighbors of an element.
+% Its not actually an approximation to our PDE.
+
+% Method 3 is yet another variation on this theme, except the PDE
+% model used is one more suited to a model of a thin plate than for
+% heat diffusion. Here the governing PDE is:
+%
+%   u_xxxx + 2*u_xxyy + u_yyyy = 0
+%
+% again discretized into a linear system of equations. 
+
+%%
+
+% Finally, method 4 of inpaint_nans has a different underlying
+% model. Pretend that each element in the array was connected to
+% its immediate neighbors to the left, right, up, and down by
+% "springs". They are also connected to their neighbors at 45
+% degree angles by springs with a weaker spring constant. Since
+% the potential energy stored in a spring is proportional to its
+% extension, we can formulate this again as a linear system of
+% equations to be solved. For the example above, we would generate
+% the set of equations:
+
+%  A(2,2) - A(1,2) = 0
+%  A(2,2) - A(2,1) = 0
+%  A(2,2) - A(3,2) = 0
+%  A(2,2) - A(2,3) = 0
+% (A(2,2) - A(1,1))/sqrt(2) = 0
+% (A(2,2) - A(1,3))/sqrt(2) = 0
+% (A(2,2) - A(3,1))/sqrt(2) = 0
+% (A(2,2) - A(3,3))/sqrt(2) = 0
+%  A(2,3) - A(1,3) = 0
+%  A(2,3) - A(2,2) = 0
+%  A(2,3) - A(3,3) = 0
+%  A(2,3) - A(2,4) = 0
+% (A(2,3) - A(1,2))/sqrt(2) = 0
+% (A(2,3) - A(1,4))/sqrt(2) = 0
+% (A(2,3) - A(3,2))/sqrt(2) = 0
+% (A(2,3) - A(3,4))/sqrt(2) = 0
+
+% Substitute for the known elements to get
+
+%  A(2,2) - 0 = 0
+%  A(2,2) - 1 = 0
+%  A(2,2) - 3 = 0
+%  A(2,2) - A(2,3) = 0
+% (A(2,2) - 0)/sqrt(2) = 0
+% (A(2,2) - 0)/sqrt(2) = 0
+% (A(2,2) - 2)/sqrt(2) = 0
+% (A(2,2) - 5)/sqrt(2) = 0
+%  A(2,3) - 0 = 0
+%  A(2,3) - A(2,2) = 0
+%  A(2,3) - 5 = 0
+%  A(2,3) - 4 = 0
+% (A(2,3) - 0)/sqrt(2) = 0
+% (A(2,3) - 0)/sqrt(2) = 0
+% (A(2,3) - 3)/sqrt(2) = 0
+% (A(2,3) - 8)/sqrt(2) = 0
+
+% This system is also solvable now:
+r2 = 1/sqrt(2);
+M=[1 0;1 0;1 0;1 -1;r2 0;r2 0;r2 0;r2 0;0 1;-1 1;0 1;0 1;0 r2;0 r2;0 r2;0 r2];
+v = M\[0 1 3 0 0 0 2*r2 5*r2 0 0 5 4 0 0 3*r2 8*r2]'
+
+A(2,2) = v(1);
+A(2,3) = v(2);
+
+% and finally plot the surface
+surf(A)
+title 'A simply inpainted surface using a spring model'
+
+%%
+
+% Why did I provide this approach, based on a spring metaphor?
+% As you should have observed, methods 2 and 4 are really quite close
+% in what they do for internal NaN elements. Its on the perimeter that
+% they differ significantly. The diffusion/Laplacian model will
+% extrapolate smoothly, and as linearly as possible. The spring model
+% will tend to extrapolate as a constant function.