Inpainting nan elements in 3-d

An extension of inpaint_nans to 3 dimensions

Description

Every once in a while I get asked for a 3-d version of inpaint_nans, so I modified my existing 2-d code to run in 3 dimensions. This code has only two methods as options. I picked what I felt were the two most useful methods from inpaint_nans to extend to 3-d.

See the included published demos for examples of use.

Acknowledgements

The author wishes to acknowledge the following in the creation of this submission:
inpaint_nans
This submission has inspired the following:
Inpaint over missing data in N-D arrays

• MATLAB release: MATLAB 7.5 (R2007b)
• Other requirements: This should run on older releases of MATLAB, I hope back as far as release 10, since the original code ran under release 10.

[Carlos Adrian Vargas Aguilera]

Thanks John, this give us hope for the GRIDFITN :) (just kidding). It's a great job.

Carlos

[Tim Rae]

Worked extremely well!

Thankyou so much, John

[Nikola Toljic]

Excellent. Thanks.

[Georgina]

 

[Alvaro Canivell]

Exactly what I was looking for !

[Pierre-Julien Trombe]

Great. It helps a lot, not talking about the gain of time. :)
John: Would you have a references (articles, etc) on the methods you implemented?
Cheers

[John D'Errico]

I discuss the two methods at some level in the original inpaint_nans submission. Perhaps that would be sufficient for you. Really, the default method in these tools is simply a boundary value solver for a PDE, defined over the domain of a hole in the supplied array. Since the function values are known around the perimeter of the hole, I simply formulate a simple partial differential equation problem on the interior. The solution involves discretizing the PDE on the lattice points of the array, which is a very natural thing to do on a regular grid. In the end, large scale, sparse linear algebra is employed, something that MATLAB does very well.

In the default method in inpaint_nans3, I have chosen Laplace's equation for the PDE. This has the virtue of making for a stable, robust, smooth solution, as well as providing a well behaved extrapolant for external holes. Had I used something else here (the more general Navier-Stokes comes to mind) the inpainting solution would now have been far less stable, far less robust, etc. And any intelligent extrapolation would be nearly impossible.

The second method in inpaint_nans3 is a spring analogue. This is something I originally put in on a lark, expecting it to have very limited utility. (I was wrong about that - as multiple people have pointed it out to be quite useful to them.) It formulates the problem in a mechanical way, connecting each node to its neighbors using "springs". Since a mechanical spring resists deformation by storing energy as a function of the square of the extension, the springs method tries not to extrapolate at all, but it still interpolates smoothly. In effect, it tries to do constant extrapolation, but to do so smoothly and in multiple dimensions. That springs problem is formulated as a large scale linear algebra one, much as the default PDE solution was solved. I don't have a reference for this method, it was simply my ad hoc solution to an interesting problem.

For those who simply wish to provide a citation, you can cite submissions on the FEX. Two schemes for doing so are posed here:

http://matlabwiki.mathworks.com/Citing_Files_from_the_File_Exchange

[Sean de]

Thank you John for this wonderful contribution!

Fast, well commented, and extremely useful for removal of noise in known locations in X-CMT images.