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# Algorithm for Two-Dimensional Phase Unwrapping Using Fast Fourier Transforms

IP.com Disclosure Number: IPCOM000111704D
Original Publication Date: 1994-Mar-01
Included in the Prior Art Database: 2005-Mar-26
Document File: 2 page(s) / 94K

IBM

## Related People

Pritt, MD: AUTHOR

## Abstract

In this disclosure an algorithm that solves the least squares two-dimensional phase unwrapping problem is presented that uses the fast Fourier Transform (FFT) rather than the fast cosine transform. It performs a mirror reflection of the phase function to yield a periodic function, and the resulting form of Poisson's equation is solved directly using the Fourier transform. Because of the way the mirror reflection is performed, this approach does not require more memory than the conventional cosine transform-based approach.

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Algorithm for Two-Dimensional Phase Unwrapping Using Fast Fourier
Transforms

In this disclosure an algorithm that solves the least squares
two-dimensional phase unwrapping problem is presented that uses the
fast Fourier Transform (FFT) rather than the fast cosine transform.
It performs a mirror reflection of the phase function to yield a
periodic function, and the resulting form of Poisson's equation is
solved directly using the Fourier transform.  Because of the way the
mirror reflection is performed, this approach does not require more
memory than the conventional cosine transform-based approach.

Two-dimensional phase unwrapping is an important processing
step in interferometry.  One of the most robust techniques for
accomplishing this step obtains a least squares solution by
minimizing the difference between the discrete partial derivatives of
the given phase function and those of the solution function.  It is
known that this solution is equivalent to the solution of Poisson's
equation on a rectangular grid with the Neumann boundary condition,
which can be solved directly (noniteratively) using the cosine
transform.  However, the implementation of this solution is
complicated by two factors.  First, the cosine transform is not
generally as well understood as the Fourier transform, and there is
more than one definition in use.  Secondly, the exact form of the
boundary condition depends on the particular cosine transform used in
[1].  For example, the IBM Engineering and Scientific Software
Library (ESSL) has eighteen FFT (fast Fourier transform) routines but
only one fast cosine transform.  It would be preferable to be able to
solve the two-dimensional phase unwrapping problem using the Fourier
transform rather than the cosine transform.  Such a solution would be
easier to implement using FFT routines in software libraries such as
ESSL or published software such as [2].  In fact, a solution that
uses FFTs has been published [4], but it is more inefficient than the
algorithm based on the cosine transform and requires four times as
much memory.

This disclosure presents an algorithm for solving the least
squares two-dimensional phase unwrapping problem using FFTs rather
than fast cosine transforms.  Conceptually, it performs a mirror
reflection of the phase function to yield a periodic function, and
the resulting form of Poisson's equation is solved directly using the
Fourier transform.  Because of the way the mirror reflection is
defined, this approach does not require more memory than the
conventional cosine transform-based...