Two-DimensionalPhaseUnwrappingProblem

Matlab

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Two-Dimensional Phase Unwrapping Problem

By Dr. Munther Gdeisat and Dr. Francis Lilley

Pre-requisite: In order to understand this tutorial it is necessary for you to have already studied and

completed the “one-dimensional phase unwrapping problem” tutorial before reading this document.

There are many applications that produce wrapped phase images. Examples of these are synthetic aperture

radar (SAR), magnetic resonance imaging (MRI) and fringe pattern analysis. The wrapped phase images that

are produced by these applications are not usable unless they are first unwrapped so as to form a

continuous phase map. This means that the development of a robust phase unwrapping algorithm is an

important topic for all these applications. In this article, we will not discuss phase unwrapping only in the

specific context of these applications, but we will instead explain the concept of the 2D phase unwrapping

problem in general terms.

1. Introduction to 2D phase unwrapping

We shall explain the 2D phase unwrapping process as follows. Suppose that we have a computer-generated

continuous phase image that does not contain any phase wraps (2π jumps). This image may be displayed as

a visual intensity array, as shown in Figure 1(a). The same image may also be plotted as a 3D surface, as

shown in Figure 1(b). The intensities from a single row of this image (row 410) are graphically plotted in

Figure 1(c). The Matlab code that is used to generate this phase image is as follows. The peaks Matlab

function is used to generate the continuous phase image. Please note that we are using the term

“continuous” here to refer not to an analogue signal, but to a discrete 1D phase signal, or a discrete 2D

phase image, that does not contain any phase wraps.

%This program is to simulate a continuous phase distribution to act as a dataset

%for use in the 2D phase unwrapping problem

clc; close all; clear

N = 512;

[x,y]=meshgrid(1:N);

image1 = 2*peaks(N) + 0.1*x + 0.01*y;

figure, colormap(gray(256)), imagesc(image1)

title('Continuous phase image displayed as a visual intensity array')

xlabel('Pixels'), ylabel('Pixels')

figure

surf(image1,'FaceColor','interp', 'EdgeColor','none', 'FaceLighting','phong')

view(-30,30), camlight left, axis tight

title(' Continuous phase map image displayed as a surface plot')

xlabel('Pixels'), ylabel('Pixels'), zlabel('Phase in radians')

figure, plot(image1(410,:))

title('Row 410 of the continuous phase image')

xlabel('Pixels'), ylabel('Phase in radians')

(a) (b) (c)

Figure 2: (a) A wrapped phase image displayed as a visual intensity array, (b) the wrapped image plotted as

a surface, (c) row 410 of the wrapped phase image.

Recall from the 1D phase unwrapping tutorial, that when we were dealing with lines of phase values, the

phase wraps appeared as multiple 2π jumps forming a saw-tooth waveform like that shown in Figure 2(c).

Note that in the 2D case, where we now have phase images in the form of a 2D array, the phase wraps

appear as contour curves, as shown in Figure 2(a), which we shall refer to as wrap curves. These curves will

appear in the form of either closed, or open, curves and you can see both types of curve in Figure 2(a).

Note that in the latter case, if an open curve enters a wrapped phase image, it must therefore also leave it.

In order to unwrap the image we can use the Itoh 2D phase unwrapper. There are two main methods by

which the Itoh 2D phase unwrapper may be implemented. The first method involves unwrapping the rows

in the wrapped image sequentially (one at a time). This produces an intermediate image that is only

partially phase unwrapped. Next we perform a similar process, but this time unwrap all the columns within

the partially unwrapped image. The resultant unwrapped phase image, as produced by this first

implementation of the Itoh unwrapper, is shown in Figures 3(a) & (b). The Matlab code to perform this task

is as follows.

%Unwrap the image using the Itoh algorithm: the first method is performed

%by first sequentially unwrapping the all rows, one at a time.

image1_unwrapped = image1_wrapped;

for i=1:N

image1_unwrapped(i,:) = unwrap(image1_unwrapped(i,:));

end

%Then sequentially unwrap all the columns one at a time

for i=1:N

image1_unwrapped(:,i) = unwrap(image1_unwrapped(:,i));

end

figure, colormap(gray(256)), imagesc(image1_unwrapped)

title('Unwrapped phase image using the Itoh algorithm: the first method')

xlabel('Pixels'), ylabel('Pixels')

figure

surf(image1_unwrapped,'FaceColor','interp', 'EdgeColor','none',

'FaceLighting','phong')

view(-30,30), camlight left, axis tight

title('Unwrapped phase image using the Itoh algorithm: the first method')

xlabel('Pixels'), ylabel('Pixels'), zlabel('Phase in radians')

Wrapped image displayed as visual intensity array

Pixels

50 100 150

200

250

300 350

400 450 500

100

150

200

250

300

350

400

450

500

0 100 200 300 400

500 600

-4

-3

-2

-1

Row 410 of the wrapped image

Pixels

Phase in radians

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