Synthetic Aperture Sonar Image Segmentation using the ...scripts.cac.psu.edu/users/j/p/jps120/ImageProcessing/JUA_2001.pdf · 1 Synthetic Aperture Sonar Image Segmentation using the

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Synthetic Aperture Sonar Image Segmentation using the Fuzzy C-Means Clustering Algorithm.

J. P. STITT R. L. TUTWILER A. S. LEWIS

Research Associate Senior Research Associate Research Associate

[email protected] [email protected] [email protected]

Autonomous Control and Intelligent Systems Division

The Pennsylvania State Applied Research Laboratory

PO BOX 30, State College, PA, 16804 USA

Key Words: Image Segmentation, Fuzzy C-Means, Clustering, Sonar, Automatic

Abstract

Synthetic aperture side-scan sonar (SAS) provides an imaging modality for detecting

objects on the sea floor. It is also an excellent tool for shallow water characterization where

immobile, submerged threats would not be detected by conventional forward-looking sonar

range-doppler techniques. SAS images provide an image of an object and its shadow, both of

which can be used in the classification and localization of potential threats.

This document discusses the development of an image segmentation algorithm that

was capable of segmenting (detecting) the image of an object and its acoustic shadow in the

presence of reverberation noise. As a component of an autonomous deployable active sonar

system, no human input was required. An unsupervised form of cluster analysis, the Fuzzy

C-Means Algorithm (FCM) was used to implement the segmentation procedure. FCM is a

generalization of the classical K-Means or Hard C-Means (HCM) clustering algorithm and

the FCM outperformed the HCM in the segmentation of SAS images. Operating in an

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unsupervised mode, the FCM image-segmentation algorithm was capable of dealing with

variability in gray level range from one image to the next as well as the low signal-to-noise

ratio (SNR) that is inherent to SAS images.

Introduction

In this paper we describe an autonomous image segmentation algorithm designed to

segment a side-scan sonar image of the echo of an object, its acoustic shadow, and the sea-

bottom reverberation (background). A number of unsupervised techniques have been

proposed to deal with segmentation of sonar images, including use of Markov Random Fields

(MRF) to approximate the differing gray level regions and the pixel correlations within the

regions [1, 2]. The latter technique provides an accurate segmentation procedure, but is

computationally intense. When multiple images are available, a method of combining MRF

clustering with image fusion can be used [3]. A self-organizing feature map method [4] also

has been effectively applied, but again involves iterative, computationally expensive

processing. These methods ignored the object’s echo that can be used to better localize (e.g.,

height above the sea floor) the object during subsequent processing. Finally, in parallel with

the development of the FCM procedure [5] described here, an unsupervised split-and-merge

algorithm capable of segmenting the acoustic shadow from the reverberation is being

developed [6].

The purpose of an image-segmentation algorithm is to partition an image into its

component regions (i.e., objects and background) [7, 8]. The K-Means or Hard C-Means

algorithm (HCM) is an example of an unsupervised clustering algorithm [9] and has been

shown to be a computationally efficient image segmentation procedure [10]. The

segmentation results of the HCM algorithm are highly dependent upon the initialization

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resulting in highly variable segmentation due to the local minima of the associated objective

function. This problem was overcome by generalizing the HCM to allow for a fuzzy

membership function for each cluster resulting in the FCM algorithm [10, 11]. The FCM has

been used to implement the image segmentation procedure [11-13]. Further, the FCM

algorithm has been employed in conjunction with multiresolution processing techniques [14]

to solve the problem of shading effects during the segmentation of magnetic resonance

images [15]. Both implementations deal with estimating the number of classes that are

present, an approach that increases the computational complexity of the procedures. In this

application, the procedure is required to estimate three clusters: echo, acoustic-shadow, and

reverberation.

Figure 1 – The SAS images of a cylinder (left) and a spheroid (right). Both objects show the brighter echo of the object itself and larger dark acoustic shadow. Throughout the paper, the eight-bit gray scale of all of the images has been normalized to the range [0, 1] and pseudo-color (color bar on the right) has been applied to enhance the appearance of the images.

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In our application, image-segmentation serves as a preprocessing step for an object

classification procedure. We expected that the segmentation algorithm would improve the

image SNR prior to the classification module, thereby minimizing the likelihood of

misclassification and maximizing the classifier’s efficacy. Because the first-step in this

procedure extracts a subimage containing the region of interest and discards the bulk of the

pixels that contain no information, the classifier’s computational complexity is reduced.

Figure 1 shows the Synthetic Aperture Sonar (SAS) images of two objects. The

images were simulated using the Shallow Water Acoustics Toolkit (SWAT), which is

available from the Office of Naval Research [16]. All figures are plotted using a linear gray

scale with 256 gray levels. Pseudo-coloring was used to make the images easier to view.

Both images were preprocessed to enhance the gray-scale contrast. The image-segmentation

algorithm must be capable of dealing with variability in gray level range from one image to

the next as well as the low SNR that is inherent to sonar images. In each case, the simulated

range was 20 meters from the platform and the object was floating 0.3 meters above a coarse

sand bottom. Our simulations employed the default horizontal array setting.

Methods

Prior to applying the FCM image segmentation procedures, the original images were

preprocessed to deal with poor contrast and low signal to noise ratio (SNR). Figure 2 shows

the effects of the two preprocessing steps on the image of the cylinder in Figure 1. Figures

2.a, 2.b, and 2.c contain the images of the cylinder at each stage of reprocessing and Figures

2.d, 2.e, and 2.f shows the corresponding gray level histogram. The original image of the

cylinder is shown in the Figure 2.a. Figures 2.b and 2.e show the results of applying a

histogram equalization algorithm to the original image [7, 8].

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Figure 2 – An illustration of the effects of applying two image processing steps to the

original image of the cylinder shown on the left side of Figure 1. The top row contains the gray scale images at three stages of processing. The bottom row shows the corresponding gray level histograms. Figures 2.a and 2.d contain the unprocessed image and its gray level histogram, respectively. Figures 2.b and 2.e illustrate the improvement following histogram equalization. In Figures 2.c and 2.f, a median filter has been applied to the histogram-equalized image resulting in an improved SNR.

The histogram-equalization algorithm distributed the gray levels within the original image

(Figure 2.d) over the entire gray scale range (Figure 2.e) resulting in an image with an

enhanced contrast (Figure 2.b). The effects of the second filtering stage are shown in Figure

2.c. A five-by-five median filter was applied to the contrast-enhanced image in the second

column [7, 8]. As shown in the gray level histogram in Figure 2.f, the gray level distribution

that is associated with the shadow region (gray level range = [0, 0.25]) is clearly separated

from the reverberation distribution which is located in the gray level range [0.4, 0.8].

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Figure 3 – Images of the two objects shown in Figure 1 following the preprocessing step.

Figure 3 shows the resultant images of both objects following the preprocessing steps

described above. To illustrate the improvement in the image quality, the same pseudo-

coloring transformation that was used on the images in Figure 1 was applied to the processed

images in Figure 3. The color-bar on the right side of the image shows the color that was

associated with each gray level.

Two unsupervised image segmentation algorithms were implemented as part of this

project. Both procedures employed the following version of the FCM algorithm [11]. In the

following statement of the FCM algorithm the vector vi is the ith class centroid, the vector uik

contains the ith class’ membership values for the kth sample, and dxy is the Euclidean distance

metric between the xth class and the yth sample. All samples are members of all classes to

some degree (i.e., 10 ≤≤ iku ). In this implementation of the FCM algorithm, the exponent in

steps (2) and (3) is fixed at m = 2.

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Fuzzy c-means Algorithm (FCM):

1. Fix the number of clusters, c. Fix m ∞<< m1 . Initialize membership function (MF) U(h=0) to random values. Select metric dik.

2. Calculate the c mean-vectors {vi(h)} using:

( ) ( )∑∑==

=N

1k

mN

1kk

m

i ikik uu xv

3. Update U(h) using: ( )1m2

c

1j jk

ik1h

ik ddu 1

−

=

+ ∑

=

4. Compare U(h) to U(h+1) : if Lε≤− (h)1)(h U- U stop; otherwise h =

h + 1 and go to Step 2.

The HCM is a special case of the FCM clustering technique where the value of the

exponent in step (2) is fixed at m = 1 and the expression in step (3) is replaced by the

following:

(3) ( ) { }

=

= ≤≤+

otherwise,0

,1 dmindu

(h)

jkcj1

(h)

ik1h

ik.

The MFs in the HCM case are actually indicator functions have values of either zero or one.

The first step in our image segmentation process uses the FCM algorithm to extract a

small subimage from the original image by processing both the gray level and location within

the image. Figure 4 illustrates the steps used to implement this procedure, referred to as the

FCM-SIE procedure in the remainder of the discussion. Figure 4.A is a synthesized image

containing a dark region (shadow) in a field of brighter pixels (reverberation). In the first

step, each column of the image was presented to the FCM algorithm as a sample, and each

row was viewed as a feature. The algorithm was initialized to find one class (C = 1). The

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resulting membership function is plotted in Figure 4.B.

Figure 4 – An example using a simple image to illustrate the steps of the FCM-SIE procedure. Figure 4.A is a synthesized gray scale image. The central region corresponds to an acoustic shadow while the surrounding region is reverberation noise. White Gaussian noise has been added to both regions. Figure 4.B contains the plot of the MF generated by the FCM-SIE algorithm using the image columns as samples. Similarly, Figure 4.C shows the MF for the rows. Figure 4.D shows the 3-D membership function matrix that was generated by forming the fuzzy T-conorm (Union) for the 3-D extended MFs in Figure 4.B and Figure 4.C. Figure 4.E shows mask image that was obtained by thresholding the complement of the Figure 4.D. Figure 4.F contains the segmented image that results from applying the mask to the original image shown in Figure 4.A

Next, the image was transposed and presented to the FCM with C = 1 resulting in the

membership function plotted in Figure 4.C. A three-dimensional (3-D) MF matrix was

formed by replicating the MF in Figure 4.B for the number of columns in the original image.

A second matrix was formed by replicating the MF in Figure 4.C for each row in the original

image. Figure 4.D plots the mask function that resulted when the two matrices were

combined pixel-wise using a fuzzy T-conorm operation [13]. Because we were interested in

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the shadow, the complement of the mask in Figure 4.D was thresholded using the statistics of

the matrix (µ + 1σ), resulting in the binary-valued mask function which is plotted in Figure

4.E. Figure 4.F is the segmented image that resulted from the pixel-wise multiplication of the

original image (Figure 4.A) and the mask shown in Figure 4.E.

Because the segmentation mask is a binary image it can be used to extract the

subimage from the original image. This is accomplished by scanning the mask for pixels

whose gray level is equal to one and saving the corresponding pixel of the original image to

the subimage. Thus, the subimage contained only the pixels within the shadow region

(central square) of in Figure 4.a.

The second step in the image segmentation process employed FCM to segment the

images based of gray level dissimilarities alone. As part of this analysis both the FCM and

the HCM were examined to determine which algorithm performed best for this portion of the

image segmentation task. Throughout the remainder of the discussion these two algorithms

will be referred to as FCM-GLS and HCM-GLS. The segmented images were formed by

assigning to each pixel the weighted value of the MF of the class with the largest degree of

membership. No additional histogram thresholding was employed. Table 1 is a listing of the

acronyms used throughout the remainder of the paper.

Table 1.1 – List of acronyms used through out the following discussion.

FCM Fuzzy C-Means Clustering Algorithm. HCM Hard C-Means Clustering Algorithm (also known as ISODATA, K-Means). FCM-GLS Fuzzy C-Means Gray Level Segmentation procedure. HCM-GLS HCM C-Means Gray Level Segmentation procedure. FCM-SIE Fuzzy C-Means Sub-Image Extraction procedure. MF Membership Function; in the case of HCM the MF reduces to an Indicator

(Characteristic) Function. SNR Signal- to-Noise Ratio. FCM Fuzzy C-Means Clustering Algorithm. SAS Synthetic Aperture Sonar.

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Results

The FCM-SIE, FCM-GLS, and HCM-GLS procedures were implemented as

MATLAB m-files. The FCM-SIE procedure was presented with the preprocessed images that

were plotted in Figure 3. Figure 5 shows the segmented images that were generated by the

FCM-SIE procedure. The FCM-SIE procedure eliminated large regions of reverberation

noise during the segmentation process. In both cases, the echo and acoustic shadow are

preserved.

Figure 5 – The resulting FCM-SIE segmentations of the cylinder and spheroid objects shown in Figure 1.

The next step in processing involves the gray level segmentation of the subimages

that were extracted the FCM-SIE procedure. The images of both objects were presented to

both the HCM-GLS and the FCM-GLS algorithms multiple times. Each presentation

involved a unique random initialization of the prototypes and membership (indicator)

functions. The HCM-GLS produced substantial variability between the resulting

segmentations due to the presence of local minima within the objective [10].

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Figure 6 – The subimages 6.a and 6.b were extracted by the FCM-SIE procedure from the original images shown on the top of Figure 1.The FCM-GLS was applied to these subimages with C = 2 classes resulting in the image in Figures 6.c and 6.d Figures 6.e and 6.f show the FCM-GLS results for the three-class (C = 3) segmentation.

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On the other hand, the FCM-GLS produced very consistent segmentations from one

initialization to the next as a result of the algorithm’s ability to eliminate local minima in the

objective function [11].

The subimages of the cylinder and the spheroid that were extracted by the FCM-SIE

algorithm are shown in Figures 6.a and 6.b, respectively. To further reduce the likelihood of

a misclassification, the FCM-GLS algorithm was applied to these subimages. The results

FCM-GLS image segmentation for the two-class case are shown Figures 6.c and 6.d. The

pixels within the shadow region are clearly segmented from the remaining pixels in both

images resulting in improved contrast and SNR, which in turn improve the likelihood of

correct classification. Figures 6.e and 6.f show the results of the FCM-GLS algorithm for the

three-class case. In both figures, in the region to the left of the shadow, the echo is clearly

segmented from both the reverberation and from the shadow. The shadow itself is also

segmented form the reverberation regions that surround it.

Discussion

The initialization of both the HCM and the FCM required randomized MFs and the

results of the HCM algorithm are highly dependent upon this random initialization. The

function of the HCM-GLS procedure was to assign each pixel of the input image to one of

the C classes. Each class label is associated with one of the standard basis of a C-dimensional

space, termed the label space. The fact that the class label space was discrete induced local

minima in the objective function of the HCM algorithm. Depending upon the initial

conditions, the HCM became stuck in different local minima resulting in highly variable

image segmentation results.

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The FCM-GLS produced very consistent segmentation results across all repetitions.

The resulting label space for the FCM-GLS is a C-dimensional unit hypercube formed from

the convex hull about the standard basis of the discrete HCM label space. Fuzzification of the

label space increases the uncertainty in the clustering problem, which has the effect of

smoothing out or eliminating the local minima of the objective function. The FCM-GLS does

not become trapped in local minima during the optimization process and thus generates

highly repeatable results.

Figure 7 – The average Figure 7.a and variance Figure 7.b of the image gray levels. In both figures, red traces are associated with the statistics of the image columns and green correspond to the statistics of the rows. Clearly there is a change in the values of both the mean and variance of the gray levels across columns. On the other hand, the row statistics are fairly stationary.

The FCM-SIE algorithm reduced the image size to the region of interest thereby

limiting the computational complexity for the classification algorithm. While the FCM-SIE

algorithm was effective in eliminating the rows that contained only noise, the FCM-SIE

procedure did not work effectively in reducing the columns. This was due to the nature of the

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reverberation noise which was stationary within a given column but which varied along the

rows of the image. Figure 7 shows a plot of the gray level mean and variance for both rows

and columns within a region of the cylinder image that contained only noise. The green plots

show the statistics for each row while the red graphs are plots of the statistics of each

column. The red plots of both in Figures 7.a and 7.b show a change in the level across the

image while the green are, on average, flat. The nonstationary nature of the column statistics

resulted in degraded performance of the FCM-SIE algorithm when the attempt was to reduce

the number of columns in the subimage.

In addition to confounding the FCM-SIE algorithm, the nonstationary nature of the

reverberation also reduced the efficacy of the three-class FCM-GLS. As can be seen in both

Figures 6.e. and 6.f, the reverberation to the left of the shadow region was assigned to the

correct class, while much of the reverberation to the right of the shadow as assigned to the

echo class (red). This misclassification was a direct result of the increasing level of the

reverberation in the region to the right of the shadow.

While the FCM-SIE algorithm, in conjunction with the FCM-GLS, resulted in

consistent gray level image segmentations of the extracted subimages, similar efforts using

the HCM were unsuccessful. Presumably, the HCM-SIE algorithm suffered from the local

minima problem that was present in the HCM-GLS procedure and was further compounded

by the complexity of the input matrix (i.e., both gray level and pixel location were included).

Conclusion

The FCM algorithm was found to be more robust in general than the HCM procedure

when applied to image segmentation. The FCM’s superior performance resulted from the

elimination of local minima in the HCM’s objective function that caused the inconsistent

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convergence. The FCM-GLS algorithm provided very consistent gray level SAS image

segmentation whereas the HCM-GLS produced highly variable results. The FCM-SIE

procedure further improved the image segmentation by not only improving the contrast of the

segmented image but also by providing the ability to directly extract a subimage that was

significantly smaller in size, thereby decreasing the computational complexity for the

subsequent classification module.

Our future work will involve the development of an adaptive segmentation algorithm

that will be able to employ location information to remove the increasing level of the

reverberation across the columns of the image while preserving the stationary nature of the

noise within the columns. This addition should improve the performance of the FCM-SIE in

extracting a subimage and should result in an improved gray level segmentation by the FCM-

GSL.

Acknowledgement This material is based upon work supported by Mr. Les Jacobi, Code 333, Office of Naval Research,

through the Naval Sea Systems Command under contract No. N00039-97-D-0042, Delivery Order

No. 186.

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