Research Article  Open Access
HighResolution Algorithm for Image Segmentation in the Presence of Correlated Noise
Abstract
Multiple line characterization is a most common issue in image processing. A specific formalism turns the contour detection issue of image processing into a source localization issue of array processing. However, the existing methods do not address correlated noise. As a result, the detection performance is degraded. In this paper, we propose to improve the subspacebased highresolution methods by computing the fourthorder slice cumulant matrix of the received signals instead of secondorder statistics, and we estimate contour parameters out of images impaired with correlated Gaussian noise. Simulation results are presented and show that the proposed methods improve line characterization performance compared to secondorder statistics.
1. Introduction
It has been shown that a specific signal generation process yields, when applied to an image containing multiple lines, to signals which follow an array processing model. Then highresolution methods, such as MUSIC [1] and ESPRIT [2] algorithms, are used. Subspacebased Line DEtection (SLIDE) algorithm [3] and recently other methods [4] have been proposed, which can be effectively exploited to estimate line parameters (orientation and offset).
However, these methods assume that the noise is white and are based exclusively on secondorder statistics. When the noise is correlated, these developed algorithms are not efficient.
That is the fact that correlated noise exists in several circumstances. For instance, correlated noise occurs in many imaging systems such as scanners and pushbroom imagers. In medical Xray imaging, while noise in an Xray beam is uncorrelated, it is correlated in observed images. This is due to the detectors, where Xray photons typically evoke multiple light photons that can contribute to different pixels.
In Section 2, we first model the image data and formulate the problem by introducing cumulantbased model. In Section 3, highresolution algorithms for estimating line orientation, which are MUSIClike algorithm and improved TLSESPRIT algorithm, are derived from fourthorder cumulant of the signal realizations received by virtual array sensors. After orientation estimation, the extension of Hough transform is applied to find a line offset. In Section 4, numerical simulations are presented to show that cumulantbased methods are efficient when the processed image is impaired by correlated noise.
2. Problem Formulation
2.1. Image Data Model
Let us consider 2dimensional binary image (see Figure 1), where a line is characterized by its axis offset and the angle that it makes with the normal to the axis at the interception point, with positive angles defined for lines having positive axis offsets. The “1”valued pixels represent useful pixels and form the outline of the estimated contour, while the background is demonstrated by “0”valued pixels. The image consists of N rows and C columns in Figure 1. In order to make the image data representation similar to sensor array processing, array sensors are supposed to be placed aside the image, each sensor receiving signals only from its corresponding row in the matrix [3]. All the pixels in image are assumed to propagate narrowband electromagnetic waves with zero initial phases. Furthermore, assume that the waves emanating from pixels in a given row of the image matrix are confined to travel only along that row towards the corresponding sensor. In this propagation environment, each straight line in the image will in effect be equivalent to a wavefront of a traveling plane wave. For this purpose some signals are generated out of the image data. When there is only one line in image and no outlier pixels, the signal received by each sensor is: where is a propagation parameter [3] and is the axis offset in th row, .
(a)
(b)
If there are straight lines and also outlier pixels in image, the signal received by each sensor can be extended where is the effect of noise in the th row. We denote and , at the same time, consider that the image is impaired by the noise, we can be express all the signals in matrix form
where , , and with th column , therein “” denotes transposition operation.
2.2. FourthOrder CumulantBased Model
For complex variable, the fourthorder cumulant can be generally defined as
where “*” denotes conjugate operation. Quite often the complex random variables are analytical signal, so the fourth item of the right side is identically zero. Note that the definition of cumulant of complex random variables is nonunique. In practical terms, continuous signals are often turned into discrete timeseries to calculate fourthorder cumulant.
Providing that there is only one source signal , that is, only one line in image data, the signals received by array sensors is defined as . So for all the source signals, the received signals at the th snapshot are , where is the th source signal at th snapshot.
We define the vector by where “” denotes Kronecker product. Based on the independence property of cumulant and that the fourthorder cumulants of the Gaussian noise are identically zero [5, 6], we can get the following fourthorder cumulant of image data after some calculations:
To make the analogy of the formulation as sensor array processing, we do not have timedependent measurement signal vectors, because the signals generated by image data is deterministic. There is only a single snapshot across a very large array (see (2)). Therefore, to exploit highresolution methods with fourthorder cumulant in array processing to image signals, the signals received by array sensors in image model are considered to be spatially smoothed and a new matrix is rearranged as
where and . For every snapshot, , where is matrix and is the noise vector acquired by the same rearrangement of the snapshot vectors. Then, all the signals generated by image data can be written in matrix form where and . In this way, a single snapshot across a very large array in (2) is transformed to snapshots. Now we are ready to get the similar signal as in sensor array processing.
3. Detection Algorithms
3.1. Angle Estimation
3.1.1. MUSICLike Algorithm
In order to reduce the computational load, a slice cumulant matrix (e.g., , here the first row of ), instead of abovementioned cumulant (see (5)), and offers the same properties [5] where is the signal received by th senor at th snapshot, and , , the diagonal elements of which are the kurtosis. Equation (8) shows that there is no noise term in the cumulant slice matrix computed out of the generated signals. Therefore, when this matrix is computed, the parameters can be better estimated. Obviously, this reshaped matrix is the Hermitian matrix and its dimension is reduced to from , so the computational load is hugely decreased.
From the definition of matrix (see (2)), which shows that the columns of are linearly independent, it is easy to get that has the rank of . The MUSIClike algorithm relies on the singular value decomposition (SVD) of matrix :
For the independent sources, the columns of span the signal subspace, and the columns of span the noise subspace [1], , where is the eigenvalue associating with the th eigenvector. Hence, is orthogonal to the steering vector of . The pseudo spectrum of the MUSIC method is given by
In conclusion, practical application of the algorithm involves the following steps.
(1)Estimate the cumulant matrix by (5), (6), and (8).(2)Find the eigendecomposition of the cumulant matrix by 8 and get the noise subspace . If the number of line is unknown, the Minimum Description Length (MDL) or Akaike Information Criterion (AIC) [7] criterion are used.(3)Compute by (10). The straight line orientations are estimated through the research of the maxima of .3.1.2. Improved TLSESPRIT Algorithm
In this subsection, we define a matrix for implementing SVD operation, which is more appropriate to the improved TLSESPRIT algorithm and is different from the fourthorder cumulant matrix in MUSIClike algorithm
After spatial smoothing procedure is applied to image data as in (6), the multiple snapshots obtained have length , and is also dimensional matrix.
In order to follow the key steps of TLSESPRIT algorithm, we firstly derive the following four submatrices from (11):
Then, the following matrix can be used to implement SVD operation in place of the covariance matrix:
The overall improved TLSESPRIT algorithmbased cumulant can be summarized as follows.
(1)Form the matrix by (5), (6), and (11).(2)Generate four submatrices from (12) and reconstruct the matrix by (13);(3)Perform SVD operation with where with the eigenvalues and corresponding to the signal subspace. Depending on the known conditions, AIC or MDL criterion is used.(4)Let be the upperleft matrix of and let be matrix formed by deleting the first row of matrix . We carry out the eigendecomposition (5)Extract the submatrices and , then find the eigenvalues of .(6)Estimate every line's orientation by3.2. Offset Detection
After we estimate the line's angles, steering matrix can be computed by (2), so that the maximum likelihood function or other similar methods can be used to find the offsets of lines. In practice, we estimate the offsets using the "Extension of the Hough Transform" [8].
4. Simulation Results
In this section, simulation results are presented and the performance of our proposed algorithms is evaluated.
4.1. Correlated Gaussian Noise Simulation
To evaluate realistically the algorithms, correlated Gaussian noise is simulated by the following steps. First, the 2dimensional random Gaussian noise matrix is generated, which has the same dimensions as the initial binary image and obeys a Gaussian distribution with mean 0 and variance . Then, let generated Gaussian noise matrix pass the spatial Gaussian low pass filter, so that we can obtain expected correlated Gaussian noise by choosing the filter's parameter. The impulse response of the filter is , where and are vertical and horizontal variances, determining the correlation strength of noise image. Specially, means an isotropic Gaussian low pass filter. In the simulation, the filter function should be discretized to get the filter template, which is center symmetric. Herein, the correlation length (CL) of correlated Gaussian noise is defined as , where and are and .
4.2. Algorithm Simulation
We set the size of detected image as . As an example in Figure 2, standard variance of white Gaussian noise is 6 and CL of correlated Gaussian noise is 6. For initial image (see Figure 2(a)), there are two straight lines whose angles are and and offsets are 95 and 55, respectively. The estimation results provided by cumulantbased MUSIClike algorithm (see Figure 2(c)) and by the improved TLSESPRIT algorithm (see Figure 2(d)) are more accurate than those provided by secondorderbased algorithms (see Figure 2(b)). The orientations of two lines are, respectively, and for MUSIClike algorithm, and and for improved TLSESPRIT algorithm. For the offsets, they are, respectively, 93.8 and 52.9 by MUSIClike algorithm, and 92.6 and 54.8 by TLSESPRIT algorithm. Especially when the images are severely corrupted by correlated Gaussian noise, the cumulantbased algorithms still correctly characterize the lines, while the algorithms using secondorder statistics yield a large bias.
(a)
(b)
(c)
(d)
For trials, the mean error and standard deviation are defined like: and , where is each estimated angle and is the mean of all estimated angles. From Table 1, we can see that, as the augmentation of correlated noise's CL, it is insensitive to the mean and standard deviation of estimated angle when using MUSIClike algorithm, compared to the improved TLSESPRIT algorithm. Moreover, we also see in the table that the performance of MUSIClike algorithm is better than improved TLSESPRIT algorithm. Since both MUSIClike algorithm and improved TLSESPRIT algorithm should choose the slice cumulant matrices to implement eigendecomposition, these different choices of the matrices have some impacts on the performance of the algorithms.

The following experiments aim at studying the highresolution capabilities of the proposed algorithm. Firstly, two lines with and are considered in the image. The standard variance of white Gaussian noise is set to be 12, and CL is 16. After we operate 100 trials, the means of estimated angles for MUSIClike algorithm are, respectively, and , and they are and for improved TLSESPRIT algorithm. Then we change the angles to be and , the mean values of estimated angles are and for 100 trials. For improved TLSESPRIT algorithm, we can also get the similar results, their means are and .
Finally, to evaluate the computational load, we also compute the time elapsed to estimate the angle for the MUSIClike algorithm and improved TLSESPRIT algorithm. Our simulation run on the same system for two algorithms: Intel 2 Quad CPU, 2.83 GHz, with the memory 4 G. In the simulation, we set two lines with the angles and in the image. The time consumed by MUSIClike algorithm is 0.29 seconds to obtain the estimations of angle values, while it is 0.10 seconds for improved TLSESPRIT algorithm.
5. Conclusion
In this paper, the problem of contour detection in image is investigated by using array processing. Considering the circumstance of correlated Gaussian noise, fourthorder cumulantbased MUSIClike and improved TLSESPRIT algorithm for nonGaussian virtual generated signal from the image are presented to estimate the orientations of lines, while the offsets of lines are estimated by “Extension of the Hough Transform.” Numerical simulation shows that using cumulantbased algorithms improves the detection performance when correlated noise exists in image. These methods can also be extended to detect other contours in image, such as a curve line and a circle.
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Copyright
Copyright © 2010 Haiping Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.