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# Balloon energy based on parametric active contour and directional Walsh–Hadamard transform and its application in tracking of texture object in texture background

- Homa Tahvilian
^{1}Email author, - Payman Moallem
^{2}and - Amirhassan Monadjemi
^{3}

**2012**:253

https://doi.org/10.1186/1687-6180-2012-253

© Tahvilian et al.; licensee Springer. 2012

**Received:**7 November 2011**Accepted:**2 November 2012**Published:**8 December 2012

## Abstract

One of the popular approaches in object boundary detecting and tracking is active contour models (ACM). This article presents a new balloon energy in parametric active contour for tracking a texture object in texture background. In this proposed method, by adding the balloon energy to the energy function of the parametric ACM, a precise detection and tracking of texture target in texture background has been elaborated. In this method, texture feature of contour and object points have been calculated using directional Walsh–Hadamard transform, which is a modified version of the Walsh–Hadamard. Then, by comparing the texture feature of contour points with texture feature of the target object, movement direction of the balloon has been determined, whereupon contour curves are expanded or shrunk in order to adapt to the target boundaries. The tracking process is iterated to the last frames. The comparison between our method and the active contour method based on the moment demonstrates that our method is more effective in tracking object boundary edges used for video streams with a changing background. Consequently, the tracking precision of our method is higher; in addition, it converges more rapidly due to it slower complexity.

## Keywords

- Tracking
- Active contour models
- Energy function
- Directional Walsh–Hadamard transform (DWHT)
- Texture feature
- Moment
- Balloon energy

## 1. Introduction

Object tracking is one of the most interesting topics in many computer vision applications such as traffic monitoring in the intelligent transportation systems, video surveillance, medical applications, military object tracking, object-based video compression, etc. [1–4]. Detection and competitions of object motion in sequence of image or video are called tracking. Various tracking methods have been proposed and improved, from the simple and rigid object tracking with static camera, to the complex and non-rigid object tracking with moving camera [5]. These methods are categorized into five groups [6, 7] namely, region-based tracking [8], feature-based tracking [9], mesh-based tracking [10, 11], model-based tracking [12], and active contour models (ACM)-based tracking [13].

Active contour method was introduced by Kass in 1987 [14]. In general, ACM can be classified into two main types: parametric and geometric active contours. Parametric ACM is an initial curve in two- or three-dimensional images. It is modified by internal and external forces and it stops at the real boundaries of the image. Although this method was proposed for segmentation and video object tracking, it faces problems such as speed and accuracy [15].

Geometric ACM, which was presented by Caselless and Malladi, are based on the theory of curve evolution and level set techniques in which curves and levels are evaluated by some geometric criteria [16, 17]. Simultaneous detection of several object boundaries is one of the great advantages of this method. However, due to its higher computational cost and complexity, geometric active contour is slower than parametric ACM.

On the other hand, in the absence of strong edges, traditional parametric ACM cannot detect object boundaries correctly. Hence, to overcome this problem, Ivins and Porrill [18] and Schaub and Smith [19] proposed different color active contours. In their method, contour curves are directed toward the target object with a specified color. This method provides the possibility of detecting and tracking of targets with weak edges. The most noticeable disadvantage of the method is that the color of object and background should be simple and the method is not capable of tracking targets with a complex color or texture.

In the method proposed by Vard et al. [15], tracking of texture object in texture background is presented by means of adding a novel pressure energy, named texture pressure energy, to the energy function of the parametric active contour. Texture features of contour and object points are calculated by a method based on moment. Then, according to the difference between these features of target object, the contour curve is shrunk or expanded in order to adapt to the target object boundaries. When the background texture is changed considerably, this method cannot track the texture object with high accuracy. In addition, calculation of texture pressure energy needs huge computations, which leads to a low convergence speed. In another research, Vard et al. [20] used texture pressure based on the directional Walsh–Hadamard transform (DWHT) for segmentation texture object in texture background. In this article, the texture images are synthetic images selected from Brodatz album. In [15, 20], the number of iterations is not provided automatically and should be selected by the user. So, the speed of algorithm is low.

Compared with [20], in which the textured feature based on the DWHT is used in order to segment the textured object, in this article we aim to modify their method in such a way that tracking the textured object against a textured background would be possible automatically and accurately. DWHT algorithm is an implementation of multi-scale and multi-directional decomposition in ordinary Walsh–Hadamard transform (WHT) domain that was first introduced by Monadjemi and Moallem [21]. This method is based on a particular sort of input image rotation before the WHT is applied. DWHT preserves all the features of WHT except for the extra advantage of preserving the directional properties of the texture. Another advantage of DWHT method is its less computations.

In summary, in this article, our focus is on “object boundary detection and object tracking” which is achieved by using a parametric ACM. In fact, by adding a new balloon energy based on texture features to the energy function of parametric ACM, we are able to detect texture objects in texture background more accurately than moment-based active contour. Moreover, the tracking process is accelerated by the proposed method. These improvements are accomplished thanks to the calculation of the texture features, of both contour and object points, by means of DWHT-based method. Also, the parameters, which are required for calculating balloon energy and the number of iteration in each frame is obtained automatically, so that the speed of algorithm is improved.

This article is organized as follows: in the next section, we review the mathematical description; in Section 3, the DWHT is explained; The proposed method is discussed in Section 4; we explain the experimental results of the proposed method compared with those of moment-based active contour in terms of accuracy and convergence speed in Section 5, and finally, conclusions are given in Section 6.

## 2. Mathematical description of ACM

*I*is the image intensity at (

*x,y*). In order to implement, the vector function

*S*(

*u*) is approximated discontinuously at {

*u*

_{ i }},

*i*= 0, 1,…,

*M*, in which

*M*is the number of points on the contour. Finally, continuous curve will be obtained from interpolation of these points. The traditional flexible parametric method is based on the application of contour, which minimizes the weighted sum of the internal and external energies. Therefore, the final contour is defined by minimizing the following energy function.

*E*

_{int}is the internal energy of the contour defined as follows:

where G_{σ} is a 2D Gaussian kernel with standard deviation σ, ∇ and * present gradient and convolution operators, respectively. *P* is the weighting parameter that controls the image energy which is constant. Equation (4) is used for noise reduction by applying a Gaussian filtering.

*M*

_{1},

*M*

_{2},

*M*

_{3},

*M*

_{4},

*M*

_{5},

*M*

_{6}will be acquired. Then, the corresponding texture features to these moment images are obtained using the nonlinear transform:

where *L* × *L* is the window size in which pixel (*i, j*) is located at its center and ε is a parameter that controls the shape of the logistic function and is determined by the user. For each pixel of image, a texture feature vector in the form of *F*(*i,j*) = [*F*_{1}, *F*_{2}, *F*_{3}, *F*_{4}, *F*_{5}, *F*_{6}] is generated and can be used for image segmentation or target object detection in tracking application.

*ρ*and

*S*are the weighting parameter and snake curve, respectively. The ⊥ indicates that the texture pressure is perpendicularly applied to the tangent of the contour.

*T*is defined as

*F*is the texture feature vector of the contour,

*Oμ*and

*Oσ*are the mean and standard deviation of the texture feature vector of the target object points, respectively.

*K*is a parameter that is defined in the following:

where *Bμ* is the mean of texture feature vector of background.

According to the following research studies presented in this study, when the texture complexity increases, this method does not work out well.

## 3. DWHT

The WHT is known for its important computational advantages. For instance, it is a real (not complex) transform, it only needs addition and subtraction operations and if the input signal is a set of integer-valued data (as in the case of digital images), it only uses integer operations. Furthermore, there is a fast algorithm for Walsh transforms proposed in [23]. The transform matrix, usually referred to as Hadamard matrix, can also be saved in the binary format resulting in the memory requirements reduction [24]. Moreover, hardware implementation of WHT is rather easier than other transforms [25].

*H*is

*sequency-ordered Hadamard*(SOH) matrix [25, 27] where the rows (and columns) are ordered according to their sequency. In other words, while there is no sign of change in the first row, there are

*n*– 1 sign changes in the

*n*th row. As an example, see Figure 2 in which SOH matrix is shown for a rank is 3 (or 8 × 8).

In fact, for a Hadamard matrix, *H* is always equal its transpose, *H*^{′}. In this article, we use the second rank of Hadamard matrix (4 × 4).

In Equation (10), *A*_{α}, *α* = {0°, 45°, 90°, 135°}, is the rotated version of *A*. The rotation is applied to each element in the top row of the image matrix. At border pixels, corresponding elements are used from a repeated imaginary version of the same image matrix (i.e., image is vertically and horizontally wrapped around).

*α*= {0°, 45°, 90°, 135°} can be defined for a simple 4 × 4 image matrix as follows

Note that this is not an ordinary geometrical rotation. For example, we create the rows of *A*_{45°} image by considering the pixels that sit in a 45° direction in the image *A*_{0°} and so on. This means that the resulting horizontal rows capture the information at the specified angles. In fact, it looks more like a pixel rearrangement rather than a geometrical rotation.

_{ α }(

*A*) =

*A*

_{ α }×

*H*

^{′}is computed and gathers the sequency information of input matrix rows into transformed matrix columns. Hence, the same half transform for a rotated matrix (e.g.,

*A*

_{45°}) will give us the sequency information of pixels with a 45°-orientation, again into the columns of transformed matrix. In transfer matrix, the number of sign changes in each column of the sequences is the same and it increases from left to right. In other words, the transformed matrix columns from left to right correspond to the lower to higher sequency elements.

_{ α }(

*A*) as the absolute value of the DWHT output along each column. Columns can be divided into a few groups that represent different

*sequency bands*. Then the statistics of each band can be extracted to configure a feature vector with reasonable dimensionality. So, a DWHT output and feature vector can be defined as

*H*is the transform’s output matrix,

*N*is the matrix size,

*F*is the feature vector,

*M*indicates the applied statistical function, and

*b*is the desired sequency band. Again, log

_{2}or semi-log

_{2}bandwidth scales could be applied. However, we mostly use a simpler $\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}$ division for a three-band and a $\left\{\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\right\}$ division for a four-band feature sets. For example, in three-band division of four-column transform matrix, the band

*b*1 is determined by a first column sequency, the band

*b*2 is determined by a second column sequency, and the third and fourth columns generate band

*b*3. For example, the sequency bands of DWHD

_{0°}(

*A*) are defined as follows:

## 4. Proposed method

In this section, first, we explain the method used for feature extraction by DWHT. Then, in Section 4.2, we introduce the DWHT-based balloon energy. After that, in Section 4.3, tracking algorithm based on the proposed method is presented. Finally, the criterion to stop the contour is explained in Section 4.4.

### 4.1. Feature extraction using DWHT

- 1.
Determine a local window (

*A*) with a size of 4 × 4 around the object and contour points. - 2.
Matrixes:

*A*_{0°},*A*_{45°},*A*_{90°},*A*_{135°}are generated by rotating*A*in four orientations*α*= {0°, 45°, 90°, 135°}*.* - 3.
For each matrix in 2, we use a $\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}$division (see Equation 15), and obtain

*b*1,*b*2,*b*3, sequency bands. - 4.
The mean of each band is calculated as the texture feature vector,

*F*, as in the following:*F*(*i*,*j*) = [*F*_{1}, …,*F*_{12}].

### 4.2. Balloon energy based on DWHT

*E*

_{SCB}is obtained by Equation (4).

*E*

_{SCB}as a texture-based energy is defined as

*B*is a threshold function which is defined as

*F*is the texture feature vector of contour, and

*Oμ*and

*Oσ*are the mean and standard deviation of the

*F_ob*vector, respectively,

*F_*ob is the texture feature vector of the target object points (see Figure 3).

*K*is defined as follows

where β_{μ} is the mean of feature vector of background. By calculating Equation (19), 12 *K* parameters are achieved, then variance and maximum of *K* vector are calculated, after that, the distance between them is obtained. Each of the *K* parameters which is bigger than this number will be selected. *K* parameters which have the above-mentioned features are used in Equation (18). It is evident that compared with [20] the *K* parameters are obtained automatically.

Balloon force allows the contour curve to expand or to shrink in order to fit to the target boundaries.

### 4.3. Tracking algorithm based on proposed method

*B*1,

*B*2,

*B*3,

*B*4).The center of these points is determined and several points, around the central point (

*O*), are calculated as the object points (

*O*1,…,

*On*) (see Figure 5).

In the next step, the texture feature vector of both contour points, (*F*), and the object points, (*F_ob*), are determined. Then, the related mean and standard deviation vectors are calculated, *Bμ*, *Oσ*, *Oμ*. Accordingly, *K* parameters are determined using Equation (19) for the first frame.

In the next step, balloon energy based on the texture feature is obtained from Equation (17) and then total energy function of active contour is calculated using Equation (20). Then, the new location of contour points is found by minimizing energy function. For the next frames, we use these contour points as initial points and follow the same procedure until the optimized location of contour points is obtained. This process is repeated to the last frame.

### 4.4. The criterion to stop the contour

- (1)The distance between the current iteration of contour points and the last and next ones are calculated and then Equation (21) is obtained:$M=\left|\frac{{d}^{\u2033}-{d}^{\prime}}{{d}^{\prime}}\right|$(21)
where

*d*^{′}is the distance between contour points in current iteration and the last iteration*d*″is the distance between contour points in the current and the next iterations.

If *M* is less than 0.01, the iteration will be stopped. The threshold value of 0.01 is obtained by trial and error.

## 5. The result of the experiments

In this section, we have implemented the proposed method (balloon energy based on DWHT added to parametric active contour) and a moment-based active contour method in MATLAB and the results are compared in terms of speed and accuracy. In this article, we have tried to make the images more realistic than [20] by preparing different texture environments and then capturing films from those environments. In this way at least the filming process is done realistically.

All experiments are performed using2.5 GHz Intel Core 2 Dou processor with windows Vista. In these experiments, different texture objects in various texture backgrounds were used.The frame size is chosen to be 352 × 288 pixels.

As Figure 6A shows, the moment-based active contour cannot detect the object boundary correctly. On the other hand, regarding to the moment-based active contour proposed method could detect and track object boundaries in all frames with high accuracy (Figure 6B).

*n*is the obtained number of contour points, SCB(

*n*) refers to the number of contour points on the correct boundary. This measure converges to 100% when the correct boundary detection is achieved. Figure 8 shows comparative diagram of

*E*

_{SCB}for these two methods. Table 1 demonstrates the average of

*E*

_{SCB}and the convergence speed of two methods for 41 frames.

**The average of**
E
_{
SCB
}
**and convergence speed for two methods obtained by three experiments**

Experiments | Tracking method | Average of E | Total tracking time (s) | Improvement of speed (%) |
---|---|---|---|---|

Experiment 1 | Proposed method | 96.44 | 10.89 | 67.80 |

In 41 frames | Moment_ACM | 71.34 | 33.82 | |

Experiment 2 | Proposed method | 94.88 | 9.96 | 74.54 |

In 50 frames | Moment_ACM | 59.28 | 39.12 | |

Experiment 3 | Proposed method | 94.91 | 14.14 | 77.42 |

In 66 frames | Moment_ACM | 61.41 | 62.62 |

*E*

_{SCB}for these two methods. Table 1 shows the average of

*E*

_{SCB}and the convergence speed of two methods for 50 frames.

*E*

_{SCB}for these two methods. Table 1 illustrates the average of

*E*

_{SCB}and the convergence speed of two methods for 66 frames.

In these three experiments, the accuracy of proposed method is 33% more than moment method and the convergence speed of DWHT is 74.18% more than moment method.

### 5.1. Noise robustness of tracking algorithms

*E*

_{SCB}for different values of SNR. Average of

*E*

_{SCB}over 41 frames for different SNR values are given in Table 2. The proposed method has higher accuracy for SNR ≥ 2.70 dB and cannot track target object for SNR = 0.49 dBand for the frame numbers of more than 27, there is no contour for tracking texture object in texture background.

**The average of**
E
_{
SCB
}
**for different values of SNR**

SNR (dB) | ||||||
---|---|---|---|---|---|---|

14.47 | 7.48 | 4.47 | 2.70 | 1.46 | 0.49 | |

Average of | 91.27 | 84.18 | 75 | 64.07 | 37.49 | 18.83 |

## 6. Conclusion

Parametric active contour based on moment was introduced for tracking texture object in texture background. This model cannot detect target object correctly, when texture background is changing. In these cases, the contour curve around the target is expanded. In this article, a new balloon energy based on the texture feature is added to parametric active contour. The texture feature is calculated based on DWHT to identify the direction of the balloon. Experimental results demonstrate that our method can achieve a higher tracking precision than that of the moment-based method. In addition, the convergence speed of DWHT is higher than moment-based active contour. We use different levels of Gaussian noise to evaluate resistance of the method against the noise. It shows that the proposed method has high robustness to noise. As a result, target object is tracked well against texture background. This leads to a better detection of object against texture background.

## Declarations

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.