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# Variational segmentation model for images with intensity inhomogeneity and Poisson noise

- Qiang Chen
^{1}Email author and - Chuanjiang He
^{1}

**2013**:28

https://doi.org/10.1186/1687-5281-2013-28

© Chen and He; licensee Springer. 2013

**Received: **29 October 2012

**Accepted: **14 March 2013

**Published: **8 May 2013

## Abstract

In this paper, we propose a variational segmentation model to deal with intensity inhomogeneity and Poisson noise. An energy functional is first proposed, which uses a data-fidelity term deduced from Poisson distribution instead of the usual *L*
_{2} norm as a measure of fidelity. Due to the new data-fidelity measure, this energy functional can fit the image intensity more accurately while it can diminish the influence of Poisson noise on segmentation results. We then reformulate the energy function as globally convex formulation, which allows for more flexible initialization. The final convex energy functional is minimized via the dual formulation instead of the usually used gradient descent method. Experimental results show that the proposed model can efficiently segment images with intensity inhomogeneity and Poisson noise.

## Keywords

## 1 Introduction

Image segmentation is one of the fundamental and important tasks in image analysis and computer vision. The segmentation problem can be formulated as follows: given an image *I* ∈ *L*^{2}(Ω) on a two-dimensional domain Ω (assumed to be bounded, smooth, and open), one seeks out a closed ‘edge set’ *C* and all the connected components Ω_{1,}…,Ω_{
k
} of Ω\*C*, so that by certain suitable visual measure, the image *I* is discontinuous along *C* while smooth or homogeneous on each segment Ω_{
i
} (*i* = 1, …, *k*). Until now, a wide variety of techniques including variational methods [1] has been proposed to solve the image segmentation problem.

The variational segmentation methods are characterized by deriving an energy functional from some *a priori* mathematical model and minimizing this energy functional over all possible partitions. The Mumford-Shah model [2] is a classical variational segmentation method, which contains a data-fidelity term, regularization on the model, and regularization on the partitioning. Based on this framework, Chan and Vese [3] developed the frequently used variational segmentation model, which simplifies the Mumford-Shah model to the case of piecewise constant approximations of the image intensity. However, the Chan-Vese model tends to rely on global information to guide contour evolution, and thus fails to segment the images with intensity inhomogeneity [4, 5]. Besides, each region (foreground or background) in Chan-Vese model is also considered as a Gaussian distribution with different mean and same deviation [6]; therefore, they are not suitable for images with Poisson noise.

Intensity inhomogeneity often occurs in real images, especially in medical images, such as X-ray radiography/tomography and magnetic resonance (MR) images [4, 5, 7]. The intensity inhomogeneity usually refers to the slow, non-anatomic intensity variations of the same tissue over the image domain. For example, in MR images, it often appears as an intensity variation across the image, which arises from radio frequency coils as well as variations in object susceptibility. Although the presence of intensity inhomogeneity is usually hardly noticeable to a human observer, variational segmentation models such as the Chan-Vese model [3] are highly sensitive to the spurious. Thus, segmentation of such medical images usually requires intensity inhomogeneity correction as a preprocessing step [8].

Poisson noise also appears in a wide class of real-world applications, e.g., positron emission tomography in medical imaging [9], fluorescence microscopy [10] and radiography images [11, 12]. It is signal-dependent and obeys a Poisson distribution, which describes a signal-dependent perturbation of an image. In particular, the radiograph images are typical images with Poisson noise, which are determined by photon counting statistics and are described as particle-limited, emphasizing the quantized; furthermore, due to technical limitations or artifacts introduced by the object being imaged, the radiograph images such as X-ray is often created with intensity inhomogeneity. In [13], Lee et al. studied the segmentation problem of images with Poisson noise. However, this model was based on intensity homogeneous (roughly constant) statistics and, thus, it was not able to deal with intensity inhomogeneity.

In order to handle intensity inhomogeneity, Li et al. [5] proposed the region-scalable fitting (RSF) model (originally termed as local binary fitting (LBF) model [4]) in a variational level set formulation. In the RSF model, the RSF energy is defined over the neighborhood of each image pixel, and the active contour is deformed to minimize the integration of the RSF energy over the whole image domain. With two extra regularization terms, this final energy minimization problem is converted to solve a level set evolution equation by using the gradient descent method. The RSF model can deal with intensity inhomogeneity accurately, but it is quite sensitive to contour initialization. Because the gradient descent is a very slow numerical method, the RSF model generally converges slowly when implemented numerically. Besides, the RSF model is derived from the Mumford-Shah model [2] that implicitly assumes the given image to be biased by additive Gaussian noise; thus it is not suitable for images with signal-dependent noise (e.g., Poisson noise).

Following the RSF variational model, various extensions and analysis have been studied [7, 14–16], among others. In [7], Wang et al. proposed a local and global intensity fitting model in a variational level set formulation. Zhang et al. [14] proposed a local image fitting (LIF) model by minimizing the difference between an original image and the fitted image. He et al. [15] presented a scheme of improvement on the RSF model in terms of robustness to initialization and noise. Wang et al. [16] proposed a novel algorithm by using a piecewise smooth approximation to image. However, these models have some common drawbacks; in particular, they are not suitable to segment images with Poisson noise because these models are also formulated in view of the Mumford-Shah energy functional [2].

In this study, we propose a variational model to segment images with intensity inhomogeneity and Poisson noise. Based on the RSF model [5] and inspired from variational Poisson denoising model [17], we first propose a new energy functional with the data-fidelity term deduced from Poisson distribution, which is more suitable for images corrupted by Poisson noise. We then reformulate this energy functional into a convex formulation to guarantee the global minima. We also use the weighted total variation (TV) norm as the regularization term to detect the boundaries more easily. Finally, in order to implement the proposed model, we adopt the dual formulation which is introduced by Chan et al. [18] and Chambolle [19] for denoising and is later adapted by Bresson et al. in [20] for segmentation.

The remainder of this paper is organized as follows. In Section 2, we briefly review the level set method, two denoising models [17, 21] and the RSF model [5]. The proposed model is introduced in Section 3. The numerical results are given in Section 4. This paper is concluded in Section 5.

## 2 Related works

### 2.1 Level set method

*C*(

*t*) is represented implicitly by the zero level set of a Lipschitz function

*ϕ*(

*x*,

*t*), called a level set function, with the following properties:

*C*in normal direction with speed

*F*amounts to solving the following level set evolution equation:

with the initial condition *ϕ* (x, 0) = ϕ _{0}(x).

where *δ*
_{
ϵ
}(*ϕ*) is the smooth Dirac function, which is the derivative of the smooth Heaviside function. A segmentation of the image is given by the two regions {*x*|*ϕ*(*x*, *t*) > 0} and {*x*|*ϕ*(*x*, *t*) < 0}. The steady state solution of the Equation 3 hopefully gives a useful edge contour extraction or segmentation of the image.

### 2.2 Two related denoising models

where BV(Ω) is the space of functions of bounded variation defined on Ω, and *μ* > 0 is a parameter to be chosen.

In (4), ∫_{Ω}| ∇ *u*|*dx* is the TV regularization term to remove the noise, and ∫ _{
Ω
}|*I* − *u*|^{2}
*dx* is the *L*
_{2} norm data-fidelity term that measures the dissimilarity between original image *I* and the reconstructed image *u*. It is shown in [23] using probability arguments that the *L*
_{2} norm data-fidelity term is most appropriate for removing additive Gaussian noise. However, many important data contain noise that is signal dependent and obeys a Poisson distribution; thus removing this noise without losing image features requires a data-fidelity term reflecting the noise characteristics.

The energy in (5) differs from the energy functional of ROF model only in the data-fidelity term; the *L*
_{2} norm data-fidelity of the original model is replaced by *F*(*u*) = ∫_{Ω}(*u* − *I* log(*u*))*dx* as a measure of fidelity. We refer the reader to [17] for the derivation of this data-fidelity term using Bayesian statistics. As pointed out in [17], *F*(*u*) is more suitable for Poisson noise. Recently, Chartrand and Staneva [24] have shown that the data-fidelity term reflecting the noise characteristics of the image can provide a better image reconstruction.

### 2.3 Region scalable fitting model

*C*be a closed contour in Ω, which separates Ω into two regions: Ω

_{1}= inside(

*C*) and Ω

_{2}= outside(

*C*). For a given point

*x*∈

*Ω*, the RSF energy is defined as [4, 5]

where *K*
_{
σ
}(*z*) is a kernel function.

*C*that minimizes the energy ${\u03f5}_{x}^{\mathrm{RSF}}$ for all

*x*∈

*Ω*. This can be achieved by minimizing the integral of ${\u03f5}_{x}^{\mathrm{RSF}}$ over Ω:

To handle topological changes, the energy *ϵ*(*C*, *f*
_{1}, *f*
_{2}) is incorporated into a variational level set formulation with two extra internal energy functionals. They then use the gradient descent method to solve the variational level set formulation.

The RSF model can deal with intensity inhomogeneity accurately and efficiently; however, it easily gets stuck in local minima for most of the contour initializations. This makes the RSF model sensitive to the selection of initial contours. Besides, the RSF model uses *L*
_{2} norm to measure the difference between the fitted image and the original image; thus, it is not suitable for images with Poisson noise.

## 3 The proposed model

### 3.1 Intensity fitting energy and its level set formulation

In this section, we first define an intensity fitting energy for a given contour *C* and then change it into the energy directly defined on level set functions.

*C*) =

*Ω*

_{1}and outside(

*C*) =

*Ω*

_{2}. For any

*x*∈

*Ω*, the local fitting energy (LFE) of the contour

*C*at

*x*is defined as

where *h*
_{
i
}(*x*)(*i* = 1, 2) are two values that approximate locally image intensities in Ω_{1} and Ω_{2}, respectively.

which is called the intensity fitting energy (IFE) of the contour *C* in this paper.

To allow contour splitting and merging naturally (i.e., a change of topology), we use a level set function to represent a contour *C*. The IFE functional of the contour *C* is thus changed into the energy functional which is directly defined on the level set functions.

*ϕ*be a level set function, and then the IFE functional in (10) can be expressed as

where *H*(*z*) is the Heaviside function, *M*
_{1}(*ϕ*) = *H*(*ϕ*), *M*
_{2}(*ϕ*) = 1 − *H*(*ϕ*).

*H*(

*z*) is approximated by a smooth function

*H*

_{ ϵ }(

*z*), which is typically defined by [3, 5]

*ϵ*is a positive constant. Therefore, the IFE functional in (11) is rewritten as

where ${M}_{1}^{\u03f5}\left(\varphi \right)={H}_{\u03f5}\left(\varphi \right)$, ${M}_{2}^{\u03f5}\left(\varphi \right)=1-{H}_{\u03f5}\left(\varphi \right)$.

### 3.2 Description of the model

The IFE functional in (13) is not convex with respect to *ϕ* ; hence, the corresponding optimization problem can sometimes get stuck in undesirable local minima. In this section, we propose to solve this problem by determining a globally convex formulation. Our idea comes from [25], which reformulated the piecewise constant Chan-Vese model [3] into a convex model.

In the following, we first derive the gradient descent flow equation of (13) and then find a simplified flow which has the coincident steady state solution with the original gradient descent flow equation and finally define a new and convex energy in view of the simplified flow.

*ϕ*, we minimize the functional

*E*(

*ϕ*,

*h*

_{1},

*h*

_{2}) with respect to the functions

*h*

_{1}(

*x*) and

*h*

_{2}(

*x*). By the calculus of variations, it can be shown that the functions

*h*

_{1}(

*x*) and

*h*

_{2}(

*x*) that minimize

*E*(

*ϕ*,

*h*

_{1},

*h*

_{2}) satisfy the following Euler-Lagrange equations:

*ϕ*. Making the interchange of variables

*x*and

*y*(i.e.,

*x*=

*y*,

*y*=

*x*) for (13), we have

*K*

_{ σ }(−

*z*) =

*K*

_{ σ }(

*z*), we obtain

*h*

_{1}(

*y*) and

*h*

_{2}(

*y*) by minimizing (17) with respect to

*ϕ*using the gradient descent method, we can obtain the gradient descent flow equation of (13) as

where *δ*
_{
ϵ
}(*z*) = *H*′_{
ϵ
}(*z*) = *ϵ*/*π*(*ϵ*^{2} + *z*^{2}) is a smooth Dirac function.

*δ*

_{ ϵ }(

*z*) is a non-compactly supported function, the gradient flow Equation 18 and the following equation have the same stationary solutions:

which is clearly the energy functional associated with the gradient descent flow (20).

*ϕ*to avoid the occurrence of small, isolated regions (e.g., noise points) in the final segmentation results. Most of models focus on the regularizations as length regularization [3], mean curvature regularization [22], and

*H*

^{1}regularization [26]. In this study, we employ the weighted TV norm [20] as the regularization term:

*g*(

*x*) = 1/(1 +

*β*|∇

*I*|

^{2}) is the edge detector function, and

*β*is a non-negative parameter. The weighted TV norm can better smooth the level set function and makes the model detect boundaries more easily. Thus, we obtain the following energy functional

where λ > 0 is a parameter. The energy *E*(*ϕ*) is a convex functional (not strictly), but it does not have a stationary solution because of homogeneous of degree one in *ϕ*; therefore, we need to restrict the solution to lie in a finite interval such as [0, 1].

*ϕ*, we arrive at the entire model as follows:

### 3.3 Minimizing the proposed model by dual formulation

We solve the constrained minimization problem (24) by the dual formulation presented in [18, 19] instead of the usually used gradient descent method. The dual formulation is proposed by Chan et al. [18] and Chambolle [19] to solve the ROF minimization problem (4) for image denoising and is later adopted in [20] for image segmentation.

with $\psi \left(z\right)=max\left\{0,2|z-{\scriptscriptstyle \frac{1}{2}}|-1\right\}$ .

*φ*and propose to minimize the following approximation to (25):

with *θ* > 0. One can note that if *θ* → 0, the functional (26) is exactly the minimization problem (25). Moreover, the functional in (26) is still convex, which means that we can compute its global minimizer. However, (26) is a minimization problem in two variables, thus, we have to perform an alternating minimization respect to *ϕ* and *φ*. The solving process of minimization is described in the following two steps:

*φ*being fixed, the solution of (26) is

*p*= (

*p*

^{1},

*p*

^{2}) satisfies the following equation:

*τ*∈ [0, 1/8] and

*p*

^{0}= 0,

*ϕ*being fixed, the solution of (26) is

## 4 Numerical results

The proposed model has been tested with synthetic and real images from different modalities. The level set function *ϕ* is simply initialized as a binary step function taking 1 inside a region and 0 outside. The set *M* = {*x*|*ϕ*(*x*) > 0.5} is used to extract the objects. Besides, unless otherwise specified, we use the following default parameter setting: the time step *Δt* = 1/8, the space step Δx = Δy = 1, σ = 5, β = 50/255^{2}, θ = 0.1, λ = 1. We record the iteration number and the CPU time from our experiments with Matlab codes run on an PC, with AMD Athlon (tm) 2.70GHz CPU, 2.00 GB memory, and Matlab 7.4 on Windows 7.

**Iterations and CPU time (in seconds) by proposed and RSF models for Figure**
1

First image | Second image | Third image | Fourth image | Fifth image | ||
---|---|---|---|---|---|---|

Image size | 125 × 121 | 161 × 122 | 111 × 110 | 200 × 210 | 156 × 155 | |

RSF model | Iterations | 240 | 250 | 150 | 300 | 150 |

Time (s) | 9.39 | 12.57 | 9.86 | 59.76 | 11.63 | |

Our model | Iterations | 70 | 30 | 80 | 90 | 30 |

Time (s) | 3.56 | 2.12 | 7.06 | 12.86 | 3.76 |

*h*

_{1}(

*x*),

*h*

_{2}(

*x*), $u\left(x\right)={\displaystyle {\sum}_{i=1}^{2}{M}_{i}}\left(\varphi \left(x\right)\right){h}_{i}\left(x\right),$ and the original image I(x). Column 3 shows that the fitting functions

*h*

_{1}(

*x*) and

*h*

_{2}(

*x*) are smooth enough for the three images without or with Poisson noise. In addition, the final fitting image

*u*(

*x*) can better fit the original image

*I*(

*x*) (see Figure 2d,h); in particular, the fitting image

*u*(

*x*) shown in Figure 2l is similar to the one given in Figure 2h, which shows that our model can more fit the original image while it can diminish the influence of Poisson noise. The segmentation results, the fitting functions (

*h*

_{1}(

*x*),

*h*

_{2}(

*x*)), and the final fitting image

*u*(

*x*) demonstrate the robustness of the proposed model to the Poisson noise.

*v*; we choose the optimal values of the parameter

*v*by using a ‘coarse-to-fine’ scheme on the test images.

Figure 3 shows the segmentation results using both models for a synthetic palm image (108 × 130) by adding Poisson noise. Figure 3a is the original image and Figure 3b shows the initial contour. The results of the RSF model and our model are shown in Figure 3c,d. Clearly, our model achieves better segmentation accuracy for this image with Poisson noise; it separates excellently the two middle fingers that stuck almost together. The final contour obtained by our model reflects accurately the true palm shape. It is shown from Figure 3 that our model can get accurate segmentation results for image with Poisson noise, while the RSF model is sensitive to Poisson noise.

Figure 4 shows the segmentation results for a synthetic image with intensity inhomogeneity which contains seven distinct gray levels and with added Poisson noise. Figure 4a is the original image and Figure 4b shows the initial contour. The results of RSF model and our model are shown in Figure 4c,d. Although the RSF model can extract part of the objects accurately, it fails to segment the white disc and is sensitive to noise (see Figure 4c). In contrast, the proposed model produces satisfactory segmentation result for all discs (see Figure 4d). The final level set functions corresponding to both models are shown in Figure 4e,f. From Figure 4f, we can see that the level set function of our model is almost close to a two-valued function (the object is 1 and the background is 0). Experiments in Figure 4 show that the proposed model can achieve better segmentation results for the images with intensity inhomogeneity and Poisson noise compared to the RSF model.

In Figure 5, we give four examples on real radiograph images. Four test images, which are shown in row 1, are a foot image (177 × 116), a cervical vertebra image (123 × 117), a hip joint image (124 × 118.), and a pastern joint image (140 × 125), respectively. In our model, we use *θ* = 0.03 for the third image and *θ* = 0.2 for the fourth image. The foot image in Figure 5a is corrupted by Poisson noise and has severe intensity inhomogeneity. The cervical vertebra image (Figure 5b) has complex object shapes and the object is also corrupted by Poisson noise. The last two images are inhomogeneous and the boundary is very weak. Figure 5a,b,c,d shows the original images along with initial contours. It can be seen from Figure 5e,f,g,h that the RSF model cannot segment correctly these images. Figure 5i,j,k,l shows the corresponding segmentation results of our model; as can be seen, our model accurately detects the objects in these images. These four examples illustrate the abilities of the proposed model to deal with intensity inhomogeneity, weak boundaries, complex object shapes, and immunity to Poisson noise.

Figure 6i,j,k,l displays the segmentation results of our model for the T-shaped image with different initial contour sizes. The initial contours are chosen as circles at the center of image but with different radius. In Figure 6m,n,o,p, we also show the segmentation results of the T-shaped image with added Poisson noise for the same initialization shown in Figure 6i,j,k,l. The segmentation results in the second and third rows of Figure 6 show that our model can obtain accurate segmentation for all of the four initial contour sizes (images without or with Poisson noise). Experiments in Figure 6 show that our model really allows for more flexible initialization even if the images were added Poisson noise.

## 5 Conclusion

Inspired from the RSF model [5] and the variational Poisson denoising model [17], we propose a new variational model to segment images with intensity inhomogeneity and Poisson noise. We first propose an energy functional based on the data-fidelity measure deduced from Poisson distribution. We then reformulate the energy functional as globally convex formulation to guarantee the global minima (not the local minima), which makes our model less sensitive to initialization. The dual method is employed to minimize the convex energy functional with an extra regularization term (weighted TV norm). The proposed model is tested on many synthetic and real images; the results demonstrate that it can efficiently cope with intensity inhomogeneity and Poisson noise.

## Declarations

### Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve this paper. In addition, this work was supported by the Fundamental Research Funds for the Central Universities grant no. CDJXS11100023.

## 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.