The framework of the proposed approach is illustrated in Figure
4. The segmentation is three-fold: a specific preprocessing of US images to obtain denosing images, coarse segmentation of the kidney. an efficient optimization process of segmentation result with smooth boundary have been implemented. Sometimes the shape model may not cover all the shapes to get the satisfied result, so we can use the alignment model to generate a new shape to cover all kinds of deformed shape.
Initial process: NLTV denoising
US images are known to be low quality due to speckle or acounstic shadow. The aim of preprocessing US images is to reduce the noise, while preserving the echogenic boundaries of the organ. In this paper, we use NLTV denoising method to get a denoised image. Zhou-Scholkopf
[19] and Gilboa-Osher
[20] have introduced the definitions about nonlocal/graph (NL/G) gradient, which are useful in explaining the NLTV denosing model. The NL/G gradient of a function
is defined for the pair of points (x, y) ∈ Ω × Ω as:
(1)
Where
is the given image, such as a noise image,
is the edge weight between the points x, y (w defines a graph), function w is defined as:
Where f(x, y, z) = ∫
Ω
G
a
(z)|u
0(x + z) − u
0(y + z)|2
dz is the distance between the points x and y, G
a
is a Gaussian function with standard deviation a, and h is a Positive constant which acts as a scale parameter. The NL/G gradient is to give a weighted value between two points x, y to indicate whether the patches around point x and patches around point y are the same.
We utilized the nonlocal image denoising model of Gilboa-Osher
[20]. Gilboa-Osher has used the NLTV operator in the form of ∫
Ω
| ∇ NL
u|. The whole image denoising model is written as follows:
(2)
where u, u
0:
, ∇ NL
u:
, and u
0 is the given noisy image and λ is a positive constant that controls the trade-off Between the regularization process and the fidelity with respect to the original image.
In order to get the denoised image u
opt
, we have to minimize Equation (2). Here, Zhang-Burger-Bression- Osher’s model
[21] was used to solve the problem of NLTV minimization. More details can be referred to
[21]. This initial process helped us to acquire an image with the kidney region in almost homogenous intensity gray scale. This is very beneficial for the following segmentation.
Segmentation process: DRLSE
In level set method, the contour concerned is embedded as the zero level set of a level set function (LSF). During the evolution of level set, LSF may not be smooth. It may become steep or flat which destroy the unique property | ∇ ϕ| = 1. As a result reinitialization is in need. The most popular method is proposed in
[22]. In this section, we used a level set formulation that has an intrinsic mechanism of maintaining this desirable property of the LSF. This formulation is called DRLSE proposed by Chunming Li
[6].
Let I be an image on a domain Ω, edge indicator function g is defined by
(3)
Where G
σ
is a Gaussian kernel with a standard deviation σ. The convolution in (3) is used to smooth the image to reduce the surplus noise. This function g usually takes smaller values at object boundaries than at other locations.
For an LSF
, energy functional ε(ϕ) is defined by
(4)
Where μ > 0 λ>0 and
are the coefficients of the energy functionals R(ϕ), L(ϕ) and A(ϕ) which are defined by
(5)
(6)
And
(7)
Where δ and H are the Dirac delta function and the Heaviside function, respectively. This energy functional (4) can be minimized by solving the Following gradient flow:
(8)
Given an initial LSF ϕ(x, 0) = ϕ
0(x), the first term on the right side in (8) is associated with the distance regularization energy R(ϕ) (5) which keeps the unique property LSF | ∇ ϕ| = 1. The second is associated with the energy terms L(ϕ) (6) which is minimized when the zero contour of ϕ is located at the object boundaries, while the third term is related to A(ϕ) (7) which is introduced to speed up the motion of the zero level contour in the level set evolution process. The segmentation process produced a binary image with the black and white region representing the kidney and background respectively.
Post process: shape prior optimization
To model the shape prior, we applied a similar shape model construction method described in
[4],
[16] and
[17]. This shape model is based on the principal component analysis (PCA). So we can obtain the main variations of a training set in which alignment model is applied,while removing redundant information. The new idea introduced in
[4] is to apply the PCA on the signed distance functions (SDF) of these contours which are implicit and parameter free representations, instead of the parametric geometric contours or active shape model. This PCA method is able to produce a new kidney shape based on the training set {ϕ
j
}:
where xpca is called the coefficients of eigen vector,
is a shape formed from the shape space,
is the mean shape of the shape space, and matrix W
p
contains p principal components of the shape space.
Shape prior is to find the interesting shape in the binary image produced by the DRLSE method. The shape energy model proposed in
[8] is as follows:
Functional F
shape is based on the fixed contour C, the vector xpca of PCA eigen coefficients and the vector x
T
of geometric transformations. This functional evaluates the shape difference between the contour C after the geometric transformations x
T
and the zero level set
of the shape function
provided by the PCA. The function
at the point C(q) is:
where | · | stands for the Euclidean norm.
The flow minimizing F
shape with respect to the vector of eigen coefficients xpca is :
(9)
and φ is a constant representing the signed distance function of the shape in the binary image.
The flow minimizing F
shape with respect to the vector of geometric transformation x
T
is:
(10)
For one iteration, the computation order is in the following order: (10), (9).
Compare with other methods
The segmentation process of
[1, 3, 4, 16–18] is shown in the Figure
5. The iteration procedure of these methods as follows: Firstly, they evolve contour level set according to the image texture, image gradient or region. Secondly, a similar shape of the evolving contour is found. Finally, reinitializing the level set is needed to keep its unique property | ∇ ϕ| = 1. In the every iteration, they find the interesting shape.
As the segmentation flow chart of our method is depicted in the Figure
6, we can observe that we only evolve contour level set in the every iteration. We do not need reinitialize the contour, because DRLSE has an intrinsic mechanism of maintaining this property of the LSF | ∇ ϕ| = 1. When the contour evolution is terminated, we find the interesting shape from the shape. Compared with the other method in
[1, 3, 4, 16–18], we do not need to search the interesting shape in every iteration and reinitialize the level set function. Thus, we save a lot of computation time.
Compared with the original level set or DRLSE methods, we use the shape prior to optimize the segmentation result. Because the ultrasound image is low quality with speckle noise, the original level set or DRLSE method usually gets the segmentation result with indent boundary. In our framework, we use the shape prior to get a satisfied segmentation result with smooth boundary.
Due to the diversity in patients’ kidney shapes, the training sets used in the construction of the kidney shape space may not be enough. In many level sets with shape prior method
[4, 16–18], they had only used the similar model like the alignment model in the process of shape space construction. Once the shape space was established, it remained unchanged. However, in many situations, we cannot cover all kinds of shape from the initial data set. Hence the alignment model should be used to generate a new shape in order to give more kind of the shape space after the segmentation. In our process, when the SN, SP and PPV of the segmentation results all fell under 90%, we used the alignment model (AM) proposed in
[18] to reconstruct the shape space and the new procduced shape was done by manual segmentations.