Automatic Segmentation of the Liver in CT Using Level Sets Without Edges J.F. Garamendi1 , N. Malpica1 , J. Martel2 , and E. Schiavi1 1
Universidad Rey Juan Carlos, M´ ostoles, Madrid (Spain) {juanfrancisco.garamendi, norberto.malpica, emanuele.schiavi}@urjc.es 2 Fundaci´ on H´ ospital de Alcorc´ on, Alcorc´ on, Madrid (Spain)
[email protected] Abstract. Liver volumetry is a required step for the planning of liver surgery and resection. It is generally based on Computerized tomography images, and segmentation of the liver is the most important step of the process. We propose an automatic segmentation algorithm based on a geometric level set method which provides an accurate segmentation of the liver, and requires no a priori information. We show results on different datasets, with and without a contrast agent. The segmentation is compared to manual delineation by a radiologist with good results.
1
Introduction
The planning of liver surgery requires accurate volumetric measures of the organ. Planning is generally based on computerized tomography (CT) and more recently on Magnetic Resonance Imaging (MRI). The main step for volumetry is the segmentation of the organ, which is generally carried out manually using a software platform. Manual delineation of the liver is time-consuming and can lack repeatability among users. The automatic segmentation of the liver is especially difficult due to several structural reasons: – Hounsfield units corresponding to the liver are the same as those of neighbouring organs, so the use of simple gray level segmentation methods give inaccurate results. – The shape and size of the liver can vary a lot between patients, making it difficult to impose a priori conditions or models. Several semiautomatic methods have been proposed and tested, based mainly on the interactive editing of the dataset by the user. Some methods use the segmentation of other structures in the image as a reference for the liver [4]. Soler et al. use a statistical shape model, trained from previous scans, which is deformed onto the new image [3]. Pan et al. proposed a method based on level sets with a new dynamic force [2]. The model must be stopped by the user and interactively corrected. Recently, level set methods have evolved from being edge-based to including statistical information about the region to segment [1]. We have developed a variation of the Chan-Vese method for the automatic segmentation of the liver, which needs no a priori shape or size information and stops automatically. J. Mart´ı et al. (Eds.): IbPRIA 2007, Part I, LNCS 4477, pp. 161–168, 2007. c Springer-Verlag Berlin Heidelberg 2007
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This paper is organized as follow, in section 2 the algorithm is described in detail. Section 3 shows some results on clinical datasets. We end with conclusions in section 4.
2 2.1
Material and Methods Preprocessing
The original CT data is encoded in twelve bits, so we have 4096 gray levels. A supervised preprocessing step is necessarily applied in order to select a gray level window around the liver. The preprocessing step consists of a windowing dependent on the mean μ and the standard deviation σ calculated from a selected region of the liver, using function (1) or (2). The proper choice of the windowing function, (1) or (2), must be done by the user clicking on the image. Let us denote by I0 the new image processed by using expression 1 or expression 2: ⎧ if I(x, y) < μ − 3σ ⎨ μ − 3σ if |I − μ| 3σ I0 (x, y) = I(x, y) (1) ⎩ μ − 3σ if I(x, y) > μ + 3σ ⎧ if I(x, y) < μ − 3σ ⎨ μ + 3σ if |I − μ| 3σ (2) I0 (x, y) = I(x, y) ⎩ μ + 3σ if I(x, y) > μ + 3σ Bottom row of figure 1 shows the effect of the preprocessing using differents μ and σ, obtained from the regions shown in the top row. 2.2
Segmentation Model
Chan-Vese Algorithm. Let Ω ⊂ IR2 be an open, bounded domain (usually a rectangle) where (x, y) ∈ Ω denotes pixel location and I0 (x, y) is a function representing the intensity image values. The Chan-Vese model for binary segmentation is based on the minimization of an energy functional expressed in terms of a level set formulation. Let ω ⊂ Ω (eventually not connected) open, a positive measured sub-region of the original domain. If the curve C represents the boundary of such a segmentation ω then, in the level set formulation, the (free) moving boundary C is the zero level set of a Lipschitz function φ : Ω → IR, that is: C = {(x, y) ∈ Ω : φ(x, y) = 0}, C = δω where ω = {(x, y) ∈ Ω : φ(x, y) > 0}, Ω\ω = {(x, y) ∈ Ω : φ(x, y) < 0}. The level set function φ can be characterized as a minimum of the following energy functional: Ecv (ein , eout , φ) = μ |∇H(φ)|dxdy + ν H(φ)dxdy Ω Ω +λin H(φ)ein dxdy Ω +λout (1 − H(φ))eout dxdy (3) Ω
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A
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C
Fig. 1. The effect of preprocessing. Left, original slice. Top row, region of interest used for calculate μ and σ, below, the slice preprocessed according with expression 1 (images A and B) and expression 2 (image C). The color bar at left of each image shows the value of Hounsfield Units in the image.
where μ, ν, λin and λout are parameters which can be considered as weight factors which control the trade off between smoothness (μ, ν) and the fidelity data terms (λin , λout ). The Heaviside function H(x), H(x) = 1 if x ≥ 0 and H(x) = 0 otherwise, allows to express the length of C and the area of ω respectively by |C| = Length(φ = 0) = |∇H(φ)|dxdy, |ω| = Area(φ 0) = H(φ)dxdy Ω
Ω
The functions ein , eout are defined as ein (x, y) = |I0 (x, y) − cin |2 , where cin and cout are I0 H(φ)dxdy cin (φ) = Ω , H(φ)dxdy Ω
eout (x, y) = |I0 (x, y) − cout |2
(4)
Ω cout (φ) =
I0 (1 − H(φ))dxdy (5) (1 − H(φ))dxdy
Ω
and represent the mean value inside the segmentation (cin ) and the mean value outside the segmentation (cout ). Following the calculus of variations, the minimum of the energy Ecv corresponds to a solution of the Euler-Lagrange equations: ∇φ 0 = δ (φ) μ∇ · − ν − λin ein + λout eout , |∇φ|
a.e. (x, y) ∈ Ω
(6)
where δ (x) is a regularized version of the Dirac delta function [1]. As usual, the equation is complemented with homogeneous Neumann boundary conditions.
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Algorithm Proposed. The expressions ein and eout (see (4)) that appear in the Chan-Vese model can be modified in order to use local information around the pixel. In order to do this, we redefine the above functions as: ⎛ ⎛ ⎞2 ⎞2 I0 dsdr I0 dsdr ⎜ D(x,y) ⎜ D(x,y) ⎟ ⎟ ⎜ − cin ⎟ − cout ⎟ ein (x, y) = ⎜ ⎝ |D(x, y)| ⎠ eout (x, y) = ⎝ |D(x, y)| ⎠ (7) where D(x, y) ⊂ Ω is a domain around the pixel (x, y). In the experiments, D(x, y) is a square centered in (x, y). 2.3
Numerical Implementation
As usual, instead of solving the elliptic equation (6) directly, we solve the associated parabolic equation: ∇φ ∂φ = δ (φ) μ∇ · − ν − λin ein + λout eout ∂t |∇φ|
(8)
The derivatives are implemented with a finite difference scheme , Dd+ is the forward difference in the direction d and Dd− is the backward difference in the direction d. The parabolic equation (8) is solved by an explicit gradient descent method: ⎛ ⎞ ⎡ + n D φ φn+1 − φn x ⎠+ = δ (φ) ⎣μDx− ⎝ Δt (D+ φn )2 + (D+ φn )2 + x
⎛
y
⎞ Dy+ φn
1
⎤
⎠ − ν − λin ein + λout eout ⎦ +μDy− ⎝ (Dx+ φn )2 + (Dy+ φn )2 + 1
(9)
where ein and eout are computed by using (7) and 0 < 1 1 is a small parameter avoiding degeneracy in the elliptic equation. 2.4
Complete Algorithm
The global procedure for liver segmentation is implemented using a multistage approach, consisting of three tightly interleaved stages: – Preprocessing step: In this stage the expert selects a Region-of-Interest (ROI) inside the liver and the image is preprocessed according with section 2.1. The value cin is fixed to the mean of the ROI to guide the segmentation to the liver. – Segmentation step: Equation (8) with ein and eout from (7) is solved using the scheme proposed in [1]:
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• Initialize φ0 as the ROI selected by the user at the preprocessing step. Set n = 0. • Compute cout (φn ) by (5). • Solve the PDE in φ from (8) to obtain φn+1 . • Reinitialize φ locally to the signed distance function to the curve. • Check whether the solution is stationary. If not, n=n+1, recalculate cout and repeat. – Postprocessing step: Small bubbles are automatically removed. Only the biggest object of the segmentation (the liver) is preserved. 2.5
Data Adquisition
The test materials are a set of multi-phase CT images of human abdomen. The images were acquired at the Alcorcon Hospital in Madrid, which were collected at three phases before and after contrast agents were injected. They are called the pre-contrast, early and portal images respectively. In the first series the dimensions are (512x512x40) in the pre-contrast phase and (512x512x38) in the early and portal phase. In the second series the dimensions are (512x512x50) ,
Fig. 2. The effect of preprocessing in the segmentation. At the top row, the image segmented is Fig. 1-A (without preprocessing step) with Chan-Vese algorithm (top left) and with our modification (top right). At the bottom row, the image segmented is Fig. 1-B (with preprocessin step) with Chan-Vese algorithm (bottom left) and with our modification (bottom right). all images are displayed within the range from -141 to 284 HU.
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Fig. 3. In black, manual segmentation and examples of segmentation result in white over images in portal phase. Paremeters are μ = 10−1 · max(I0)2 , ν = 0, λin = 0.6, λout = 1, and Δt = 0.1, = 1. All images are displayed within the range from -110 to 190 H.U.
(512x512x48) and (512x512x46) in pre-contrast, early an portal phase respectively and the dimensions of the third series are (512x512x46) in the three phases. Spatial resolution is (0.74x0.74) mm and slice thickness is 5 mm in all studies.
3
Experimental Results
The algorithm has been tested on the series explained in section 2.5 to take into account different signal to noise ratios and differents shapes of the liver. A first set of experiments was carried out to evaluate the diferences between the Chan-Vese method and the modification proposed in this work, and to evaluate the effect of the windowing in the method. Figure 2 shows the segmentation results over the same slice with the Chan-Vese algorithm (left column) and our modification (right column) using differents windowing preprocess. Figure 3 shows four slices of the liver and the manual segmentation and the semi-automatic segmentation. The first row is a manual segmentation done by
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Fig. 4. Segmentation result over images in portal phase. At the left, two slices of a liver hipertrofied and at the right two slices of a cutted liver. Paremeters are μ = 10−1 · max(I0 )2 , ν = 0, λin = 15, λout = 1, and Δt = = 1. All images are displayed within the range from -110 to190 H.U.
an expert. The middle row shows the result obtained with our algorithm and the bottom row shows both segmentations on the same image. Parameters in the experiments are μ = 10−1 · max(I0 )2 , ν = 0, λin = 0.6, λout = 1, and Δt = 0.1, = 1. In order to test the algorithm with ’non standard’ livers, we use a CT of two patients, one with the liver bigger than normal and the other with the liver previously rejected. Figure 4 shows two slices of this CT. Parameters are μ = 10−1 · max(I0 )2 , ν = 0, λin = 15, λout = 1, and Δt = = 1.
4
Conclusions
We have proposed a new method for liver segmentation in CT images which requires no a priori information. The method is based on the Chan-Vese model, with a previous adaptive gray level windowing.. The algorithm stops automatically and is robust with respect to the initial contour. The method has been applied to clinical datasets with good results and has been tested on data sets with different liver shapes and sizes. Segmentation of the liver in different phases of contrast administration has also been achieved. Further clinical validation of the algorithm is warranted.
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References 1. Chan, T.F., Vese, L.A.: Active Contours Without Edges. In: IEEE Transactions on Image Processing, vol. 10(2), IEEE, NJ, New York (2001) 2. Pan, S., Dawant, B.: Automatic 3D segmentation of the liver from abdominal CT images: a level set approach. In: Proc. SPIE Medical Imaging 4322, 128–138 (2001) 3. Soler, L., Delingette, H., Malandain, G., Montaganat, J. et al.: Fully automatic anatomical, pathological and functional segmentation from CT scans for hepatic surgery. Computer Aided Surgery 6, 131–142 (2001) 4. Seo, K., Ludeman, L.C., Park, S., Park, J.: Efficient liver segmentation based on the spine. In: Yakhno, T. (ed.) ADVIS 2004. LNCS, vol. 3261, pp. 400–409. Springer, Heidelberg (2004)
