**4. Removement of vessels by applying Maximum likelihood method**

Before tumor detection step, we first remove vessels from CT images. In the conventional method [5], as the intensity of vessels is higher than those of health liver tissues and tumor tissues, intensity threshold method is used to remove vessels. We classify the CT volume into 3 classes by using Maximum likelihood method. And then, voxels of the class with the highest mean are removed as vessels. After this process, CT images only include tumor and healthy liver tissues. The tumor detection problem can be simplified as a 2-class classification problem. This process will also significantly reduce the detection time.

### **5. Tumor candidate detection by using EM/MPM algorithm**

To extract tumor candidate, we used the EM/MPM algorithm [9]. It is based on a Bayesian framework that assumes a Gaussian mixture to model intensity distribution and concurrently estimates both the labels of the voxels and the model parameters. In MPM, the

 (a) (b)

(c) Fig. 3. (a) Estimated intensity PDFs for tumor (curve on the left), liver (curve in the center), and vessel (curve on the right); (b) Overlap the Curve Pattern A to the new image's liver

( ) (255 1) 1 ( ) ( )

Before tumor detection step, we first remove vessels from CT images. In the conventional method [5], as the intensity of vessels is higher than those of health liver tissues and tumor tissues, intensity threshold method is used to remove vessels. We classify the CT volume into 3 classes by using Maximum likelihood method. And then, voxels of the class with the highest mean are removed as vessels. After this process, CT images only include tumor and healthy liver tissues. The tumor detection problem can be simplified as a 2-class

To extract tumor candidate, we used the EM/MPM algorithm [9]. It is based on a Bayesian framework that assumes a Gaussian mixture to model intensity distribution and concurrently estimates both the labels of the voxels and the model parameters. In MPM, the

*x T if T x T T T*

  min max

*i*

*i i*

*if x T x T*

0( )

min max

(1)

volume histogram; (c) Histogram obtained after our histogram transformation

: 3, : 3 

**4. Removement of vessels by applying Maximum likelihood method** 

classification problem. This process will also significantly reduce the detection time.

**5. Tumor candidate detection by using EM/MPM algorithm** 

*tumor vessel*

min

max min

*i*

min max

*T T*

cost function is defined to minimize the total number of misclassified voxels. It can be proved that minimization of the cost function is equivalent to maximization of the posterior marginal probability of the label fields (Eq. 2) [10].

$$\hat{\mathbf{x}}\_{s\_{\text{MAP}}} = \arg\max\_{\mathbf{x}} P\_{X\_s \mid \mathbf{Y}} \{ \mathbf{x}\_s \mid y\_\prime \theta \} = \arg\max\_{\mathbf{x}} \sum\_{\mathbf{x} \in \Omega\_{k,s}} P\_{\mathbf{X} \mid \mathbf{Y}} \{ \mathbf{x} \mid y\_\prime \theta \} \tag{2}$$

In Eq. (2), *x*, *y*, and *θ* are the label vector, feature vector, and model parameters, respectively, *s* is a pixel, *k* is the label of pixel *s*, and refers to all possible labels of the image in which the label of pixel *s* equals *k*. The posterior probability is composed of two factors, namely the likelihood function and the prior probability. The likelihood function is a multiplication of the Gaussian distribution function and the prior probability is modeled by Markov random field (Eq. 3).

$$\begin{split} P\_{\boldsymbol{X}\_{s}|\mathcal{Y}}\{\boldsymbol{x}\_{s}|\boldsymbol{y},\boldsymbol{\theta}\} &= \\ \sum\_{\boldsymbol{X}\in\boldsymbol{\Omega}\_{k,s}} P\_{\boldsymbol{X}|\mathcal{Y}}\{\boldsymbol{x}\mid\boldsymbol{y},\boldsymbol{\theta}\} & \propto \sum\_{\boldsymbol{X}\in\boldsymbol{\Omega}\_{k,s}} \left[ \left[ \prod\_{i=1}^{N} \frac{1}{\sqrt{2\pi\sigma\_{\boldsymbol{x}\_{i}}^{2}}} \right] \cdot \exp\left(-\sum\_{i=1}^{N} \frac{\left(\boldsymbol{y}\_{i}-\boldsymbol{\mu}\_{\boldsymbol{x}\_{i}}\right)^{2}}{2\sigma\_{\boldsymbol{x}\_{i}}^{2}} - \sum\_{\{r,s\}\neq\boldsymbol{\alpha}} \frac{1}{T(\boldsymbol{n})} \delta(\boldsymbol{x}\_{r},\boldsymbol{x}\_{s}) \right) \right] \tag{3} \end{split}$$

In Eq. (3), *N* is the total number of voxels, ( *xi* , <sup>2</sup> *xi* ) are model parameters of class *xi*, *yi* is the intensity of a voxel, *T*(*n*) is called temperature, and δ(*x*r, *xs*) is a function that contributes to the labels of the neighboring voxels (r) when determining the label of *s*. For all neighboring voxels whose labels (*xr*) same as *xs*, the output of the weighting function is zero. Otherwise, a value of 0.5 or 1 is assigned according to Fig. 3. In our method, we consider six neighboring voxels; four in the same slice and two in the upper and the lower slices (Fig. 4.)

Fig. 4. Weight functions for 6-neighbours of a voxel.

Optimization of Eq. (2) is not simple. We use the simulated annealing method to iteratively determine an estimate [9], which is given by <sup>1</sup> *Tn T n c* ( ) / log( ) . Here, *T1* is the initial temperature which is a large constant, *c* is a constant number and *n* is the iteration number. After determining the MPM estimate (the E-step), the model parameters are calculated (the M-step). We continue the iteration of the two steps until convergence is achieved. As a result, we obtain tumor candidates that may include both true and false positive regions. In our experiments, we assign the values of 1.4 to *c*, 50 to *n*, and 2.0 to *T1* through several testing runs. To remove false results, we use the shape information described below.
