**3. Contrast enhancement**

As tumor detection is mainly based on the intensity of CT images, the contrast of the images is very important. Two typical histograms of CT images having high and low contrasts are shown in Figs. 2(a) and 2(b), respectively. As shown in Fig.2(a), if the contrast of CT images is high, the tumor is in a different intensity range (left small peak) with the liver (right large peak) and the tumor can be easily detected by the intensity threshold, while if the contrast of

We propose a new method for detecting tumors in CT images. Our method is based on adaptive contrast enhancement and the expectation maximization / maximization of the posterior marginal (EM/MPM) algorithm. User interaction is not required and both large and small tumors can be accurately found. Compared with our previously reported method [6], the newly proposed method is also suitable for images with poor contrasts. We describe the method in Sections 2–6 and present the experimental results in Section 7, followed by

Our method is composed of seven steps: (1) read the CT images; (2) extract the liver region using a well-established liver region segmentation program [8]; (3) smooth out the noise from the CT images; (4) enhance the CT image contrast by using probability density functions (PDFs) estimated from the training data; (5) remove vessels by applying Maximum likelihood method; (6) detect tumor candidates by employing the EM/MPM

algorithm; and (7) detect and segment the tumor regions by using a shape filter.

Fig. 1. The flowchart of our proposed method for tumor detections

As tumor detection is mainly based on the intensity of CT images, the contrast of the images is very important. Two typical histograms of CT images having high and low contrasts are shown in Figs. 2(a) and 2(b), respectively. As shown in Fig.2(a), if the contrast of CT images is high, the tumor is in a different intensity range (left small peak) with the liver (right large peak) and the tumor can be easily detected by the intensity threshold, while if the contrast of

**3. Contrast enhancement** 

our conclusions.

**2. Overview of the proposed method** 

CT images is low, the tumor is in the same narrow intensity range as the liver as shown in Fig.2(b) and it is difficult to detect the tumor from the liver volume. Density value of all objects is in a narrow range as shown in Fig.2(b). So we have to enhance the contrast of CT images as a preprocessing.

Fig. 2. Histograms with (a) high (b) low contrasts

A piecewise linear histogram transformation is usually employed to enhance the intensity contrast. However, it is difficult to determine a fixed set of lower and upper limits of the transformation slope for all images because they are data dependent. In this study, we automatically and adaptively determine the parameters from each image. In liver CT images, there are three classes of tissues: tumor, healthy liver, and vessels. First, sample voxels of the three classes are manually selected from the training data. The intensity PDFs of each class is estimated.

We also compute their mean values ( *<sup>A</sup> tumor* , *<sup>A</sup> liver* , *<sup>A</sup> vessel* ), their standard derivations ( *tumor* , *liver* , *vessel* ), and *<sup>A</sup> M* . *<sup>A</sup> M* is the intensity value which has the highest probability in the liver class. The mean values have the property of *A AA tumor liver vessel* . In Fig. 3(a), the three curves, from left to right, represent the PDFs of the tumor, the (healthy) liver, and the vessel classes. The mean and standard deviation values may differ among images, but the mean value of the (healthy) liver is always larger than that of the tumor and smaller than that of the vessels. This can be used for the classification of the three classes. The pattern of these three curves is called Curve Pattern A.

Given a new image and its segmented liver volume, we first compute the intensity histogram of all voxels in the liver volume. We find the intensity value *<sup>B</sup> M* , which has the most component in the liver class, and assume that *<sup>B</sup> M* corresponds to the probability density peak of the class of healthy liver tissues in the new liver volume. Such an assumption is viable because the healthy tissues normally dominate the volume.

As a result, the mean values of the tumor and vessels are estimated as *B AA tumor M M tumor* and *B AA vessel M M vessel* , respectively. Now, we set the lower limit as min 3 *T tumor tumor* and the upper limit as max 3 *T vessel vessel* . The intensity transform formula for the range 0–255 is given in Eq. (1). Using the formula, the intensity histograms of the liver images are re-estimated [Fig. 3(c)] for later use.

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

$$\begin{cases} \begin{array}{c} (\mathbf{x}\_{i} - T\_{\text{min}}) \\ (T\_{\text{max}} - T\_{\text{min}}) \end{array} (2.55 - 1) + 1 \quad \text{if } (T\_{\text{min}} \le \mathbf{x}\_{i} \le T\_{\text{max}})\\ \mathbf{0} \qquad \text{if } (\mathbf{x}\_{i} < T\_{\text{min}} \cup \mathbf{x}\_{i} > T\_{\text{max}})\\ T\_{\text{min}}: \boldsymbol{\mu}\_{\text{max}} - 3\boldsymbol{\sigma}, T\_{\text{max}}: \boldsymbol{\mu}\_{\text{wval}} + 3\boldsymbol{\sigma} \end{array} \tag{1}$$
