# Lesion Detection in the Abdominal Region

## Local Intensity Structure Analysis

The local intensity structure analysis technique is widely used for analysis of object structures in images. Applications of this technique to medical images were originally proposed by Sato et al. [89] andFrangi et al. [90]. Local intensity structure analysis technique enhances blob, line, and sheet structures in images. Blob structure enhancement is commonly used for lesion detection in CAD. For instance, polyp detection in the colon [91], ulcer detection in the small and large intestines [92], and enlarged lymph node detection [93] methods were developed using the blob structure enhancement technique. Line and sheet structure enhancements are

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used in segmentations of blood vessels and lung lobes. This technique utilizes eigenvalues of the Hessian matrix to enhance the target structures.

Enhancement processes based on local intensity structure analysis consist of three steps including (1) local region clipping, (2) a Hessian matrix elements calculation, and (3) an enhancement filter value calculation. Details of each step are described below.

In the local region clipping step, a small local region is clipped from an input image. Local region clipping is performed for a local region centered at each voxel in the image. Local region size is defined based on the size of the detection target. If the detection target is a sphere of diameter 1 cm, a local region will be a square of =1 cm or a sphere of diameter =1 cm. The local region size should be chosen so as to include the target. A large local region size will reduce the detection performance of the enhancement filter.

In the Hessian matrix elements calculation step, the matrix elements are calculated for each local region. The Hessian matrix is written as

The Hessian matrix elements are the second-order partial differential coefficients of a function *f*. The function *f* represents image intensity values in the local region. The function is obtained based on the image intensity values. For 3D medical images, a second-order polynomial

can be used as the function. a = (аь...,а_{10}) is a coefficient vector of the polynomial. The coefficient vector a is calculated by minimizing squared errors between the image intensity values and the polynomial. Instead of obtaining the function that represents image intensity values, the second-order partial differential coefficients of the function can be directly calculated from image intensity values. The second-order partial differential coefficients of the function are estimated from difference values of the image.

Finally, the enhancement filter value calculation step is performed. In this step, three eigenvalues of the Hessian matrix Я_{1},Я_{2},Я_{3}(Я_{1} > Я_{2} > Я_{3}) and their corresponding eigenvectors e_{1;} e_{2}, e_{3} are calculated. The intensity structure in the local region can be classified into blob, line, or sheet categories by checking the magnitude relationship of the three eigenvalues. If the intensity structure in the local region shows a blob structure, three eigenvalues have nearly equal values and they are smaller than o. Eigenvalue conditions of three local intensity structures are shown in Table 4.2. A blob structure enhancement filter [89] can be defined that gives high outputs when the eigenvalues satisfy the eigenvalue condition of the blob

**Table 4.2 **Eigenvalue conditions of three local intensity structures [89]

Local intensity structure |
Eigenvalue condition |

Blob |
A3 22 A2 — |

Line |
A3 22 A2 |

Sheet |
A3 A2 — |

**Fig. 4.4 **An example of output of the blob structure enhancement filter applied to the colon including a polyp. *Red* and *yellow colors *indicate low and high output values, respectively. High output values are shown in a blob structure

structure. The blob structure enhancement Alter is given by where

i

The blob structure enhancement filter outputs high values in blob structures. Figure 4.4 shows an example of blob structure enhancement filter output. The line and sheet structure enhancement filters are defined similarly based on each eigenvalue condition. After applying the blob structure enhancement filter to all voxels in the image, voxels having high output values of the filter can be classified into lesion candidate voxels.