# Overview of Lymph Node Segmentation from Abdominal CT Images

This section introduces the lymph node extraction process on abdominal CT images. The basic flow of abdominal lymph node detection consists of four steps: (1) preprocessing, (2) blob-like structure enhancement, (3) initial candidate detection from enhanced image, and (4) false-positive reduction.

## Preprocessing

In this step, a Gaussian smoothing filter of kernel size о is applied. Because the sizes of enlarged lymph nodes vary, it is necessary to adjust the threshold target size. This size control can be achieved by changing the size of the kernel of the Gaussian smoothing filter and the size of the hypersurface fitting area explained in the next subsection.

## Blob-Like Structure Enhancement

As stated in the previous sections, enlarged lymph nodes are spherical or elliptical in shape, which can be described as blob-like structures. Intensity structure analysis based on a Hessian matrix can distinguish between sheet-, line-, and blob-like shapes. As enlarged lymph nodes may have a blob structure, a Hessian-based blob structure enhancement filter is used. A blob structure enhancement filter was developed by Sato and Frangi in 1999. The basic idea of this method is that specific patterns emerge from the eigenvalues of the Hessian matrix.

The partial derivative of the intensity function *f* can be derived by simple numerical differentiation of *f* or hyper-curve fitting to the function f. After computing the Hessian matrix at (x,*y,*z), three eigenvalues X_{b} X_{2}, X_{3} (0 > X_{1} > X_{2} *>* X_{3}) are computed. Before applying this blob-like structure enhancement filter, the Gaussian filter whose standard deviation is о is executed. The eigenvalues obtained by applying the Gaussian filtering of о are described as X^, X2,^{CT/}, X^{<}^^ ^{)}.

Although there are several variations in blob-like structure enhancement, Nakamura et al. (2013) [212] used the following blob-like structure enhancement filter:

for a CT voxel x = (x, y, z), the blobness value at x can be expressed as

where X_{1},X_{2}, and X_{3} are the eigenvalues of the Hessian matrix computed at the voxel x. The Hessian matrix of the voxel located at (*x*, *y*, *z*) can be calculated by

where *f* means a function that approximates intensity distribution around the point p. The function *f* can be expressed as

where *a*, *b*, *c*, *d*, *e*, *f*, *g*, *h*, l, and *m* are coefficients of the function. A set of these coefficients are denoted as w. These coefficients can be obtained by solving the least mean square problem that minimizes a residual error *e* defined as

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Optimum w is obtained as the solution of

As stated above, it is necessary to enhance lymph nodes of different sizes. If the input image is smoothed by the Gaussian smoothing filter of kernel size о, a hypersurface fitting process is performed in the area of 3 о. Blob structure-enhanced images of different scales are generated by computing G_{B}(x) for all voxels of the input image.