## Figure 431

Algorithm pipeline for the ellipsoidal filter.

for this conversion. The second step in this preprocessing is image resizing. We used the standard wavelet transform method to down-sample the volume to half, primarily for speed concerns. Convolution of the image volume with the higher-order Gaussian derivative operators: The computation of the second Derivatives of Gaussian in the Hessian matrix is implemented using three separate convolutions with one-dimensional kernels given as:

Ixiyizk (x, a f) = j-G (x, a f) ® j j- G (y, a f) ® j (z, a f) ® 1 (x) jj convolution-1

convolution-2

convolution-3

where I (x) is the interpolated gray-scale volume; i, j, and k are the nonnegative integers satisfying i + j + k = 2; and 3ct was used as the radius of the kernel. G is the Gaussian kernel in the x, y, and z directions. j-fG (z, ct f) <g> I (x) is the convolution of the Gaussian kernel with a standard deviation of ctf. Note that three sets of convolutions are done to obtain the scale-space representation of the gray-scale volume, rather than one convolution in three dimensions.

1. Running the directional processor for computing the eigenvalues. This is computed using Jacobi's method.
2. Computation ofthe vessel score to distinguish the vessels from non-vessels using the eigenvalues based on connectivity, scale and contrast. This can be computed using the combination of components that are computed using the geometry of the shape, which in turn is a function of the eigenvalues \i, \2, and \3,

where y is the Lindeberg constant and a, p, and € are the thresholds that control the sensitivity of the filter for the measurements of features of the image such as area, blobness, and distinguishing property given by notations R2A, R2B, and S2. The first two geometric ratios are gray-level invariants. This means they remain constant under intensity rescaling. The geometrical meaning of R2B is the derivation from a blob-like structure, but cannot distinguish between a line and a plate-like pattern. The R2B is computed as the ratio of the volume of the ellipsoid to the largest cross section-area, which came out to be a traction yj=Xri• The "blob-term" is at a maximum for a blob-like structure and is close to zero it Xi, or X1 and X2, tend to vanish. R2A referred to the largest area of the cross section of the ellipsoid in the plane that was perpendicular to the vessel direction (the least eigenvalue direction). This is computed as the ratio of the two largest second-order derivatives. This ratio basically distinguishes between the plate and line-like structures. Mathematically, it was given as j^]. The third term helps in distinguishing the vessel and non vessel structures. This term is computed as the magnitude of the derivatives, which is the magnitude of the eigenvalues. This is computed using the norm of the Hessian. The Frobenius matrix norm was used because it was straightforward in terms of the three eigenvalues and was given as S = ^JXi + X2, X|.

1. Repeating the above steps from starting scale, smin, to ending scale, smax.
2. Scale optimization to remove the nonvascular and background structures. The filter optimization was done by finding the best scale ct. This is computed as:

The volume corresponding to the best scale sopt is the filtered volume.

4.5 Comparisons: Three-Dimensional Median vs.

### Three-Dimensional Black-Blood Ellipsoidal Filtering

This section presents briefly the previous methods that have been used for filtering the angiographic volumes. They are primarily the Alexander et al. [27] and Sun et al. [26] directional filtering approaches. Alexander et al.'s algorithm is discussed using the object process diagram (OPD), as shown in Figure 4.32. This method exploits the fact that most intracranial vessels are narrower than other structures that appear dark in the image (called black-blood angiography). The main reasons that complicate the vessel segmentation in BBA include:

1. Bone, air, and vessels are all seen as black in intensity.
2. Two neighboring bones can touch each other, thereby making the black region thicker. If they do not touch each other, then they can have gray-level tissues in between.
3. Air pockets are irregular shapes. These can be very large or very small. The very large ones can be longitudinal or circular in nature.

Median Filtering Algorithm

Range of kernels W

Black-Blood Angio. Volume

3-D Separable Median Filter ___small vessel and narrow

Range of kernels W

3-D Separable Median Filter ___small vessel and narrow