Multiscale Computation and Integration of Filter Responses

Local structures can exist at various scales. For example, vessels and bone cortices can, respectively, be regarded as line and sheet structures with various widths. In order to make filter responses tunable to a width of interest, the derivative computation for the gradient vector and the Hessian matrix is combined with Gaussian convolution. By adjusting the standard deviation of Gaussian convolution, local structures with a specific range of widths can be enhanced. The Gaussian function is known as a unique distribution optimizing localization in both the spatial and frequency domains [20]. Thus, convolution operations can be applied within local support (due to spatial localization) with minimum aliasing errors (due to frequency localization).

We denote the local structure filtering for a volume blurred by Gaussian convolution with a standard deviation af as

where % e {sheet, line, blob}. The filter responses decrease as af in the Gaussian convolution increases unless appropriate normalization is performed [21-23]. In order to determine the normalization factor, we consider a Gaussian-shaped model of sheet, line, and blob with variable scales.

Sheet, line, and blob structures with variable widths are modeled as lsheet(x; ar) = exp ^- 20^ , (10.12)

and fx2 + y2 + z2\ lbiob(x; Or) = exp ^--202-J ' (10.14)

respectively, where ar controls the width of the structures.

We determine the normalization factor so that {l%(x; ar); af} satisfies the following condition:

• max0r {l% (0; ar); af} is constant, irrespective of af, where 0 = (0, 0, 0).

The above condition can be satisfied when the Gaussian second derivatives are computed by multiplying by af as the normalization factor. That is, the normalized Gaussian derivatives are given by fxpyqzr(x; af) = j af • gxP Qyq dzr Gauss(x; af ) J * f (x) (10.15)

where p, q, and r are non-negative integer values satisfying p + q + r = 2, and Gauss(x; a) is an isotropic 3D Gaussian function with a standard deviation a given by (V2na)-1 exp(-|x|2/(2a2)) (see the Questions section at the end of

(a) Line

Figure 10.3: Plots of normalized responses of local structure filters for corresponding local models, S%{I% (0; ar); af}, where ar is continuously varied, and af = ais1-1 (a1 = 1, s = 1.414, and i = 1, 2, 3, 4). See "Brain Storming Questions" at the end of this chapter for the theoretical derivations of the response curves. (a) Response of the line filter for the line model (% = line). (b) Response of the blob filter for the blob model (% = blob). (c) Response of the sheet filter for the sheet model (% = sheet). (© 2004 IEEE)

this chapter for the derivation). Figure 10.3 shows the normalized response of S% {¿%(0; ar); af} (where af = aisi-1, ai = 1, s = V2, and i = 1, 2, 3, 4) for % e {sheet, line, blob} when ar is varied.

In the line case, the maximum of the normalized response Siine{hine(0,ar); af} is 4(= 0.25) when ar = af [7]. That is, SHne{f; af} is regarded as being tuned to line structures with a width ar = af. A line filter with a single scale gives a high response in only a narrow range of widths. We call the curves shown in Fig. 10.3 as width response curves, which represent filter characteristics like frequency response curves. The width response curve of the line filter can be adjusted and widened using multiscale integration of filter responses given by

1<i<n where ai = si-1 a1, in which a1 is the smallest scale, s is a scale factor, and n is the number of scales [7]. The width response curve of multiscale integration using the four scales consists of the maximum values among the four single-scale width response curves, and gives nearly uniform responses in the width range between ar = a1 and ar = a4 when s = V2 (Fig. 10.3(a)). While the width response curve can be perfectly uniform if continuous variation values are used for af, the deviation from the continuous case is less than 3% using discrete values for af with s = a/2 [7]. Similarly, in the cases of Ssheet{^sheet(0, ar); af} and Sbiob [hiob(0, ar); af}, the maximum of the normalized response is (-2)3 (^ 0.385)

when ar = f (Fig. 10.3(b)), and f^/f)5^ 0.186) when ar = (Fig. 10.3(c)), respectively (see the Question section at the end of this chapter for the derivation). For the second-order cases, the width response curve can be adjusted and widened using the multiscale integration method given by

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