Af

We define the height measure of the multiscale filter response as the peak response hM(ar) = Mline{lline(0, 0, 2; ar)}. Since the filter response is normalized, hM(ar) is constant regardless of ar. That is, hM(ar) = hMc, (10.23)

where hMc = 0.25 (see the "Brain Storming Question" at the end of this chapter for the derivation). We define the width measure wM(ar) of the multiscale filter response as the distance ^Jxfi + y^ from the z-axis to the circular locus where Mline{lline(x0, y0, z; ar)} gives half of the peak response, that is, h|c. Let wM(ar) be the ratio of the observed width wM(ar )to ar. The width ratio wM (ar) is constant regardless of ar, that is,

where w^, ^ 1.0 when y23 = 1 in the formulation of Eq. (10.2). Similarly, we define the height measure hR(ar; af) of the single-scale filter response as the peak response hR(ar; af) = Sline{lline(0, 0, z; ar); af} and the width measure wR(ar, af) as the distance ^x^ + y2 from the z-axis to the circular locus where Sline{lline(xo, y0, z; ar); af} gives the half of the maximum response, h|c. To compare the widths of the filter response and the original profile, we also introduce the width measure wL(ar) of the original line image as the distance + y2 from the z-axis to the circular locus where tline(x0, y0, z; ar) gives half of lline(0, 0, z; ar). While ar is introduced for the convenience of generating line profiles, wL(ar) is for the convenience of comparing the widths of various profile shapes.

Figure 10.10(a) and 10.10(b) show the variations in the height and width measures. Figure 10.10(a) gives the plots of hR(ar; af) at three values of af and hM(ar), and Fig. 10.10(b) shows the plots of wR(ar; af) at three values of af, wM(ar), and wL(ar). The width measure of the multiscale response is proportional to that of the original line image. In the case of the line image lline(x; ar) with a Gaussian cross section, wL(ar) « 0.9wM(ar). Although the filter responses make the lines a little thinner than the original lines, the multiscale line-filter can be designed so that the width of its responses becomes approximately proportional to the original one.

Figure 10.10: Height and width measures of filter responses in multiscale integration with ai = si-1 a1 (a1 = 1.5, s = 1.5, i = 1, 2, 3). The height measure is normalized so that hMc is one. (a) Height measures hM(ar) = hMc and hR(ar; ai) for lline(x; ar). (b) Width measures wM(ar) = wMc, wL(ar), and wR(ar; ai) for tiine(x; ar) with y23 = 1. (c) Height measures for ¿elliptic(%; ax, ay) with y23 = 1. Solid lines denote the height measures for the discrete scales. Dashed lines denote the height measures for the continuous scales from a1 to a3. (d) Height measures for ¿elliptic(x; ax, ay) with y23 = 0.5.

Figure 10.10: Height and width measures of filter responses in multiscale integration with ai = si-1 a1 (a1 = 1.5, s = 1.5, i = 1, 2, 3). The height measure is normalized so that hMc is one. (a) Height measures hM(ar) = hMc and hR(ar; ai) for lline(x; ar). (b) Width measures wM(ar) = wMc, wL(ar), and wR(ar; ai) for tiine(x; ar) with y23 = 1. (c) Height measures for ¿elliptic(%; ax, ay) with y23 = 1. Solid lines denote the height measures for the discrete scales. Dashed lines denote the height measures for the continuous scales from a1 to a3. (d) Height measures for ¿elliptic(x; ax, ay) with y23 = 0.5.

10.3.3 Multiscale Responses of Discrete Scale Integration

We assumed continuous scales for multiscale integration in the previous subsection. The response at each scale, however, has to be computed at discrete values of af .In 3-D image filtering, a large amount of computation is necessary to obtain the filter response at each value of af. The maximum and minimum values of af can be essentially determined on the basis of the width range of the anatomical structure of interest. The interval of af should be sufficiently small for the filter response to work uniformly for every line width within the width range, while it is desirable that it should be large enough to minimize the amount of computation required. Thus, we need to determine the minimum number of discrete samples of af which satisfy the following conditions:

1. The height measure of the response should be approximately constant within the width range.

2. The width measure of the response should be approximately proportional to the original one within the width range.

Given the discrete samples of af and the assumption of the cross-section shape (here, we use the Gaussian cross section), the accuracy of the approximation can be estimated. Let ai = si-1a1 (i = 1, 2,..., n) be discrete samples of af, where a1 is the minimum scale and s is the scale factor determining the sampling interval of af. The multiscale filter response using the discrete samples of a f is given by

Similarly, the multiscale filter response using the discrete samples of af for the line image lline(x, ar) is given by

Mline{lline(x, ar)} = max SHne{£Hne(x, ar); ai}. (10.26)

Given the scale factor s, we can determine hMmdn and kp satisfying hM„in = hR(kpai; ai) = hR(kpsai; ai), (10.27)

where ai = si-1a1 (i = 1, 2,..., n), hMmin is the minimum of the height measure of the multiscale response Mline{lline(x, ar)} within the range kpa1 < ar < kpsna1, and the minimum is taken at ar = kpsia1 (i = 0, 1, 2,... n) (Fig. 10.10(a)). hMmdn can be regarded as a function of the scale factor s. The height measure of the multiscale response should be sufficiently close to h^ within the width range of interest. The values of hMmdn and kp at typical scale factors are summarized in

Table 10.2: Height measure hM(ar) and width ratio wM(ar) minima in multi-scale integration at discrete scales using typical scale factors, and ar where the minima are taken

Scale factor s

Min. height hMmn

at Gr — kpGi

Min. width ratio

at ar

— kwai

s ^ 1

hMml

, ^ hMe

kp ^ 1

wM _

„ ^ wMc

kw

^ 0.65

s = 1.2

hMml

„ * 0.99hMc

kp * 0.92

, * 0.99wMc

kw

* 0.59

II 2

hMml

„ * 0.97hMc

kp * 0.84

WM m„

„ * °.97wM c

kw

* 0.56

s = 1.5

hMml

„ * 0.96hMc

kp * 0.82

WM _

„ * 0.96wM c

kw

* 0.55

s = 2.0

hMml

„ * 0.89hMc

kp * 0.71

, * 0.88wMc

kw

* 0.50

Table 10.2. When s = 1.5, hM & 0.96hM , which means that the deviation from

J wi-min iV1c'

the continuous case is less than 4%.

With regard to the width measure of the filter response, given the discrete samples of af and the assumption of the profile shape, the accuracy of this approximation can also be estimated. Given the scale factor s, we can determine and kw satisfying

where ai = si-1a1(i = 1, 2,..., n), w^ is the minimum of the ratio of wM(ar) to ar within the range kwa1 < ar < kwsna1, and the minimum is taken at ar = kwsia1 (i = 0,1, 2, ...,n) (Fig. 10.10(b)). w^^ can be regarded as a function of the scale factor s. wM (ar) should be sufficiently close to w^, within the width range of interest. The values of wM^ and kw at typical scale factors are summarized in Table 10.2. When s = 1.5, w^.^ ^ 0.96w^ .When the parameters for the discrete scales of a f are s = 1.5and n = 3, the ranges of deviation within 4% for the height and the width measures are 0.55ai < ar < 1.86ai and 0.82ai < ar < 2.77ai, respectively. The range of deviation within 4% for the width measure is shifted to a smaller ar than that for the height measure. As a result, the range of deviation of less than 4% for both the height and the width measures is 0.82a1 < ar < 1.86a1.

We now extend the experimental analysis of the multiscale integration to the response to lelliptic(x; ax, ay) shown in Eq. (10.21). We define the hight measure of ^elliptic(X; ax, ay) as hR^(ax, ay; af) = SHne[£eiiiptic(0, 0, z; ax, ay); af}. Fig. 10.10(c) and 10.10(d) show the multiscale integration of the responses at continuous and discrete scales with a1 = 1.5, s = 1.5 and n = 3 for y23 = 1 and

Y23 = 0.5, respectively. The multiscale integration of these discrete scales gives a good approximation of that of the continuous scales.

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