Spatial Statistics and Digital Image Analysis (1991) / Chapter Skim
Currently Skimming:
10. Stereology
10. Stereology
Pages 181-216
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 181... ...
It was initially developecl in biology and materials science as a quick way of analyzing threedimensional solid materials (such as rock, living tissue, and metals) from information visible on a two-dimensional plane section through the material.
|
From page 182... ...
Here E denotes expectation with respect to a suitable random sampling design (not the most obvious one) ; we give details in §10.2.
|
From page 183... ...
TABLE 10.1: Standard Notation for Geometrical Quantities Space dimension n set X Letter Meaning 3 solid domain V volume curved surface S (surface) area space curve ~ curve length finite set of objects N number of objects curved surface M,K integral of mean curvature 2 plane domain A area curve [,B curve length finite set of points I,P number of points finite set of objects N,Q number of objects curve C total curvature These methods are summarized in Table 10.2 with notation listed in Table 10.1.
|
From page 184... ...
, we estimate the volume fraction of tissue as Vv = AA = 7/40 = 17.5%. There are 16 positions where a line segment crosses the tissue-airway boundary, so the surface area of lung/air interface per unit volume of lung is estimated at Sv = 21~ = 2 x i6/~42/1500)
|
From page 185... ...
to dramatically reduce the overall sampling variance. In biological applications, the variance contributions associated with variation between animals, and between parts of the same animal, are usually far greater than the variance due to stereological sampling [17,244.
|
From page 186... ...
It is impossible to estimate Nv, the number of points or objects per unit volume, from plane sections in the sense of Table 10.2. One indication of this is the mismatch of dimensions or units.
|
From page 187... ...
One example is disector sampling [79,28,273. A disector is a pair of parallel plane sections a fixed distance apart; often these are two consecutive slices through the material.
|
From page 188... ...
~ (b) (A f FIGURE 10.1: Two biased counting rules for planar profiles: (a)
|
From page 189... ...
Plate 10.2 (preceding page 71) shows the unbiased estimation of Nv for nuclei in human renal giomeruTus using a combination of Gundersen's tiling rule and the disector.
|
From page 190... ...
As we have seen, plane sections and rectangular sampling windows generate biased samples of a particle population, since larger particles have a greater probability of being "caught." Other more subtle biases are caused by selecting a particular orientation for the section plane (for example, always slicing muscle tissue transverse to the muscle fibres) or selecting sections where a particular feature is visible.
|
From page 191... ...
have made it possible to estimate parameters such as the mean squared particle volume, without any assumptions about particle shape. The list of parameters that can be reliably estimated without shape assumptions now includes some quantities related to curvature, orientation, and "shape."
|
From page 192... ...
Explicit Moclels At one extreme, we could build a probability moclel for the entire spatial structure X using random set models from stochastic geometry t34,4S,53, 77,804. An explicit, parametric model would contain information about the sizes, positions, shapes, relative arrangement, and topological relationships of components in X, which conic!
|
From page 193... ...
, and justifications must remain largely empirical, because it is difficult to derive any distributional theory from probabilistic models of the structure or the sampling design. 10.3 Statistical Theory Stereological methods can be applied with minimal knowledge of the threedimensional structure under study.
|
From page 194... ...
Sampling designs and estimators that are unbiased or optimal for estimating .Bv may not be appropriate for JIB and vice versa. 10.3.2 Inference Statistical inference is called design-based if it relies on the randomness in the sampling design.
|
From page 195... ...
This would be inappropriate for objects such as biological organs, which have many levels of internal organization. 10.3.3 Geometrical Identities Unbiased estimation of properties of a set X from observations of the intersection X n T is possible thanks to the mean-content formulas or section formulas of integral geometry [76,89]
|
From page 196... ...
of a curved surface Y C R3 can be determined from the lengths of plane sections, I/ ICY n TW,h)
|
From page 197... ...
Again, the orientation distribution of a curved surface Y in R3 is the probability distribution of the unit normal vector at a uniformly distributed random point on Y This is a distribution Q on the unit sphere
|
From page 198... ...
The set X is fixed (Miles's restricted or extended models) ; the probe T is generated by a random sampling design.
|
From page 199... ...
However, note that for a general set X the coordinates of an {UR line are not independent and their marginal distributions are not uniform. A practical method of generating {UR lines through an arbitrary set X is to enclose X by a larger disc D ~ X, generate a sequence of JUR lines intersecting D, and take the first line that happens to intersect X
|
From page 200... ...
showed that a solution is to take T with the weighted distribution with probability density proportional to c~'(X n T)
|
From page 201... ...
Different plane sections of a bounded three-dimensional object have different size and shape. Thus, simple random sampling does not generalize easily to most stereological situations.
|
From page 202... ...
(10.15) Suppose the random process Z is such that for any x ~ R3 the indicator variable ~ ~ { 0 if not is a weD-defined random variable.
|
From page 203... ...
The resulting formula gives the variance as an integral in terms of r: this is equivalent to the basic variance result of geostatistics (see chapter 5~. · c~ ~ w ~ ~ Characteristics of "infinite order" can be considered exactly as in the design-based case, for example, the orientation distribution of a curved surface.
|
From page 204... ...
Finally the total for the entire animal is estimated by the subsample total estimate times the product of the successive inverse sampling fractions kit · kn. Clearly this estimator is unbiased.
|
From page 205... ...
First take an area-weighted plane section of the sample material; superimpose a point grid on the section, and at every grid point which hits a particle profile, place a line in a random direction through the grid point and measure the cubed intercept (i.e., length of the intersection between the line and the particle profile)
|
From page 206... ...
This can be estimated on plane sections by the mean cubed length of an intercept through a point in the section. The star volume gives us an interesting measure of the average "local size" of holes in a porous material.
|
From page 207... ...
As remarked in §10.3.3, the general formulas for estimating quantities other than volume require random section planes with (roughly speaking) uniform distribution over all possible orientations and all possible positions.
|
From page 208... ...
These are one step more complex than completely random Poisson processes, in that a stochastic interaction is alTowed between "neighbouring" elements of the process, for example, pairwise interactions between the points in a point process. Markov point processes and random sets can easily be simulated using Monte CarIo methods, and they are convenient for likelihood-based inference [684.
|
From page 209... ...
Boyde, and S Reid, Threedimensional analysis of the spatial distribution of particles using the tandem-scanning reflected light microscope, Acta Stereologaca 6 (supplement II, 1987)
|
From page 210... ...
R Weibel, Sampling designs for stereology, '7.
|
From page 211... ...
L Smith, The kernel method for unfolding sphere size distributions, '7.
|
From page 212... ...
B., and R Sundberg, Statistical models for stereological inference about spatial structures; on the applicability of best linear unbiased estimators in stereology, Biometrics 42 (1986)
|
From page 213... ...
t53] Matheron, G., Random Sets and Integral Geometry, John Wiley and Sons, New York, 1975.
|
From page 214... ...
D., Statistical Inference for Spatial Processes, Cambridge University Press, Cambridge, 1988.
|
From page 215... ...
S., Characteristic statistical problems of stochastic geometry, in Geometrical Probability and Biological Structures: Buffon's 200th Anniversary, Lecture Notes in Biomathematics, no. 23, SpringerVerIag, New York, 1978.
|
From page 216... ...
[92] Weil, W., Translative integral geometry, pp.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.