This chapter demonstrates the use o f surface detection operators for purposes of surface extraction on real world data sets. The application of such operators is used to demonstrate the concepts described previously and not necessarily to find optimum solutions. In particular, no attempt is made to analyse the accuracy of resultant sur faces, to investigate alternate solutions, or to test for reproducibility on numerous data sets.
7.1 B ack g ro u n d
Given the results o f chapter 6, the Sobel operator is applied to four different real
world data sets. The choice o f the Sobel is due to its overall best performance (resis tance to noise, etc.) as determined previously.
The output of the Sobel will be treated as a probability in the following examples. In particular, for situations where a simple threshold on the input data is not sufficient to accurately locate objects o f interest, it may also be the case that thresholding the output o f the Sobel does not work either. In such cases, treating the Sobel output as if it is a probability is sometimes useful. For example, given a voxel with value v representing density (which we will think o f as being the probability that the voxel is part o f an object rather than part of the background) and a corresponding Sobel value s
representing the probability that the voxel is in a surface, then the combined probabil ity that a voxel is in the interior of an object is v ( l -5 ) (assuming v and s are in the
fractional range 0..1) and that a voxel is on the surface o f an object is vs. A loss of accuracy is incurred with such an approach, as portions o f objects whose diameter is less than 2 voxel units may be lost, depending on local voxel neighbourhood
configurations.
The data sets exhibit various conditions under which object segmentation is difficult. In particular, the spectrum of conditions range from good to bad sampling, small to large amounts o f object correlation over Z, small to large objects (field of view), and few to many different types o f objects present in the same data set. Other conditions also include various forms o f imaging artifacts including tissue tear, smear, noise, misalignment, etc. (see chapter 2).
7.2 Data description
The data sets used in this thesis are all biologically based, and thus this thesis has a bias towards medical applications. The first two data sets operated on are MRI voxel slices through the human body and the latter two data sets are serial section slices through the organs o f a rat.
7.2.1 M RI data
The original condition of the two M RI voxel data sets is uncertain. They were acquired from a visiting Professor who acquired them, for development o f visualisation algorithms, from a colleague at John Hopkins Hospital in the Washington D.C. area. The data sets were initially labeled "bone" and "brain 1", with the first data set consist ing of slices through the pelvic region o f a patient and the second of slices through a skull. Both data sets are of resolution 128x128x50 where each voxel is not cubic and consists o f 8-bit values: thus each data set comprises 800 Kbytes.
The two MRI data sets exhibit various properties which make them useful initial examples for demonstrating object extraction using surface detection operators. The first is that the objects under consideration are large and the overall contrast between background and object is high. The slices are also in alignment due to the nature o f the acquisition method. Both data sets exhibit streaks radiating from the centre o f each slice which may be due to the back-projection reconstruction of the initial scan data. For both the brain and bone data the objects under consideration exhibit good fields of view, although portions of the bone are only a few voxels thick. The brain data con tains many different structures (eg. skull, brain, spine, etc.) whereas the bone data con tain only one object. For both the brain and bone data it appears that the dimensions of the voxels are 2X - 2 Y =Z and thus the voxels are not cubic.
7.2.2 Serial Section Microscopy Data
The serial section microscopy data operated upon in this chapter present a greater challenge than the MRI data. The slices exhibit properties which are far from ideal: they contain many very small objects whose diameter in the X - Y plane is o f the order o f the physical slice thickness. As a result it is necessary to work with non-cubic vox els whose X and Y dimensions are many times smaller than their Z dimension. The data was originally prepared for visualisation and understanding o f the morphology of the branching nature of the small objects, although we use them here for purposes of demonstrating surface extraction. Another difficulty with this data is that the slices must be registered (aligned) as they are digitised.
For both data sets, there are significantly more slices available than have been used in this chapter. Due to significant tissue tears in some o f the physical slices, it is not possible to treat the data contiguously. Although it is possible to interpolate slices to "fill in the gaps", for one o f the data sets used (the carotid body), as many as three consecutive slices are damaged beyond recognition, and thus interpolation becomes impracticable (because the thickness of the slices is much larger than the object diame ters, significant changes occur from one slice to the next and thus there is little correla tion).
7.2.2.1 Acquisition
Slice registration (alignment) involves determining and correcting undesirable spatial transforms. Two images are said to be registered if their respective image space coordinate systems are the same with respect to some fixed origin. A function of two images, called the cross correlation (or simply correlation) function is used to produce a measure of the similarity o f the two images in determining if they occupy the same coordinate system. The registration method used for this work uses a simple and pseudo correlation function which is perhaps best described as being "the brute force method". A difference function was chosen as its implementation maps well onto the SIMD parallel image processing computer called CLIP4 [Duff76] used throughout this thesis.
• Many different correlation functions exist, some of which explicitly determine the desired transform by solving systems of simultaneous equations for some (or all) of the pixels. Others correlate a reduced set of extracted primitives (features) of the original data. The interested reader is referred to [Barnea72], [Goshtasby87], and [Mitra8 8] for
variations on this theme.
The brute force difference function produces a single integer number describing the similarity o f two images. The number is simply the unsigned (absolute value) of the sum o f the differences between "corresponding" pixels in both images. Mathemati cally it is expressed as
for two images A and B of resolution N x N , where x and y are Euclidean coordinates in the plane of the slices. Implementation on SIMD parallel machines, where there is a processor per pixel, is straightforward for the subtraction and absolute value. The actual summation is implemented using an algorithm which simply counts the number o f Is in each bit-plane o f the resultant (absolute value) image (which is a hardware feature o f CLIP4 taking 69|is), multiplies this number by 2 raised to the plane number
p , and adds the result into an accumulator. Mathematically this is expressed as