Some processes are carried out on scenes. These include sub-regioning, binarisation,
interpolation and segmentation. The first three operate on a scene to produce another scene. Segmentation operates on a scene to produce an object. This is such a large topic that it will be dealt with separately in section 2.3.
2.2.1 Sub-regioning
A scene is formed as a subset of the voxels in the original scene. For example a
slice may be taken by reformatting :
z-sliceK = { V(i,j,k) I 0 < i < L, 0 < j < M, k = K }
y-slicej = { V(i,j,k) I 0 < i < L, j = J, 0 < k < N } (2.13) x-slice! = { V(i,j,k) I i = I, 0 < j < M, 0 < k < N }
More flexibly, a two-dimensional digital image may be formed by any oblique slice through the data or even through a curved plane. The technique appropriate to octree
data has been described by Yau [YAU84]. The current implementation of UCL3D produces only orthogonal (i.e. coronal, sagittal, and transverse) reformatting.
Another meaning of sub-regioning is to choose a three dimensional Volume o f Interest
(VOI), within the data. For example, the MIPG packages allow a rectangular parallepiped sub volume to be specified, which allows a simple form of spatial dissection.
Another operation that is sometimes required is to reduce the resolution of the scene, if, for example the system has a limited memory [GOLDWASSER86]. This can be achieved by averaging over a block of voxels.
2.2.2 Binarisation
This is the process of producing a binary DDS from a grey DDS. Any segmentation that effectively defines a hard object will have a characteristic function <E>D that can be represented in a binary DDS. However some 3D methodologies work entirely in binary DDS structure (MIPG, Vannier and Marsh), including the segmentation algorithms. In this sense binarisation means a mapping B on the voxel values that reduces them to binary. The simplest operation is thresholding at a value p:
b(ij,k) = 1 if value(i,j,k) > p
b(ij,k) = 0 otherwise (2.14).
In early work, which was primarily interested in bone tissues from CT data, such a mapping was sufficient. A simple extension is to allow a density window in the grey values
b(ij,k) = 1 if value(i,j,k) > p ^ and value(i,j,k) < pHigh
b(i,j,k) = 0 otherwise (2.15).
This is used in the binary implementation of UCL3D, in common with in other systems [e.g. TROUSETT88].
Since the function b(i,j,k) is a binary function defined for each voxel, it is an example of a characteristic function d>D defining an Object. The term binarisation is used to describe an operation mapping a scene to a scene. The term thresholding will be used
to describe the mapping of a scene to an object. This is an example of a segmentation operation, described in section 2.3.1 below.
2.2.3 Interpolation
This is the process by which a scene with different voxel dimensions is produced from the original scene. It is most usually done because data acquisition methodologies produce data that is more sparsely sampled in one direction than another. Often this is done just to reduce the "blockiness" of derived images. Many topological results utilised in the MIPG approach require cubic voxels [HERMAN79]. Usually nearest neighbour [VANNIER83, LENZ86], or linear [GOLDWASSER87, HERMAN88b] interpolation is carried out in the undersampled direction, to give the same sampling as in the other directions. Linear interpolation is that used in UCL3D.
A correct analysis of interpolation procedures necessitates consideration of the actual sampling process of the data acquisition. Ideally, data would come from a band-limited object that was sampled at or above the Nyquist frequency. Then the original space function f(r) can really be recovered to any accuracy using, ideally a low-pass filter in the frequency domain [PARKER83]. Since this low-pass filter is a sine function in the spatial domain, and therefore strictly infinite, some approximations to it are required. It is this approximation that gives rise to nearest neighbour or linear interpolation as mentioned. Some investigations have been made using higher order interpolants, such as cubic splines [PARKER83] and restoring splines [LEE83]. In practice the real function f(r) is not of course band-limited, but its convolution with the point-spread function (PSF) of the imaging device will be. This suggests that a correct interpolation, to the resolution of the acquisition device is indeed possible if the PSF is known. Unfortunately the correct sampling in th e: coaxial direction would require overlapping slices that (for CT) impart an unnecessarily dangerous radiation dose to the patient and, secondly, take a long time. Experience on phantoms with such overlapping slices shows the improvements that may result [POMMERT89b]. An indication of the "interpolation problem" can be seen from figure 2.2, where a dry skull was scanned at maximum resolution (1.5mm) in the
j
coaxial direction (pixel size 0.8mm square). The left hand images show the object segmented from the (interpolated) scene producedfrom these slices, while the right hand images show the same object reconstructed from every alternate slice. The linear interpolation o f the sharp edges produces an unrealistic "steppy" effect.
The interpolation schemes mentioned above operate on a grey DDS to produce a grey DDS. However, a different approach is to interpolate in a binary DDS to produce a binary D DS. Here an attempt is made to interpolate the shape o f the object Qc. The idea is to assign a distance map to the segmented regions in 2D, where a pixel is labelled with its distance from the boundary. Then this distance map is interpolated, and new 2D regions created. A quantitative study o f this technique where slices were deleted from the original scene, and recreated by interpolation, found a much greater accuracy over the grey scene methods IRAYA88J. Such evaluations are very dependent
on the type o f data, and further work is necessary in this area.
( DATA) DRY-SKULL-1
Another meaning to interpolation is to improve the resolution of the surface of objects. This has been done by producing an interpolated scene at finer resolution than the original sampling in all directions [HEMMY87, CLINE88], or by using the Partial Volume effect to ’’undo" the digitisation step (see Chapter 5). Of course these interpolation processes also suffer from the limitations of sampling theory [TIAN86].
2.2.4 Other Scene Operations
If the data is noisy it may need to be smoothed, for example with a median filter [VANNIER83]. In NMR applications some non-uniform correction may need to be made for anisotropy and inhomogeneity of the magnetic fields. This is also a 3D operation.