After reconstruction from µCT, the 3D volume data is made up of an array of [x, y, z] voxels. Each voxel will contain a greyscale value ranging from 0 to 256 in 8 bit data or 0 to 65536 for 16 bit data depending on how much X0ray energy reaches the scintillator. Regardless, the result from filtered back0projection will generally require some level of enhancement in order to make datasets comparable and to extract features as automated and unbiased manner as possible. As a voxelised dataset is an approximation of the original 3D object, there will be ‘fuzzy edges’ where the material only partially intersects a voxel, giving a greyscale value between the two phases. This is represented in Figure 2.9 in a 3x3 array, whereby the red line represents the true edge of a feature. Pixels that fall fully inside the feature are given the value 1.00 and pixels that fall outside are given the value of 0.00.
When the edge of a feature dissects a pixel, the pixel is only partially filled to which its value is proportional to the area filled by the feature. When these are represented in greyscale, as in Figure 2.9, these particular pixels need to be differentiated into pixels that are considered either part of the feature or not, commonly known as thresholding (thresholding techniques are discussed in more detail in section 2.3.4).
In order to improve image quality before analysis, there are a number of image enhancement techniques. Generally these range from artefact removal, noise reduction and intensity rebalancing from a variety of filtering and manipulation techniques, the most important of which are discussed below.
2.3.3.1 Histogram manipulation
A histogram of a dataset represents its distribution of greyscale values. Different phases will be represented as peaks in the histogram, their relative positions depending on its X0ray attenuation.
Despite the same scan settings, the position of peaks in the histogram will vary from scan to scan.
Normalisation of histograms involves stretching and translation of the dynamic range of the histogram such that peaks in separate histograms are aligned to a pre0defined reference (Figure 2.10).
Stretching the dynamic range can improve contrast in the image but also has the drawback of simultaneously increasing background noise.
Figure 2.9 Example of fuzzy voxels that lead to uncertain edges
Figure 2.10 Histogram manipulation of original histogram (dotted black line) such that the peaks are aligned to a reference value, such as a reference histogram, by scaling (dotted red line)
and translating (solid red line)
The normalisation of histograms makes future analysis steps much easier, such as in thresholding, where a common greyscale value can be determined as the threshold value.
2.3.3.2 Filtering
There are a large number of different filtering techniques that can be employed in the enhancement of 3D volumes. For the sake of simplicity, the function of these filters will be described in 2D, although most filters will be applied in 3D. The most common and easiest type of filtering is using a smoothing filter to reduce noise in an image. A simple and effective smoothing filter would be a median filter that is based on the use of a kernel to identify the neighbouring cells. A 3x3 (or larger) kernel represents the 8 immediate neighbours in contact with the central pixel (Figure 2.11). The median value of all 9 values is calculated and the result is input as the value for the central pixel.
Similarly, for a mean filter, the mean value of the values in the kernel is calculated. Larger kernel sizes can be used to increase the smoothing effect, but this leads to greater blurring of edges, which can make features indistinguishable.
Figure 2.11 Showing a 3x3 kernel for a pixel; neighbour pixels in 2D are shown in grey
A variation of the median filter is the Gaussian filter, which is a smoothing filter where the weighting of each value in the kernel is based on a Gaussian curve. For example, in a 3x3 kernel used in a median filter, the weight of all 9 pixels would be equal, i.e. 1/9th weighting given to each for a sum of 1. In the Gaussian filter, the weight of each pixel in a 3x3 kernel would be determined by the discretisation of the Gaussian distribution, which is defined by its width or σ value: see Equation (2.5) for the 2D Gaussian filter. A high value of σ leads to a larger blurring effect as neighbouring pixels are given more weight. The Gaussian filter removes ‘high frequency’ components from an image and therefore is classified as a low0pass filter.
2"%, 4& = 1 278/ 9
:;<:
/=: (2.5)
Where x and y are the points along the two directions along which the filter is applied (axes) and σ is the standard deviation of the distribution.
The weakness with using the basic low0pass filters described above is that these filters are not sensitive to features, which means that they will remove noise but also smooth the edge, a problem that is highlighted by Canny (Canny, 1986). This is especially important when studying very thin features. To improve upon this, edge0preserving (EP) and ‘anisotropic diffusion (AD) filters were developed. Both filters act in a similar fashion; that is, to blur regions that are considered the same phase, whilst retaining the edge information as much as possible. The EP filter is essentially a
smoothing filter which has reduced effect when near edges. The edges are determined by the gradient of the image and a threshold value below which no smoothing occurs. When above the gradient threshold, a Gaussian filter is applied. More details of the EP filter is given by (Weickert et al., 1998).
The AD filter works by comparing the value of the current pixel to its neighbours by a diffusion stop value. When the value difference between the pixel and its neighbour is less than the diffusion stop value, Gaussian smoothing occurs, whilst if the difference is greater, no smoothing occurs. In this way, areas with a high gradient (i.e. strong edge), are retained, whilst areas of low gradient are effectively flattened over a number of iterations. Anisotropic diffusion is outlined in more detail in (Perona and Malik, 1990).
2.3.3.3 Binning
Binning refers to the compression of data by combining information from several voxels into one. An image or volume can be reduced in size by a factor specified by the user. There are several methods to combine voxel information; mean, median, minimum, maximum, etc., all of which have their own merits. As an example, if a 3D volume of dimensions 512x512x512 were to be binned by a factor of 2, the resultant volume would be 256x256x256 in size, but with half the resolution. In terms of tomography information, this is akin to doubling the voxel size. Although binning leads to information being lost, the biggest advantage to binning datasets is the reduction in file size, especially important when using computationally expensive operations such as anisotropic diffusion.