4.6.1 Materials
For this study we have use 7 MRI datasets for proposed curvature descriptor. The MRI datasets are same as those mentioned in section 3.4.1 of chapter 3 and are used to validate the proposed method for curvature computation of the cartilage tissue. All the datasets underwent cartilage detection using 3D undecimated wavelets proposed in chapter 3, section 3.3 prior to shape analysis. The algorithm for cartilage curvature descriptor using wavelet based Riemannian geometry was written in MATLAB R2013b.
4.6.2 Experimental Results
For the computation of the geometric descriptor for cartilage shape, each MRI dataset initially underwent cartilage detection with the proposed local tL and global thresholds
`
γ described in Chapter 3. The segmented volume was reconstructed using the inverse undecimated wavelet transform. This volume was further used to determine the high pass filtered gradients vital for computation of the Riemannian metric gi,j using the
wavelet decomposition of the segmented volume.
E G F
e g f
Figure 4.3: first fundamental form for cartilage tissue and second fundamental for
the cartilage tissue
The wavelet coefficientscψ are selected across these two scales. The wavelet coefficients
a b c
Figure 4.4: a) original volume b) cartilage topology 1 and c) cartilage topology 2
across the cartilage surface. The horizontal and vertical coefficients of wavelet decom- position at scale 1 are very similar to the first order gradients for point P across the shape surface SC. The second order gradients for the surface points is obtained as the
second order wavelet decomposition and are given by the gradient information along the horizontal, vertical and diagonal wavelet decomposition. In this study as we have utilized un-decimated wavelet transform which up samples the original signal content as a result, we have resized and normalized the original wavelet coefficients before the Riemannian metricgi,j computation.
The wavelet coefficients of the first and second scale are thus used to compute the first I and second II fundamental form. For the curvature computation using Riemannian metricgi,j we require the first and second fundamental forms of the manifold. The coef-
ficients of first fundamental forms E,G, F are given as the dot product of the horizontal, vertical and diagonal wavelet coefficients Wp. The normal for the 2D shape is obtained
using the horizontal and vertical surface gradients as given by Eq.(4.13). Product of the normalnwith second order gradients provide us with the elements of the second funda- mental form II. The subsequent intrinsic Gaussian curvatureW G and extrinsic mean curvatureW H for the cartilage surface is then computed using Eq.(4.14) and Eq.(4.15) respectively. Principal curvatures of the shape is computed using Eq.(4.16).
This study is performed only for the femoral cartilage for simplicity of analysis. For computation of the curvature, we select a small region at the base of the femoral car- tilage which correlates more closely with the tibial surface used in most of the studies
Table 4.1: Riemannian metric computation for different cartilage topologies
Dataset Gaussian Curvature Mean Curvature
D7 original 51.689 43.259 topology 1 52.708 67.478 topology 2 59.933 56.495 D11 original 53.894 56.939 topology 1 54.694 20.954 topology 2 44.758 48.001
Table 4.2: Curvature computation for MRI volume using Riemannian metric
Datasets Gaussian curvature Mean Curvature
D1 41.246 25.262
D2 45.250 47.624
D5 50.664 48.229
D6 41.582 17.563
D12 89.286 53.400
(Folkesson et al.,2007;Tummala and Dam,2010;Iranpour-Boroujeni et al.,2011). Tib- ial surface is selected as it is the most weight bearing structure of the cartilage tissue and undergoes degradation earlier as compared to the rest of the surface. In correlation the femoral cartilage at the base surface may also be subjected to high degree of mechanical stress and may be more sensitive to deformation changes.
Fig.(4.3) demonstrates the firstI and second fundamentalII forms for the cartilage sur- face of an MRI dataset respectively. While Fig.(4.4) demonstrates the different topolo- gies for cartilage surface. The different cartilage topologies are used to mimic changes in cartilage surface during the initial stages of deformation in OA. While these may not be an accurate depiction, it includes common variations observed in the femoral shape. For this purpose the cartilage geometry is rotated by +/- 10 degrees to mimic variations in the femoral cartilage. These modified cartilage topologies were later used to compute the Riemannian curvature information for the tissue.
The computed values of these Riemannian curvatures for cartilage topologies and MRI datasets are given in Tab. 4.1. and Tab. 4.2 respectively. The objective of the study is to compute the intrinsic and extrinsic curvature value and to determine variations in curvatures. As demonstrated by Tab. 4.1. and 4.2. the Gaussian curvatures W Gshow very limited fluctuations in their curvature value and may be used as an indicator of the global structure of the cartilage. This is also supported by literature that intrinsic parameters for shape may be more invariant to topological shape change (Griffin,1994). The mean curvatureW Hvalues as indicated by Tab. 4.1. show considerable fluctuations in their values as the cartilage undergoes deformation. This extrinsic local curvature information may be more sensitive to changes in cartilage topology and can be used for study of cartilage deformation (Chung et al., 2003). As compared to Gauss curvature W Gvalues the difference may be not be huge but it is still sensitive to local deformation with the structure. Tab. 4.2gives us the Gaussian and mean curvature values for other MRI datasets.