Aplicación del Modelo de Cees Van Riel

The primary function of this unit is to perform image and wavelet analysis of the MRI image volume for cartilage detection and to provide the processed data to the wavelet GUI. This sub-system largely works with 3D image volumes and performs wavelet anal- ysis on volumetric data. 3D assessment of the MRI volume help compensate for the lack of sampling along the third direction (z-axis) by the MRI machine and helps in improving the thickness estimation of cartilage geometry (Folkesson et al., 2007). Also 3D volume provides complete structural information of the cartilage which might get overlooked when processing 2D images slice by slice. This sub-system is also designed

to carry out other 3D image analysis tasks such as smoothing, thresholding, histogram generation etc. which may support other image processing operations using wavelets. As wavelets offer better ‘spatial localization’ they are used for the cartilage detection process (Burrus et al., 1998; Starck et al., 2007). The primary tasks of this proposed sub-system is 1) to obtain image spatial and edge details with wavelet decomposition and 2) multiresolution analysis of this wavelet information.

For wavelet analysis, this sub-system decomposes the 3D image volume I(x, y, z) ob- tained by successive stacking of 2D images slices, read by the DICOM image reader. This 3D image function is given as I(x, y, z) ∈ L2 (R3) integrable space (Gonzalez et al.,2004). Low and high frequency components of the image are obtained by volume decomposition using scaling ϕj,k(x) and wavelet functions respectively ψj,k(x) (Gon-

zalez et al., 2004). In the proposed diagnostic system we have used 3D undecimated Haar wavelet transform for cartilage detection. Due to the non-decimate nature of this wavelet transform we retain all coefficients obtained by wavelet decomposition which may prove useful in the cartilage detection process and improve accuracy of diagnosis. The forward and inverse wavelet transform with undecimated wavelets have been pre- viously mentioned in Chapter 3. The values ofj are identified by the user to determine the necessary decomposition level and its corresponding inverse operation.

The wavelet decomposition output is further provided for operations such as binary thresholding, matrix transpose etc. necessary to carry out other imaging tasks as re- quired by the cartilage detection process. Adaptive thresholding of wavelet coefficients along with multiscale information are used to perform articular cartilage detection. As the high frequency components contain edge information they are used for localized cartilage detection in the wavelet domain as given in Fig. (5.2). Also the histogram of wavelet coefficients are used to compute the cartilage biomarkers such as the smoothness and entropy for cartilage tissue assessment. Other than wavelet operation, the image analysis system can also carry out standard image processing tasks such as 3D noise removal, image binarization, thresholding, user-defined filtering, geometric data anal- ysis, polydata mesh generation etc. with help of the currently available VTK inbuilt functions for image processing and visualization.

In addition a copy of the wavelet volume is also provided to the widget GUI for visualiza- tion at multiple wavelet resolutions. For a wavelet multiresolution analysis, coefficients at higher resolution are obtained by further decomposition of the image volume with scaled and dilated versions of scalingϕj,k(x) and wavelet functionsψj,k(x) (Burrus et al.,

contains data that is present at the lower subspace (Burrus et al.,1998;Gonzalez et al., 2004).

Figure 5.2: Wavelet analysis system

The widget uses the output of the image analysis system to display the cartilage volume and update itself on user-interaction. For a faster visualization operation we have re- stricted the widget GUI to provide only two consecutive wavelet resolutions. As a result once the image analysis system computes the wavelet transform for the user provided input resolution level; it automatically also computes the higher resolution level and provides the information of both the wavelet resolutions to the GUI for display. The image analysis system also uses the same output to determine the smoothness and en- tropy for cartilage tissue at the two wavelet resolution for assessment.

In addition the 3D volume can be used to generate cartilage models using the volume rendering process inbuilt in VTK 5.8.0. The basic rendering equation can be written as (wikipedia)

Lo(x, wo, λ, t) =Le(x, wo, λ, t) +

Z

Ω

fr(x, wi, wo, λ, t)

Li(x, wi, λ, t)(wi.n) dwi (5.1)

whereλis particular wavelength of light,tis time,xis location space,nis surface normal at that location,wois direction of outgoing light,wiis the negative direction of incoming

light,Lo is total spectral radiance,Le is emitted spectral radiance, Ω is unit hemisphere

containing all possible values of wi,fr is bidirectional reflectance distribution function

and Li is incoming spectral radiance (wikipedia). Volume rendering is a computer

visualization technique for 3D data and displays 2D projection of discretely sampled 3D data (wikipedia). The volume is considered as extraction of the iso-surface of the volume and rendered as a polygonal mesh. Information content obtained at wavelet decomposition can be used to generate a 3D model of the cartilage tissue for different resolution levels. These models are used to capture changes on the cartilage surface for assessment of the articular cartilage. Then the wavelet information for a given particular level is provided to the multiresolution GUI for an interactive assessment of the geometry of the tissue as shown in Fig. (5.1). If multiple cartilage resolution is required, the user has to enter the wavelet decomposition level and cartilage model for the desired resolution level and higher is generated which is given to the wavelet widget as shown in Fig. (5.3). The wavelet coefficients can also be used to visualize 3D edges of cartilage tissue for a complete assessment or only just the grey scale volumetric data.