DAY #1 September 15, 2006

The fusion of medical images taken at the same slice/part of the body by differentmodalities is very useful technique in medical diagnosis. Medical image fusion tried withwavelet transform methods proved to be image dependent and preservation of highfrequency contents of the image by wavelets are also not satisfactory. In this work, curvelettransform is applied for fusion with absolute value maximum and addition of coefficient asfusion rules. The edge, singularities and other high frequency contents of the fused imageare well represented by the curvelets. The fused result is better by visual appearance andquantitative analysis than their wavelet equivalents tried with the same fusion rules. IntroductionMedical Image fusion can be defined as the process by which several images or someof their features are combined together to form a single image for better diagnosis andthereby a better treatment planning. Compared to the input images the new image includesmore comprehensive, more accurate, more stable information. So with the availability ofmultimodal tissue information on a single image, clinicians are provided the salient featuresof each medical imaging modality and thus eliminating their individual limitations. Formedical diagnosis Computed Tomography (CT) uses X-Ray on its way through the body toreconstruct a two dimensional image of the absorption coefficient within an axial slice. CTshows highly detailed anatomical information on the distribution of the absorptioncoefficient with high contrast in bone but little in soft tissue. Magnetic Resonance Imaging(MRI) uses nuclear spin interaction with the magnetic field and resonance phenomena togenerate an image of the tissue of the human body. But MRI does not show the bonystructure. Positron Emission Tomography (PET) shows physiological processes but littleanatomical information .Likewise each and every medical imaging modality providesimages which are complementary depending on their mode of tissue reconstruction. Hence alocal integration of the complementary information of various modalities is needed foraccurate diagnosis. Most of the conventional image processing software has been designedfor optimal operation on single images and backed up by a large increase in computationalpower .An alternative to this costly solution is image fusion algorithms which provide aneffective way of reducing the total amount of information presented without loss in imagequality and content. It is required that the fused image should preserve as closely as possibleall relevant information obtained in the input images and the fusion process should notintroduce any artifacts or inconsistencies, which can distract or mislead the medicalprofessional, thereby a wrong diagnosis. Respective anatomical structures are matchedagainst each other to make the fusion meaningful. Image registration is the fundamental taskof image fusion. Before image fusion, the multimodality medical images must be registeredperfectly. Registration is a one to one mapping function applied to modify correspondingphysical image points whose information can then be combined. Only registered images are considered for fusion. Some of the image fusion methods have been introduced in theliteratures including simple pixel by pixel averaging using SNR [1], Laplacian pyramidmethod [2], conditional probability networks [3],Neural network methods[4] and transformdomain fusion approaches including wavelet transform methods[5,6] with different fusionrules. In transform domain fusion, a transform is applied on the registered images to identifythe vital details in the image. The image transform coefficients are obtained. Fusion rule isapplied over the transform coefficients, fusion decision map is obtained, taking inversetransform over the decision map, and the fused image is reconstructed as shown in Figure 1. Figure 1. Block diagram of transform domain image fusion [7]. Transform domain operations have several advantages over other methods which sometimesdo not guarantee fused output like simple addition of registered input images, averaging ofpixels etc.The advantages include energy compaction, larger SNR, collection ofcharacteristic features, easy manipulations as transform coefficients are representatives of allthe pixels of the image etc. Though wavelet transform exhibits time frequency localizationand yields acceptable fused output, the edges and singularities are not well represented .Alsoit suffers from limited directionality. The point singularity is better suited for wavelets in 1dimensional signals ,but 2 dimensional signals like images have curve or line singularitieswhere wavelets fails to approximate. Wavelets in 2 dimension are good at isolating the edge,but will not see the smoothness along the edge. In 2 dimension the discontinuities arespatially distributed and wavelets are not suitable for sparse nature of discontinuities. Thisdisappointing behavior of wavelets indicates that more powerful bases/structure of supportis needed for the transform in higher dimensions. Hence in order to achieve sparse imagerepresentations, where maximum information is packed into a small number of samples,curvelets are introduced. Materials and Methods Discrete curvelet transform Conceptually the curvelet transform is a multiscale pyramid with many directionsand positions at each length scale and fine ridges at fine scales. These elements have manyuseful geometric multiscale features than wavelets. In other words curvelet transform isdirectional wavelets transform; by making use of directional transform (Radon transform)and Multi resolution transform (wavelet transform). Figure 2. Curvelet spatial decomposition [8]. The curvelet transform proposed by Donoho [8] opens the possibility to analyze an imagewith different blocksizes, but with a single transform. The idea is to first decompose theimage into set of wavelet bands and to analyze each band with an inbuilt ridgelet transform.The block size can be changed at each scale. The curvelet decomposition [9] can bedescribed1. Subband decomposition of the object into sequences of subbands.2. Partitioning each subband into blocks of appropriate size, depending on itscenter frequency.3. Applying ridgelet transform on these blocks. Ridgelet transform is an inbuilttransform within curvelet transform, which is a combination of radontransform and 1 dimensional wavelet transform which makes the higherdimensional singularities into point singularities where perform well. The flow graph of curvelet transform is shown in the figure where radon transform isrealized with 1 dimensional inverse of Fast Fourier transform according to central slicetheorem. Figure 3. Curvelet flow graph [9]. Anisotropic scaling of curveletsCurvelets obey a parabolic scaling law which say that at scale 2 , each element hasan envelope which is aligned along a ‘ridge’ of length 2 and width 2 . For the analysis ofobjects with discontinuities along curves, parabolic scaling law holds good where analysisof frame elements are supported in elongated regions of obeying the relation width isproportional to square of length .Parabolic scaling ought to be extremely helpful in betterresolving the edge like components of the images ,it gives better accuracy in the vicinity ofedges while using fewer terms in an approximation ,when compared to nonparabolic scalingmethods like Fourier and wavelets.At higher Frequencies higher anisotropy occurs. The curvelet basis is governed bytriple factors scale, orientation (through radon transform) and translation. Parabolic scaling matrixDj =⎟⎟⎠⎞⎜⎜⎝⎛