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This method uses the spherical harmonics (SH) expansion and is based on generating rank-1 contravariant tensors using the SH coefficients, and contracting them with covariant tensors to obtain invariants. The proposed technique enables the systematic construction of invariants for SH expansions of any order using simple mathematical operations. In addition, it allows construction of a large set of invariants, even for low order expansions, thus providing rich feature vectors for image analysis tasks such as classification and segmentation. In this paper, we use this technique to construct feature vectors for eighth-order fiber orientation distributions (FODs) reconstructed using constrained spherical deconvolution (CSD). Using simulated and in vivo brain data, we show that these invariants are robust to noise, enable voxel-wise classification, and capture meaningful information on the underlying white matter structure.","first_page_t":"718","rights_management_t":"(c) 2014 IEEE. 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However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.","title_t":"Generalized hardi invariants by method of tensor contraction","id":713034,"publication_type_t":"pre-print","parent_i":0,"type_t":"Text","thumb_s":"/8e/20/8e20038b873a81d6ca31f452622f87d75bcc8090.jpg","last_page_t":"721","oldid_t":"uspace 11051","metadata_cataloger_t":"CLR","format_t":"application/pdf","modified_tdt":"2015-05-04T00:00:00Z","school_or_college_t":"","language_t":"eng","file_s":"/52/ab/52ab71b48494b207141fa8f3204ee6cc7deb87f8.pdf","format_extent_t":"967,534 bytes","other_author_t":"Johnson, Chris R.","created_tdt":"2015-05-04T00:00:00Z","_version_":1664094557869965312,"ocr_t":"GENERALIZED HARDI INVARIANTS BY METHOD OF TENSOR CONTRACTION Yaniv Gur and Chris R. Johnson SCI Institute, 72 S. Central Campus Dr., University of Utah, Salt Lake City, UT 84112, USA ABSTRACT We propose a 3D object recognition technique to construct rotation invariant feature vectors for high angular resolution diffusion imaging (HARDI). This method uses the spherical harmonics (SH) expansion and is based on generating rank-1 contravariant tensors using the SH coefficients, and contract- ing them with covariant tensors to obtain invariants. The pro- posed technique enables the systematic construction of invari- ants for SH expansions of any order using simple mathemati- cal operations. In addition, it allows construction of a large set of invariants, even for low order expansions, thus providing rich feature vectors for image analysis tasks such as classifi- cation and segmentation. In this paper, we use this technique to construct feature vectors for eighth-order fiber orientation distributions (FODs) reconstructed using constrained spheri- cal deconvolution (CSD). Using simulated and in vivo brain data, we show that these invariants are robust to noise, enable voxel-wise classification, and capture meaningful information on the underlying white matter structure. Index Terms- Diffusion MRI, HARDI, invariants 1. INTRODUCTION Invariants play an important role in diffusion MRI (dMRI). They represent tissue properties, such as diffusion anisotropy, and are used for registration, tissue segmentation and classi- fication, as well as white matter integrity measures in clinical studies of debilitating brain diseases. Importantly, these fea- tures need to be rotation invariant to capture orientation in- dependent tissue properties, as well as to enable comparisons across images that are not completely aligned. Modelling dif- fusion using second-order tensors, as in DTI, enable construc- tion of diverse anatomically meaningful scalars, such as frac- tional anisotropy (FA), mean diffusivity (MD), and relative anisotropy (RA), which capture tissue microstructure, and are used to indicate anatomical changes [1]. Most of these scalars are derived using the tensor eigenvalues, hence, they are natu- rally rotation invariant as the eigenvalues remain intact under rotation. In HARDI, however, the situation is more compli- cated as the diffusion profile is described by higher-order ten- sors or orientation distribution functions (ODFs), and there This work was funded in part by the NIH/NCRR Center for Integrative Biomedical Computing, P41RR12553. is no straightforward generalization of DTI invariants to these models. The generalized FA (GFA), for example, aims to rep- resent anisotropy in HARDI models, but has limited classifi- cation power, and is sensitive to noise being directly com- puted from a discrete representation of orientation distribu- tion function (ODF) [2]. In order to take advantage of the enhanced modelling ca- pabilities of HARDI compared to DTI, it is important to de- rive new invariants that capture tissue properties, and can be used as white matter biomarkers. Over the last years, re- searchers have created new invariants for HARDI-based mod- els, for example, the generalized anisotropy (GA) and scaled anisotropy (SE) [3], as well as several approaches that are based on second and fourth-order tensor representations [4, 5]. A recent approach uses the Gaunt coefficients to construct invariants for ODFs and HARDI signals using the more gen- eral SH representation [2]. Our proposed technique follows a similar path of using the SH representation. We then leverage the idea of invari- ants constructed by tensor contraction used in computer vi- sion for 3D pattern recognition [6]. The original idea emerged from the theory of angular momentum addition in quantum physics, and although it relies on deep and complex theoreti- cal foundations, the formulation enables systematic construc- tion of invariants in an elegant and simple way. This method is general, as it enables extraction of invariants from any 3D object represented as a SH series. Therefore, it can be used to construct invariants from the dMRI signal, or from any dif- fusion modelling object, such as ODF or FOD. In addition, it enables direct construction of invariants for any expansion order, thus, it is not limited to the common second or fourth order expansions. This method generalizes the SH descrip- tors used in [7] to classify autism spectrum disorder (ASD) patients and controls, and to segment brain tissue [8, 9]. It is based on constructing contravariant rank-1 tensors (vec- tors) using the SH and Clebsch-Gordan (CG) coefficients, and contracting them with covariant vectors to obtain rank-0 ten- sors (invariants). This process can be continued repeatedly to build as many invariants as desired regardless of the SH or- der, therefore, enables the construction of long feature vectors with strong classification capabilities. This is an advantage over the method presented in [2] in which the maximal num- ber of invariants is bounded by the rank of the Toeplitz-like matrix. 978-1-4673-1961-4/14/$31.00 ©2014 IEEE 718 We demonstrate the strengths of our approach in both syn- thetic and in vivo experiments. Using simulated data we show that these invariants are robust to noise and can classify vox- els based on the number of fiber compartments and their dif- fusivities. Using in vivo brain data, we show that they cap- ture anatomically meaningful information, and may be used as white matter integrity measures. 2. CONSTRUCTION OF INVARIANTS Given a basis feig, one can write a vector in this basis us- ing the expansion coefficients, such that v = ciei. The ex- pansion coefficients are components of a contravariant rank-1 tensor (vector), and are denoted with upper indices. On the other hand, the elements of a covariant vector (dual vector) are denoted with lower indices: v = ciei. When feig is an orthonormal basis as in our case, contravariant and covariant vectors are simply complex conjugates of each other. The tensor contraction operation multiplies a contravariant with a covariant tensor, such that repeated lower and upper indices are summed in the Einstein summation convention. This op- eration is equivalent to an inner product, thus, a contraction of two vectors yields a scalar (an invariant): s = cici. Let f be a continuous spherical function. According to Fourier's theorem on the sphere, it can be expanded in a spher- ical harmonics (SH) basis, such that f(θ, ϕ) = 1Σ l=0 Σl m=l cml Y m l , (1) where the Y m l denotes the SH, and the cml 's are complex ex- pansion coefficients. In practice, when representing 3D ob- jects, the infinite sum is truncated at a finite order L. In our case, the function f may represent the dMRI signal given at a discrete set of gradient directions, an orientation distribu- tion function (ODF), a fiber orientation distribution (FOD), or even a higher-order tensor representation of the diffusion pro- file using the appropriate coefficients transformation. Thus, this representation is not restricted to a particular reconstruc- tion technique. While in general there is no restriction on L, most HARDI reconstruction techniques are based on even or- der expansions. While the common HARDI reconstruction technique uses the real SH expansion, the method described here is based on the complex SH representation. Therefore, the real coeffi- cients computed by the various reconstruction algorithms are transformed to the complex domain using the following rela- tion: cml = p1 2 (^clm + i^cl m) if m > 0 ^cl0 if m = 0 (1)m(^c m l ) if m < 0, (2) where ^clm denotes the real coefficients, and stands for the complex conjugate. The SH form an orthonormal basis of the Hilbert space of square-integrable functions. An important observation is that this space can be decomposed into a set of subspaces, each associated with a specific l value and is spanned by the spherical harmonics Y m l , l m l. We will denote these subspaces using the subscript l as Vl, that is, Vl = SpanfY l l , Y l+1 l , . . . , Y l1 l , Y l l g. These subspaces correspond to irreducible representations of the rotation group, SO(3). Thus, each of these subspaces is globally rotational invariant, that is, given a vector f 2 Vl and a ro- tation R, we have R(f) 2 Vl. This property enables us to construct rotation invariants by contracting contravariant and covariant vectors defined in Vl. The first set of invariants are constructed by contracting the SH coefficients, cml , for a given l. Being expansion coef- ficients, they are contravariant components of a vector in Vl. Therefore, by contracting them with their covariant counter- parts, we construct a tensor of order zero (a scalar), which is rotation invariant: Il = cml clm = Σl m=l cml(cml ) . (3) These set of invariants are the well-known power spectrum SH descriptors used in various biomedical applications as an input to the SVM classifier [7, 9], as well as in 3D object recognition in computer vision [10]. Despite their usefulness, as we show here, these invariants are sensitive to noise and may lead to misclassification of objects. For example, ODFs describing similar white matter structure may be classified into separate groups due to the impact of noise. Since the SNR in HARDI scans is generally low, robustness to noise is an important property. In addition, given the maximal ODF order L, only L/2 + 1 such invariants exist (for even L val- ues), resulting in limited classification power. However, it turns out that these invariants are a particular case of a richer set of invariants that are constructed by mixing tensors from different subspaces, and contracting them. This idea origi- nated from the theory of angular momentum addition in two- particle systems in quantum mechanics [11]. It also corre- sponds to decomposing reducible representations of the rota- tion group, SO(3), into irreducible ones. The main elements here are the CG coefficients (known also as the Wigner coef- ficients), which are used to couple the subspaces. Using these coefficients, one can construct new contravariant rank-1 ten- sors in Vl as follows: Tm l;l1;l2 = km;m1;m2 l;l1;l2 cl1m1cl2m2 (4) = Σl1 m1=l1 Σl2 m2=l2 km;m1;m2 l;l1;l2 (cm1 l1 ) (cm2 l2 ) , where km;m1;m2 l;l1;l2 are the CG coefficients. The computed ten- sors can be contracted with the SH coefficients to create a new 719 set of invariants: Jl;l1;l2 = Tm l;l1;l2clm = Σl m=l Tm l;l1;l2 (cml ) . (5) In addition, they can be contracted with each other to build another set of invariants: Kl;l1;l2;l3;l4 = Tm l;l1;l2Tl;l3;l4;m = Σl m=l Tm l;l1;l2 (Tm l;l3;l4 ) . (6) As shown here, the computation of these invariants only in- volves multiplications of coefficients and a summation. The CG coefficients are available in tables, and can also be com- puted using recurrence relations. The process of constructing tensors in Vl and contract- ing them can be continued as desired to produce more invari- ants. For example, one can use the tensors Tm l to generate new tensors in Vl, and then contract them with the SH coeffi- cients, the Tm l 's, or with themselves. However, the three sets of invariants I, J and K already provide long feature vec- tors, even for low expansion orders (e.g., L = 2). Finally, the conditions under which these invariants can be computed are jl1 l2j l (l1 + l2) for J, and the additional condition jl3l4j l (l3+l4) forK, whereas the I invariants can be computed for any l (l = 0, 2, . . . ,L). Further symmetry rela- tions can be used to reduce the number of invariant, however, we will not consider this aspect in this short paper. 3. RESULTS 3.1. Synthetic data In this experiment, we simulated the signal using the multi- tensor model with diffusivities [1.7e-3, 3e-4, 3e-4] for each tensor. The signal was corrupted by Rician noise with a base- line SNR of 20. We generated 100 repetitions of randomly rotated ODFs for each of the following cases: one fiber, two fibers crossing at 60 degrees with equal mixing weights, and three fibers with equal mixing weights (the two fibers case with a third orthogonal fiber). For each ODF we computed 10 different invariants: Il, l = 0, 2, . . . , 8 and J0;2;2, J2;2;2, J4;2;2, J6;4;4, J8;4;4. Then, we plotted the first three invari- ants as vectors in R3 for both the I and J invariants (Fig. 1). We used the variances computed using PCA to measure the concentration of the point clouds in R5 generated by these in- variants. Fig. 1 and Table 1 show that both sets of invariants could classify the different fiber configurations into three dis- tinct groups corresponding to voxels with one, two, or three fibers. However, these results also show that the J invariants are more robust to noise as their associated point clouds are more compact. Fig. 1. Number of fiber classification using the I invariants (left), and the J invariants (right). The point clouds corre- spond to one fiber (blue), two fibers (green), and three fibers (red). Variance 1F I 0.0336 0.0176 0.0065 0.0040 0.0013 J 0.0190 0.0063 0.0011 0.0008 0.0008 2F I 0.0246 0.0193 0.0153 0.0049 0.0040 J 0.0063 0.0038 0.0034 0.0019 0.0007 3F I 0.0263 0.0125 0.0074 0.0052 0.0035 J 0.0040 0.0030 0.0011 0.0010 0.0007 Table 1. Variance comparisons using PCA. The labels 1F, 2F and 3F correspond to one, two, or three fibers, respectively. The labels I and J indicate the invariant type, and each row presents the vector of invariants beginning with l = 0 (left). 3.2. Simulated phantom In this experiment, we extracted the invariants J0;2;2, J2;2;2, and J4;2;2 from 8th-order FODs reconstructed from a slice of the 3D simulated phantom used for comparisons at the HARDI reconstruction ISBI challenge [12]. This phantom is composed of voxels containing fiber configurations differing by the number of compartments, and the compartment diffu- sivities used to simulate the signal. To compute the invariants we used the phantom simulated at SNR=30. After comput- ing the set of 3 invariants for each FOD, we plotted them as vectors in RGB space. The results presented in Fig. 2 show that using these invariants as feature vectors we were able to classify the voxels not only based on the number of fibers, but also based on differences in compartment diffusivities. The previously discussed robustness to noise of the J invariants compared to the I invariants is clearly shown. 3.3. In vivo brain data The brain data was acquired on a 3T Siemens Alegra machine using 60 diffusion weighted directions, 10 baseline scans, and a b-value of 2000. For the voxels within the brain mask, we reconstructed 8th-order FODs using CSD, and computed a set of J invariants. In addition, we computed the GFA map for comparison. The results, presented in Fig. 3, show that these invariants describe white matter structures, such as the cor- pus callosum, and the superior and inferior longitudinal fasci- 720 Fig. 2. Voxel-wise classification of the simulated ISBI'12 challenge phantom. From left to right: the ground-truth num- ber of fibers map, the same map segmented according to the principal diffusivities, voxel-wise classification using the I in- variants, and the J invariants. culi (ILF and SLF). Interestingly, these results also show that these invariants describe anisotropy by high intensity values, for example, in the corpus callosum area. Thus, in addition to their classification abilities, these invariants can be used as rotation invariant integrity measures and may indicate struc- tural changes in white matter. For comparison, the GFA map (bottom right) appears to be noisy and presents white matter structures less clearly. It is also shown that the invariant cor- responding to the 6th-order show less structure and captures mostly noise. This may indicate that the data is very noisy, but it also raises a question whether higher-order invariants can be used for denoising or for extracting features from a noisy background. Fig. 3. Maps of the J invariants extracted from in vivo brain data. Top (L to R): J0;0;0, J0;2;2, J2;2;2, and J2;2;0. Bottom: J4;2;2, J4;4;2, J6;4;4, and GFA. 4. CONCLUSIONS In this paper, we applied tensor contraction techniques used in 3D object recognition to generate invariants for HARDI models. This technique allows us to compute invariants for the dMRI signal, or any HARDI reconstruction model that uses SH expansion. It provides a systematic procedure to construct invariants for any given SH truncation order using simple operations, and it can generate as many invariants as desired, regardless of the expansion order. We used simulated and in vivo brain data to show that these invariants are robust to noise, enable voxel-wise classification, and capture white matter structure. 5. REFERENCES [1] Denis Le Bihan, Jean-Franois Mangin, Cyril Poupon, Chris A. Clark, Sabina Pappata, Nicolas Molko, and Hughes Chabriat, \"Diffusion tensor imaging: Con- cepts and applications,\" Journal of Magnetic Resonance Imaging, vol. 13, no. 4, pp. 534-546, 2001. [2] Evan Schwab, H. Ertan etingl, Bijan Afsari, Michael A. Yassa, and Ren´e Vidal, \"Rotation invariant features for HARDI,\" in Information Processing in Medical Imag- ing, vol. 7917 of LNCS, pp. 705-717. 2013. [3] Evren O¨ zarslan, Baba C. Vemuri, and Thomas H. Mareci, \"Generalized scalar measures for diffusion MRI using trace, variance and entropy,\" Magn. Reson. Med., vol. 53, pp. 866-876, 2005. [4] A. Fuster, J. van de Sande, L.J. Astola, C. Poupon, J. 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