Tensor Field Regularization Using Normalized Convolution

Tensor Field Regularization Using Normalized Convolution Carl-Fredrik Westin1 and Hans Knutsson2 1

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Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston MA, USA [email protected] Department of Biomedical Engineering, Link¨ oping University Hospital, Link¨ oping, Sweden [email protected]

Abstract. This paper presents a filtering technique for regularizing tensor fields. We use a nonlinear filtering technique termed normalized convolution [Knutsson and Westin 1993], a general method for filtering missing and uncertain data. In the present work we extend the signal certainty function to depend on locally derived certainty information in addition to the a priory voxel certainty. This results in reduced blurring between regions of different signal characteristics, and increased robustness to outliers. A driving application for this work has been filtering of data from Diffusion Tensor MRI.

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Introduction

This paper presents a filtering technique for regularizing vector and higher order tensor fields. In particular we focus on filtering of volume data from Diffusion Tensor Magnetic Resonance Imaging (DT-MRI). Related works include [20,19, 16,22,21,15,17,6,4,1]. DT-MRI is a relatively recent imaging modality that calls for multi-valued methods for data restoration. DT-MRI measures the diffusion of water in biological tissue. Diffusion is the process by which matter is transported from one part of a system to another owing to random molecular motions. The transfer of heat by conduction is also due to random molecular motion. The analogous nature of the two processes was first recognized by [7], who described diffusion quantitatively by adopting the mathematical equation of heat conduction derived some years earlier by [8]. Anisotropic media such as crystals, textile fibers, and polymer films have different diffusion properties depending on direction. Anisotropic diffusion can be described by an ellipsoid where the radius defines the diffusion in a particular direction. The widely accepted analogy between symmetric 3 × 3 tensors and ellipsoids makes such tensors natural descriptors R. Moreno-D´ıaz and F. Pichler (Eds.): EUROCAST 2003, LNCS 2809, pp. 564–572, 2003. c Springer-Verlag Berlin Heidelberg 2003


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for diffusion. Moreover, the geometric nature of the diffusion tensors can quantitatively characterize the local structure in tissues such as bone, muscle, and white matter of the brain. Within white matter, the mobility of the water is restricted by the axons that are oriented along the fiber tracts. This anisotropic diffusion is due to tightly packed multiple myelin membranes encompassing the axon. Although myelination is not essential for diffusion anisotropy of nerves (as shown in studies of non-myelinated garfish olfactory nerves [3]; and in studies where anisotropy exists in brains of neonates before the histological appearance of myelin [23]), myelin is generally assumed to be the major barrier to diffusion in myelinated fiber tracts. Using conventional MRI, we can easily identify the functional centers of the brain (cortex and nuclei). However, with conventional proton magnetic resonance imaging (MRI) techniques, the white matter of the brain appears to be homogeneous without any suggestion of the complex arrangement of fiber tracts. Hence, the demonstration of anisotropic diffusion in the brain by magnetic resonance has paved the way for non-invasive exploration of the structural anatomy of the white matter in vivo [13,5,2,14]. In DT-MRI, the diffusion tensor field is calculated from a set of diffusionweighted MR images by solving the Stejskal-Tanner equation. There is a physical interpretation of the diffusion tensor which is closely tied to the standard ellipsoid tensor visualization scheme. The eigensystem of the diffusion tensor describes an ellipsoidal isoprobability surface, where the axes of the ellipsoid have lengths given by the square root of the tensor’s eigenvalues. A proton which is initially located at the origin of the voxel has equal probability of diffusing to all points on the ellipsoid.

2

Methods

In this section we outline how normalized convolution can be used for regularizing scalar, vector, and higher order tensor fields. Normalized convolution (NC) was introduced as a general method for filtering missing and uncertain data [10,19]. In NC, a signal certainty, c, is defined for the signal. Missing data is handled by setting this signal certainties to zero. This method can be viewed as locally solving a weighted least squares (WLS) problem, were the weights are defined by signal certainties and a spatially localizing mask. A local description of a signal, f , can be defined using a weighted sum of basis functions, B. In NC the basis functions are spatially localized by a scalar (positive) mask denoted the “applicability function”, a. Minimizing




Wa Wc (Bθ − f ))
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results in the following WLS local neighborhood model: f0 = B(B ∗ Wa Wc B)−1 B ∗ Wa Wc f,

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where Wa and Wc are diagonal matrices containing a and c respectively, and B ∗ is the conjugate transpose of B.

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Fig. 1. Filtering of a scalar signal: Original scalar field (upper left) and the result without using the magnitude difference certainty, cm (upper right). The amount of inter region averaging can be controlled effectively by including this magnitude certainty measure. The smaller the sigma, the smaller inter the region averaging: lower left σ = 1, lower right σ = 0.5.

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Certainty Measures

In the present work, a regularization application, the local certainty function, c, consists of two parts: 1. A voxel certainty measure, cv , defined by the input data. 2. A model/signal similarity measure, cs : cs = g(T0 , T ), where T0 is the local neighborhood model. For simplicity we have constructed cs as a product of separate magnitude and angular similarity measures, cm and ca : cs = cm ca . For the magnitude certainty a Gaussian magnitude function has been used in our examples below:   2  |T0 | − |T | cm = exp − .) (3) σ

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Fig. 2. Tensor field filtering: Original tensor field (upper left) and the result using the proposed method using α = 0 (upper right), α = 2 (lower left), and α = 8 (lower right). Notice how the amount of mixing of tensors of different orientation can be controlled by the angular similarity measure.

The angular similarity measure, ca , is based on the inner product between the normalized tensors: ca =< Tˆ0 , Tˆ >α , where Tˆ = T /|T |. The final certainty function is calculated as the product of the voxel certainty and the similarity certainty: c = cv cs

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In general the voxel certainty function, cv , will be based on prior information about the data. The voxel certainty is set to zero outside the signal extent

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to reduce unwanted border effects. If no specific local information is available the voxel certainty is set to one. As described above, the second certainty component, cs , is defined locally based on neighboring information. The idea here is to reduce the impact of outliers, where an outlier is defined in terms of the local signal neighborhood, and to reduce the blurring across interfaces between regions having very different signal characteristics. 2.2

Simple Local Neighborhood Model

The simplest possible model in the normalized convolution framework is to use only one constant basis function, simplifying the expression for normalized convolution to [10]: T0 =

< a0 , cv T > < a0 , cv >

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To focus on the power of introducing the signal/model similarity certainty measure, this simple local neighborhood model is used in our examples below. The applicability function a0 was set a Gaussian function with standard deviation of 0.75 sample distances.

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Scalar Field Regularization

In this section we present a scalar example to show the effect of the the voxel and magnitude certainty functions. This concept can be seen as generalization of bilateral filtering [16] into the signal-certainty framework of normalized convolution. Figure 1 shows the result of filtering a scalar signal using the proposed technique. The upper left plot shows the original scalar signal: a noisy step function. The upper right plot shows the result using standard normalized convolution demonstrating that reduction of noise is achieved at the expense of unwanted mixing of features from adjacent regions. The amount of border blurring can controlled effectively by including the new magnitude certainty measure, cm (equation 3). The smaller the sigma, the smaller the inter region averaging as shown by the lower left (σ = 1) and lower right (σ = 0.5) plots.

4 4.1

Tensor Field Regularization Synthetically Generated Tensor Field

Figure 2 shows the result of filtering a synthetic 2D tensor field visualized using ellipses. The original tensor field is shown in the upper left plot. In this example, the voxel certainty measure, cv , was set to one except outside the signal extent where it was set to zero. The applicability function a was set to a Gaussian function with standard deviation of 3 sample distances.

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Fig. 3. Original tensor field generated from DT-MRI data.

When filtering tensor data, the angular measure ca is important since it can be used to reduce mixing of information from regions having different orientations. This is demonstrated in figure 2 using α = 0 (upper right), α = 2 (lower left), and α = 8 (lower right). Notice how the degree of mixing depends on the angular similarity measure. 4.2

Diffusion Tensor MRI Data

Figure 3 shows a tensor field generated from DT-MRI data. In this work we applied a version of the Line Scan Diffusion Imaging (LSDI) technique [9,11, 12]. This method, like the commonly used diffusion-sensitized, ultrafast, echoplanar imaging (EPI) technique [18] is relatively insensitive to bulk motion and physiologic pulsations of vascular origin.

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Fig. 4. Result of filtering the DT-MRI tensor field using the proposed method.

The DT-MRI data were acquired at the Brigham and Women’s Hospital on a GE Signa 1.5 Tesla Horizon Echospeed 5.6 system with standard 2.2 Gauss/cm field gradients. The time required for acquisition of the diffusion tensor data for one slice was 1 min; no averaging was performed. Imaging parameters were: effective TR=2.4 s, TE=65 ms, bhigh =1000 s/mm2 , blow =5 s/mm2 , field of view 22 cm, effective voxel size 4.0×1.7×1.7 mm3 , 4 kHz readout bandwidth, acquisition matrix 128×128. Figure 4 shows the result of filtering the DT-MRI tensor field in figure 3 using the proposed method. In this example, the voxel certainty measure, cv , was set to one except outside the signal extent where it was set to zero. An alternative to this is to use for example Proton Density MRI data defining where the MR signal is reliable. For the angular certainty function, ca , α = 4 was used. The applicability function a was set to a Gaussian function with standard deviation of 3 sample distances.

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Acknowledgments. This work was funded in part by NIH grant P41-RR13218, R01-MH 50747 and CIMIT.

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