Data C
Resume:
Tensor Field Regularization Using Normalized
Convolution
Carl-Fredrik Westin1 and Hans Knutsson2
1
Laboratory of Mathematics in Imaging,
Brigham and Women s Hospital,
Harvard Medical School,
Boston MA, USA
abpxxn@r.postjobfree.com
2
Department of Biomedical Engineering,
Link ping University Hospital,
o
Link ping, Sweden
o
abpxxn@r.postjobfree.com
Abstract. This paper presents a ltering technique for regularizing ten-
sor elds. We use a nonlinear ltering technique termed normalized con-
volution [Knutsson and Westin 1993], a general method for ltering miss-
ing 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 di erent signal characteristics, and increased robustness to
outliers. A driving application for this work has been ltering of data
from Di usion Tensor MRI.
1 Introduction
This paper presents a ltering technique for regularizing vector and higher order
tensor elds. In particular we focus on ltering of volume data from Di usion
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 di usion of water in bio-
logical tissue. Di usion 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 rst recognized by [7], who described di usion
quantitatively by adopting the mathematical equation of heat conduction de-
rived some years earlier by [8]. Anisotropic media such as crystals, textile bers,
and polymer lms have di erent di usion properties depending on direction.
Anisotropic di usion can be described by an ellipsoid where the radius de nes
the di usion in a particular direction. The widely accepted analogy between
symmetric 3 3 tensors and ellipsoids makes such tensors natural descriptors
R. Moreno-D and F. Pichler (Eds.): EUROCAST 2003, LNCS 2809, pp. 564 572, 2003.
az
c Springer-Verlag Berlin Heidelberg 2003
Tensor Field Regularization Using Normalized Convolution 565
for di usion. Moreover, the geometric nature of the di usion tensors can quan-
titatively 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 ber tracts. This anisotropic
di usion is due to tightly packed multiple myelin membranes encompassing the
axon. Although myelination is not essential for di usion anisotropy of nerves (as
shown in studies of non-myelinated gar sh 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 di usion
in myelinated ber 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 homoge-
neous without any suggestion of the complex arrangement of ber tracts. Hence,
the demonstration of anisotropic di usion 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 di usion tensor eld is calculated from a set of di usion-
weighted MR images by solving the Stejskal-Tanner equation. There is a physical
interpretation of the di usion tensor which is closely tied to the standard ellipsoid
tensor visualization scheme. The eigensystem of the di usion 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 di using 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 elds.
Normalized convolution (NC) was introduced as a general method for ltering
missing and uncertain data [10,19]. In NC, a signal certainty, c, is de ned 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 de ned by signal certainties and a spatially localizing mask.
A local description of a signal, f, can be de ned 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 (B f )) (1)
results in the following WLS local neighborhood model:
f0 = B (B Wa Wc B ) 1 B Wa Wc f, (2)
where Wa and Wc are diagonal matrices containing a and c respectively, and B
is the conjugate transpose of B .
566 C.-F. Westin and H. Knutsson
2.5 2.5
2 2
1.5 1.5
1 1
0.5 0.5
0 0
16 16
14 14
16 16
12 12
14 14
10 10
12 12
8 8
10 10
8 8
6 6
6 6
4 4
4 4
2 2
2 2
2.5 2.5
2 2
1.5 1.5
1 1
0.5 0.5
0 0
16 16
14 14
16 16
12 12
14 14
10 10
12 12
8 8
10 10
8 8
6 6
6 6
4 4
4 4
2 2
2 2
Fig. 1. Filtering of a scalar signal: Original scalar eld (upper left) and the result
without using the magnitude di erence certainty, cm (upper right). The amount of inter
region averaging can be controlled e ectively by including this magnitude certainty
measure. The smaller the sigma, the smaller inter the region averaging: lower left
= 1, lower right = 0.5.
2.1 Certainty Measures
In the present work, a regularization application, the local certainty function, c,
consists of two parts:
1. A voxel certainty measure, cv, de ned 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)
Tensor Field Regularization Using Normalized Convolution 567
16
16
8
8
1 1
1 8 16
1 8 16
16 16
8 8
1 1
1 8 16 1 8 16
Fig. 2. Tensor eld ltering: Original tensor eld (upper left) and the result using the
proposed method using = 0 (upper right), = 2 (lower left), and = 8 (lower right).
Notice how the amount of mixing of tensors of di erent 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 =,
where T = T / T .
The nal certainty function is calculated as the product of the voxel certainty
and the similarity certainty:
c = cv cs (4)
In general the voxel certainty function, cv, will be based on prior informa-
tion about the data. The voxel certainty is set to zero outside the signal extent
568 C.-F. Westin and H. Knutsson
to reduce unwanted border e ects. If no speci c local information is available
the voxel certainty is set to one. As described above, the second certainty com-
ponent, cs, is de ned locally based on neighboring information. The idea here
is to reduce the impact of outliers, where an outlier is de ned in terms of the
local signal neighborhood, and to reduce the blurring across interfaces between
regions having very di erent 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 con-
volution to [10]:
T0 = (5)
To focus on the power of introducing the signal/model similarity certainty mea-
sure, 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.
3 Scalar Field Regularization
In this section we present a scalar example to show the e ect of the the voxel
and magnitude certainty functions. This concept can be seen as generalization
of bilateral ltering [16] into the signal-certainty framework of normalized con-
volution.
Figure 1 shows the result of ltering a scalar signal using the proposed tech-
nique. 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 e ectively 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 ( = 1) and lower right ( = 0.5) plots.
4 Tensor Field Regularization
4.1 Synthetically Generated Tensor Field
Figure 2 shows the result of ltering a synthetic 2D tensor eld visualized using
ellipses. The original tensor eld 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.
Tensor Field Regularization Using Normalized Convolution 569
Fig. 3. Original tensor eld generated from DT-MRI data.
When ltering tensor data, the angular measure ca is important since it can
be used to reduce mixing of information from regions having di erent orienta-
tions. This is demonstrated in gure 2 using = 0 (upper right), = 2 (lower
left), and = 8 (lower right). Notice how the degree of mixing depends on the
angular similarity measure.
4.2 Di usion Tensor MRI Data
Figure 3 shows a tensor eld generated from DT-MRI data. In this work we
applied a version of the Line Scan Di usion Imaging (LSDI) technique [9,11,
12]. This method, like the commonly used di usion-sensitized, ultrafast, echo-
planar imaging (EPI) technique [18] is relatively insensitive to bulk motion and
physiologic pulsations of vascular origin.
570 C.-F. Westin and H. Knutsson
Fig. 4. Result of ltering the DT-MRI tensor eld 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
eld gradients. The time required for acquisition of the di usion tensor data for
one slice was 1 min; no averaging was performed. Imaging parameters were: ef-
fective TR=2.4 s, TE=65 ms, bhigh =1000 s/mm2, blow =5 s/mm2, eld of view 22
cm, e ective voxel size 4.0 1.7 1.7 mm3, 4 kHz readout bandwidth, acquisition
matrix 128 128.
Figure 4 shows the result of ltering the DT-MRI tensor eld in gure 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 de ning where the MR
signal is reliable. For the angular certainty function, ca, = 4 was used. The
applicability function a was set to a Gaussian function with standard deviation
of 3 sample distances.
Tensor Field Regularization Using Normalized Convolution 571
Acknowledgments. This work was funded in part by NIH grant P41-RR13218,
R01-MH 50747 and CIMIT.
References
1. D. Barash. A fundamental relationship between bilateral ltering, adaptive
smoothing and the nonlinear di usion equation. IEEE Trans. Pattern Analysis
and Machine Intelligence, 24(6):884 847, 2002.
2. P.J. Basser. Inferring microstructural features and the physiological state of tissues
from di usion-weighted images. NMR in Biomedicine, 8:333 344, 1995.
3. C. Beaulieu and P.S. Allen. Determinants of anisotropic water di usion in nerves.
Magn. Reson. Med., 31:394 400, 1994.
4. C. Chefd hotel and O. Faugeras D. Tschumperl e, R. Derich. Constrained ows of
matrix-valued functions: Application to di usion tensor regularization. In ECCV,
pages 251 265, Copenhagen, June 2002.
5. T.L. Chenevert, J.A Brunberg, and J.G. Pipe. Anisotropic di usion in human
white matter: Demonstration with MR techniques in vivo. Radiology, 177:401
405, 1990.
6. O. Coulon, D.C. Alexander, and S.R. Arridge. A regularization scheme for di usion
tensor magnetic resonance images. In XVIth IPMI, Lecture Notes in Computer
Science, volume 2082, pages 92 105. Springer-Verlag, 2001.
7. A. Fick. Uber di usion. Ann Phys, pages 94 59, 1855.
8. J. Fourier. Th orie analytique de la chaleur. Acad mie des Sciences, 1822.
e e
9. H. Gudbjartsson, S.E. Maier, R.V. Mulkern, I. A. M rocz, S. Patz, and F.A. Jolesz.
o
Line scan di usion imaging. Magn. Reson. Med., 36:509 519, 1996.
10. H. Knutsson and C-F. Westin. Normalized and di erential convolution: Methods
for interpolation and ltering of incomplete and uncertain data. In Proceedings of
Computer Vision and Pattern Recognition, pages 515 523, New York City, USA,
June 1993. IEEE.
11. S.E. Maier and H. Gudbjartsson. Line scan di usion imaging, 1998. USA patent
#5,786692.
12. S.E. Maier, H. Gudbjartsson, S. Patz, L. Hsu, K.-O. Lovblad, R.R. Edelman,
S. Warach, and F.A. Jolesz. Line scan di usion imaging: Characteriazation in
healthy patients and strok patients. Am. J. Roentgen, 17(1):85 93, 1998.
13. M. E. Moseley, Y. Cohen, J. Kucharczyk, J. Mintorovitch, H. S. Asgari, M. F.
Wendland, J. Tsuruda, and D. Norman. Di usion-weighted MR imaging of
anisotropic water di usion in the central nervous system. Radiology, 176:439 445,
1990.
14. C. Pierpaoli, P. Jezzard, P. J. Basser, A. Barnett, and G. Di Chiro. Di usion tensor
MR imaging of the human brain. Radiology, 201:637, 1996.
15. C. Poupon, C. A. Clark, F. Frouin, J. R gis, I. Bloch, D. Le Bihan, I. Bloch, and
e
J.-F. Mangin. Regularization of di usion-based direction maps for the tracking
brain white matter fascicles. NeuroImage, 12:184 195, 2000.
16. C. Tomasi and R. Manduchi. Bilateral ltering for gray and color images. In Proc.
of ICCV 98, Bombay, India, 1998.
17. D. Tschumperl e and R. Derich. Di usion tensor regularization with constraints
preservation. In CVPR, Hawaii, 2001.
18. R. Turner, D. le Bihan, J. Maier, R. Vavrek, L. K. Hedges, and J. Pekar. Echo
planar imaging of intravoxel incoherent motions. Radiology, 177:407 414, 1990.
572 C.-F. Westin and H. Knutsson
19. C.-F. Westin. A Tensor Framework for Multidimensional Signal Processing. PhD
thesis, Link ping University, Sweden, S 581 83 Link ping, Sweden, 1994. Disser-
o o
tation No 348, ISBN 91 7871 421 4.
20. C.-F. Westin and H. Knutsson. Extraction of local symmetries using tensor eld
ltering. In Proceedings of 2nd Singapore International Conference on Image Pro-
cessing. IEEE Singapore Section, September 1992.
21. C.-F. Westin, S. E. Maier, H. Mamata, A. Nabavi, F. A. Jolesz, and R. Kiki-
nis. Processing and visualization of di usion tensor mri. Medical Image Analysis,
6(2):93 108, 2002.
22. C.-F. Westin, S.E. Maier, B. Khidhir, P. Everett, F.A. Jolesz, and R. Kikinis.
Image Processing for Di usion Tensor Magnetic Resonance Imaging. In Medical
Image Computing and Computer-Assisted Intervention, Lecture Notes in Computer
Science, pages 441 452, September 1999.
23. D. M. Wimberger, T. P. Roberts, A. J. Barkovich, L. M. Prayer, M. E. Moseley,
and J. Kucharczyk. Identi cation of premyelination by di usion-weighted MRI.
J. Comp. Assist. Tomogr., 19(1):28 33, 1995.
Contact this candidate