Estimation of surface normals for use in 3D rendering ... - IEEE Xplore

METHODS
This study compared the gray-level
gradient, the eigenvector and the
modified eigenvector methods for
determining the surface normals and hence
the surface rendered images.
For the gray-level gradient method two
adjacent pixels for each axis were used
in the calculation; the normal being
N = (dx, dy, dz)
where dx, dy, dz are the central
differences and N is normalized to a unit
vector.
For the eigenvector method either 2, 4,
or 6 neighbors in each axis direction and
the gray scale values of these pixels
were used. The surface normal is
N = V1 = (XI, yl, 21) '
where V1 is the normalized eigenvector
associated with the largest eigenvalue,
11, of the covariance matrix A.
For the modified eigenvector method all
of the pixel above a threshold in a cubic
neighborhood of the pixel under
consideration and the re 1 at i ve
coordinates of these pixels, rather than
their gray scale values, are used to find
the normal. The normal is
N = V3 = (xl, yl, zl)
where V3 is the normalized eigenvector
associated with the smallest eigenvalue,
h, of the covariance matrix A.
The above three methods to compute
the surface normal were evaluated on data
sets containing a spherical object. Two
of the data sets had a step transition
between the object and the background,
one with noise and one without. The
other two sets had a linear transition
between the object and the background,
one with noise and one without. The
techniques also were tested on human CT
data acquired using a Picker
International, Inc. CT scanner.
RESULTS
Both the eigenvector method and the
modified eigenvector method produced
images with comparable quality to the
gray-level gradient method. The
quantitative studies showed that the
eigenvector method was more accurate than
the gray-level gradient method when 4 or
6 neighbors were used in the calculation
and that the modified eigenvector method
was more accurate than the gray-level
gradient method and the eigenvector
method for the step transition between
the object and the background. The
eigenvector method was more accurate than
the modified eigenvector method when the
transition between the object' and the
background was linear. For both the
eigenvector method and the modified
eigenvector method calculations using 4
neighbors were more accurate than using 2
neighbors.
The studies using human CT data
showed that the eigenvector method
revealed some fine structures that could
not be seen using the gray-level gradient
method; however, this method tended to
exaggerate the small structures. The
modified eigenvector method produced
images that looked smooth, but flat.
Both the eigenvector method and the
modified eigenvector method were
computationally more expensive than the
gray-level gradient method.
REFERENCES
[lIM. Magnusson, R. Lenz, P.E. Danielsson
(1988). Evaluation of Methods for Shaded
Surface Display of CT-Volumes.
Proceedings of the 9th ICPR, 1287-1294.
121 B.T. Phong (1975) . Illumination for
Computer Generated Pictures.
Communications of the ACM, 18(6) :311-317.
[SlU. Tiede, K.H. Hoehne, M. Bomas, A.
Pommert, M. Riemer, G. Wiebeck (1990).
Investigation of Medical 3D-Rendering
Algorithms. IEEE Computer Graphics and
Applications, 10:41-53.
[4]C.H. Wood, X. Ding, W. Sattin (1991).
Eigenvector Surface Normal Estimation in
Medical Imaging. 9th Annual Meeting of
the Society for Magnetic Resonance
Imaging, Chicago, IL.