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

ESTIMATION OF SURFACE NORMALS FOR USE IN 3D RENDERING OF
MEDICAL IMAGES
Xiaoping Ding', Karen All. Mudryl, Christopher H. Wood2
'Department of Biomedical Engineering, The University of Akron, Akron OH 44325
2Ficker Intemational, 595 Minor Road, Highland Hts., OH 44143
&.tract - Three dimensional (3D) surface
rendering techniques have been applied to
various medical images in recent yeazs.
The quality of the rendered image 1.5
dependent on the estimation of the
surface normal. A new method for
estimating the surface normal, called the
modified eigenvector method, is
described. The modified eigenvector
method was developed in order to overcome
some of the shortcomings of other
methods. The modified eigenvector method
is compared quantitatively using
simulated data and qualitatively using
human CT data to the gray-level gradient
method and the eigenvector method of
surface normal estimation.
RFTRODUCTIOH
Medical imaging devices, such as CT
and MR, generate a series of 2D slices.
The analysis of these images is usually
through the sequential observation of the
slices and mental reconstruction of the
object of interest by the interpreting
physician. With many imaging systems it
is possible to generate a 2D projection
of a 3D image through surface rendering.
The object of interest must first be
segmented from the image and then the
gray scale of the displayed surface
pixels must be determined in order to
generate the illusion of the 3D object.
The most cormnon methods to locate the
surface of the object of interest are
gtay-scale threshold, gray-scale
gradient, and zero-crossing of the second
derivative (11. For the gray-scale
threshold method, the surface points that
will be displayed correspond to pixels
with the same gray scale value or a gray
scale value larger than the threshold in
the original data. A shading function is
then used to determine the displayed
shade of these surface points so that the
illusion of a 3D object is generated.
The most often used shading function was
proposed by Phong [21, a commonly used
version of which is:
[l] where I is the intensity of the
surface point, Ib is the ambient light
intensity, Rb is the reflectivity of the
ambient light, Id is the intensity of a
point source of light, Rd is the diffuse
reflectivity coefficient of the surface,
and R, is the specular reflectivity
coefficient. The viewing distance is r,
and k is a constant. The 8 is the angle
between the direction of the point light
source and the surface normal. The angle
between the direction of the reflected
light and the viewing direction is a. If
only the diffuse reflectivity is
considered, the first term and the last
term in the above equation become zero.
If the viewer is at the same point as the
light source and the light source is at
infinity, the equation is simplified to I
= constant cos 6 . This result implies
that the projection of the surface normal
on the viewing direction is proportional
to the shade of a pixel on the rendered
surface. Since this value is determined
from the dot product of the unit surface
nom1 vector with the viewing direction
vector, the importance of knowing the
surface normal is critical.
A number of methods have been used
to find the surface normal, among which
are the gray-scale gradient method, the
adaptive gray-scale gradient method, the
depth-gradient method 131 and the
eigenvector method 141. The eigenvector
method was developed to attempt to
overcome some of the shortcomings of the
gradient methods; e.g., thin surfaces and
noisy images. The modified eigenvector
method, described below, was developed to
attempt to overcome the dependence on the
coordinate system found with the
eigenvector method.
I42

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.