Implicit modeling and surface
reconstruction
Implicit models
•
Algebraic surfaces

Algebraic methods define surfaces with polynomial
functions of three variables. Typically, these are low
degree polynomials (2 to 4).
The most popular algebraic surfaces are surfaces
with polynomials of degree 2 or so-called “quadrics”
(sphere, cylinder, cone, ellipsoid, etc.).
They can be used as separate shapes or as initial


primitives in Constructive Solid Geometry
Implicit models
•
•
•
•
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Skeleton-based implicit surfaces
A surface is considered as a zero set of the
scalar field generated by some “skeleton”
field source (discrete set of points, lines,
triangles, etc.).
Combining individual field functions of
skeletal elements can be implemented with
algebraic sum or integration technique
(convolution surfaces).
Skeleton-based implicit surfaces
A skeleton-based implicit surface is defined
as follows:
with F ( X )  0
F(X ) 

f ( rk )  T ,
k
where
is a field function of an individual skeletal
element, is distance between a given point
and the k-th skeletal element, and is a
threshold value. Figure illustrates the
simplest case of the point skeleton.
Implicit models
•
Convolution surfaces

Convolution surface is a generalization of skeletonbased implicit surfaces. A field function for a
convolution surface is obtained by spatial
convolution between a skeletal element and a
Gaussian-type kernel:
f (X ) 

R
3
s ( P ) h ( X  P ) dP ,
where X is a vector of point coordinates, S(X) is a
predicate function defining geometry of the skeletal
element ( for points of the element and for other
points in space), h(X) is a convolution kernel.
Implicit models
•
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Real function representation
The function representation (or F-rep) defines a whole geometric
object by a single real-valued continuous function of several
variables as F(X)0. F-rep combines together many different models
like algebraic and skeleton based implicits
Modeling concepts include sets of objects, operations and relations.
The main restriction of well-known min/max operations is that they
are C1 discontinuous. This can yield unexpected results in further
operations on the object.
Any operation has to be closed on the representation, i.e., generate
a continuous real function as a result. Set-theoretic operations are
closed on F-rep with the use of - C k continuous R-functions:
f1 & f 2  f1  f 2 
f1& f2  f1  f2 
2
f1  f 2
2
f1  f2
2
2
for the intersection of two objects described by f1 and f2, and for the
union. These functions have C1 discontinuity only in the points
where f1 = f2 = 0.
Implicit models
•
•
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Real function representation

A convex polygon can be represented as the intersection of half-planes.
Then, to convert this set-theoretic representation to a single real function
one can use min/max or more complex R-functions. A concave polygon
needs to be represented by set-theoretic operations on convex polygons or
half-planes.
Example
Defining a “reasonable” boundary shape of the restoration area: (a) Control
points of concave polygon. (b) Result of set-theoretic subtraction.
Implicit models
•
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Real function representation
The monotone formula
approach discussed in the
next section provides the
defining function without
internal
and
external
zeroes.
(a) A concave polygon with
the segment S25 used for
the cell partition; (b) the
surface of the defining
function constructed for the
separation example (a); (c)
a contour map of the
surface (b) for the nonnegative function values
with “internal zeroes”; (d) a
contour map of a function
without “internal zeroes”
generated
using
a
monotone formula.
Implicit models
•

Real function representation
A counter-clockwise ordered sequence of coordinates of polygon
vertices A1(x1, y1), A2(x2, y2),..., An(xn, yn) serves as the input. Also
coordinates are assigned to a point An+1 (xn+1,yn+1) coincident with A1
(x1,y1). It is obvious that the equation
f i   x ( y i  1  y i )  y ( x i  1  x i )  x i  1 y i  x iy i  1  0
defines a line passing through points Ai(xi,yi) and Ai+1(xi+1, yi+1); fi is a
function positive in an open region and negative in an open region
located respectively to the left and to the right of the line. When the
region is bounded by a convex polygon, it can be given by the logical
formula


 

1


2

 ...   n .
If is an external region of a convex polygon then





  1   2  ...   n .
Implicit models
•
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Real function representation

In practice, the monotone formula construction results in a tree structure
Consider a concave polygon shown in Figure a and a tree representing its
monotone formula in Figure b.
Implicit models
 Real function representation (example)
•
Generation of depth data and synthetic relief carving

Depth (height) data are extensively used in cutting simulation,
digital terrain modeling, prototyping and computer vision. We
apply our approach here to generate depth data from a polygonal
contour. Then we apply these data to model the process of carving
an arbitrary solid.

To define depth data in the functional form z = h(u,v) we construct
a monotone formula for the given polygonal contour. The resulting
defining function F(u,v) has positive values inside the contour,
zero values on the contour and negative values outside the contour.
So, we can model depth data by assigning
h(u,v) = 0, if F(u,v) < 0 and
h(u,v) = k F(u,v), if F(u,v)  0,

where k > 0 is a scaling coefficient. Values of the function h(u,v)
can be calculated at grid points and stored in a two-dimensional
array.
Implicit models
 Real function representation (example)
•
Generation of depth data and synthetic
relief carving
•
Figure(a) shows the original image. The
polygonal contour was taken from, the
depth data generated from the contour
(Figure(b)) and its application in carving
(Figure (c)). We model relief carving with
the “offsetting along the normal”
operation defined as follows:
•
G’(X) = G(X + hN),
where X = (x,y,z), G(X) is a defining
function of an arbitrary three-dimensional
solid, N is a gradient vector of the
function G in X,
h = h(T(X)) defined by ( ), and T(X) is a
mapping of X onto the uv-plane. In this
case, we use the parallel projection as a
mapping. Figure (c) shows the result of
this carving operation with the depth data
in Figure (b).
Implicit models
•
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Operations. Offsetting along
normal
(a) Initial “Horse” model
(b) Offset model
Implicit models
Real function representation (REFERENCES)
[1] Bajaj C., Ihm I., Warren J. Higher order interpolation and least-squares approximation using implicit algebraic
surfaces, ACM Transactions on Graphics, Vol. 11, No. 4, 1992, pp. 327-347.
[2] Blinn J.F. A generalization of algebraic surface drawing, ACM Transactions on Graphics, Vol. 1,
No. 3, 1982, pp.235-256.
[3] Dischler J.-M., Ghazanfarpour D. A geometrical based method for highly complex structured textures generation,
Computer Graphics Forum, Vol. 14, No. 4, 1995, pp.203-215.
[4] Dobkin D., Guibas L., Hershberger J., Snoeyink J. An efficient algorithm for finding the CSG representation of a
simple polygon, SIGGRAPH’88, Computer Graphics, Vol. 22, No. 4, 1988, pp.31-40.
[5] Dobkin D.P., Souvaine D.L. Computational geometry in a curved world, Algorithmica, vol.5, 1990, pp.421-457.
[6] Hoffmann C. On the separability problem of real functions and its significance in solid modeling, Computational
Algebra, Lecture Notes in Pure and Applied Mathematics, vol.151, Marcel Dekker, 1993, pp.191-204.
[7] Jones M.W., Chen M. A new approach to the construction of surfaces from contour data, EUROGRAPHICS'94,
Computer Graphics Forum, Vol. 13, No. 3, 1994, pp.75-84.
[8] Lim C., Turkiyyah G., Ganter M., Storti D. Implicit reconstruction of solids from cloud point sets, Third Symposium
on Solid Modeling and Applications, C.Hoffmann and J.Rossignac (Eds.), 1995, pp. 393-402.
[9] Muraki S. Volumetric shape description of range data using ‘blobby’ model, SIGGRAPH’91, Computer Graphics,
Vol. 25, No. 4, 1991, pp. 227-235.
[10] Pasko A., Adzhiev V., Sourin A., Savchenko V. Function representation in geometric modeling: concepts,
implementation and applications, The Visual Computer, vol.11, No.8, 1995, pp.429-446.
See also http://www.u-aizu.ac.jp/public/www/labs/sw-sm/FrepWWW/F-rep.html
[11] Peterson D. Halfspace representation of extrusions, solids of revolution, and pyramids, SANDIA Report SAND840572, Sandia National Laboratories, Albuquerque, NM, 1984.
[12] Preparata F.P., Shamos M. I. Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.
[13] Ricci A. A constructive geometry for computer graphics, The Computer Journal, Vol. 16, No. 2, 1973, pp. 157-160.
[14] Rvachev V.L. Methods of Logic Algebra in Mathematical Physics, Naukova Dumka, Kiev, 1974.
Implicit models
•
Real function representation (REFERENCES)
[15] Savchenko V.V., Pasko A.A. Collision detection for functionally defined deformable objects, Implicit Surfaces’95,
Eurographics Workshop Proceedings, B.Wyvill and M.-P. Gascuel (Eds.), INRIA, 1995, pp. 217-221.
[16] Savchenko V., Pasko A. Reconstruction from contour data and sculpting 3D objects, Journal of Computer Aided
Surgery, Vol. 1, Suppl., Proceedings of Second International Symposium on Computer Aided Surgery
ISCAS’95, 1995, pp.56-57.
[17] V. Shapiro. Real functions for representation of rigid solids, Computer Aided Geometric Design, Vol. 11, No. 2,
1994, pp. 151-160.
[18] Shapiro V., Vossler D.L. Separation for boundary to CSG conversion, ACM Transactions on Graphics, Vol. 12, No.
1, 1993, pp.35-55.
[19] Singh K., Parent R. Polyhedral shapes as general implicit surface primitives, Technical Report
OSU-CISRC-5/94-TR24, Ohio State University, 1994.
[20] Sourin A.I., Pasko A.A. Function representation for sweeping by a moving solid, IEEE Transactions on
Visualization and Computer Graphics, Vol. 2, No.1, 1996, pp.11-18.
[21] Tor S.B., Middleditch A.E. Convex decomposition of simple polygons, ACM Transactions on Graphics, Vol. 3, No.
4, 1984, pp.244-265.
[22] Velho L., Terzopoulos D., Gomes J. Constructing implicit shape models from boundary data, Graphical Models and
Image Processing, Vol. 57, No. 3, 1995, pp.220-234.
[23] Woodwark J.R., Wallis A.F. Graphical input to a Boolean solid modeller, CAD 82, A.Pipes (Ed.), Brighton, U.K.,
1982, pp.681-688.
[24] Wyvill G., McPheeters C., Wyvill B. Data structure for soft objects, The Visual Computer, Vol. 2, No. 4, 1986,
pp.227-234.
Example: Designing a lens
• Designing a lens by hand or by trial and error is a time-consuming and
tedious task. A dream of the end user is to define desired photometric
distribution and geometric constrains to design automatically an optical
shape satisfying them
• Requirements or constrains become more and more difficult to be met
using traditional optimization techniques because analytical solutions do
not exist for complicated shapes
• In techniques based on the error functional minimization, the question may
occur to solve highly non-linear problems
Example: Designing a lens (Cont.)
•
The issue of optimal design for optical systems for functionally
represented geometric objects has not yet been addressed
•
We focus on an approach to modeling shapes through the use
of evolutionary optimization or genetic algorithms for
functionally represented geometric objects
•
The motivation is
- to describe a case study of a “Trilobite eye” and “four-eyed”
fish
- to discuss problems of using the optimization procedure
RELATED WORKS
J. Loos, Ph. Slusallek, H.-P. Seidel, Using wavefront tracing for the
visualization and optimization of progressive lenses,
EUROGRAPHICS’98, Computer Graphics Forum, vol.17, No. 3,
1998, 255-265
J. Loos, G. Greiner, H.-P. Seidel, A variational approach to
progressive lens design, Computer-Aided Design, vol. 30, No. 8,
1998, 595-602
M. Halstead, B. Barsky, S. Klein, R. Mandell, Reconstructing curved
surfaces from specular reflection patterns using spline surface
fitting of normals. SIGGRAPH 96 Conference Proceedings, August
1996, 335-342
RELATED WORKS
M. Goel, R.L. Thompson, The evolution of the trilobite eye,
Computer Simulations of Self-Organization in Biological
Systems, part 11, 1988, 275-290
M.I. Sobel , Light, 1987
V.V. Savchenko, A.A. Pasko, Transformation of functionally
defined shapes by extended space mappings, The Visual
Computer, vol. 14, Nos. 5/6, 1998, 257-270
J.H. Holland, K.J. Holyoak , R.E. Nisbett, P.R. Thagard , Induction:
Processes of Inference Learning and Discovery, 1986
SHAPE MODELING SCHEME for Trilobite eye
•
•
Phacopids had eyes containing large biconvex lenses made of durable
minerals.
The lens consists of two subunits separated by a concave surface
SHAPE MODELING SCHEME for Trilobite eye (Cont.)
•
•
•
•
•
It has a spherical upper surface of radius R1 and a spherical lower surface
(“bowl”) of radius R2
Between these two surfaces, there is an intermediate refracting surface
that turns the lens shape into a non-manifold object
The shape of the lens allows any light ray passing from point f1 through the
lens to hit focus f2
The region above the lens has a refractive index of n1 , and the region
below the lens has a refractive index of bodily fluid of n4. The lens subunits
have different refractive indexes: n2 for the upper lens and n3 for the lower
lens
We would like to reconstruct the refracting surface using the combination
of the ray tracing technique, functionally based shape modeling and GAs
VOLUME MODELING for shape reconstruction
•

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
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Set-theoretic and many other operations have found quite general
solutions for geometric solids represented by continuous real functions as
f(x,y,z) >= 0
There are several reasons for using this representation in optical shape
design:
Shapes can be designed as combinations of analytically defined geometric
objects and volumetric ones that can be obtained from measurements of
natural objects
Spline-controlled global and local deformations are supported
The point membership relation based on the three-valued predicate lets us
quite easily extend applications to non-manifold objects with intermediate
refracting surfaces
Parametrized constructive solids (e.g., a lens with holes of variable shapes
and positions) can be modeled and optimized
VOLUME MODELING for shape reconstruction (Cont.)
• Naturally, approximation of the curved refractive interface by linear
segments can cause C1 discontinuities in the defining function.
• Using polynomial approximation we have to be cautious about highorder interpolation and errors depending on closeness of the point x to
the point xi.
• To provide a more smooth surface and a C1 continuous defining function,
we use least square fitting to approximate a sequence of the points (xi,zi),
i = 1,2, ... 12 by the Fourier series. The unknown refractive interface is
described as
z(x) = b0 +b1 cos(x) + b2cos(2x) + b3cos(3x) + a1 sin(x) + a2sin(2x) + a3sin(3x)
The FUNCTIONAL MODEL of the phacopid eye
1. Lens(xt,zt){
2. shift = Fz(xt);
//refractive interface
3. vf1 = sphere(xt,0,zt,R1); //upper sphere
4. vf2 = sphere(xt,0,zt,R2); //lower sphere
5. f_fi = vf1 & vf2;
//intersection of spheres
6. f_fo = f_fi & (zt-shift); //upper lens unit
7. f_so = f_fi & (shift-zt); //lower lens unit
8. f_glob = f_fo | f_so; //union of two lens units
//Assign a tag for point membership identification
9. if(f_fo >= 0.) id_obj = 0;
10. if(f_so >= 0.) id_obj = 1;
11. if(f_glob < 0.) id_obj = 2;
12. return f_glob;}
The FUNCTIONAL MODEL of the phacopid eye
•
•
•
•
To provide a more smooth surface and a C1 continuous
defining function, we use least square fitting to
approximate a sequence of the points (xi,zi), i = 1,2, ... 12
by the Fourier series
The problem of the curved refractive interface design
concerns the main feature of the eye to focus a series of
light rays emanating at many angles qk from a point f1 to
another point f2
The z-intercept of a light ray can be calculated by tracing
the ray emanating from f1
We calculate a “fitness” function that expresses the wellknown fact that the more effectively the lens can focus
lights from f1 to f2 the less residual of z-intercepts from the
fixed position of focus f2 we can attain
RESULT of evolutionary optimization of the refractive interface
CASE STUDY of the “four-eyed” fish
• The “four-eyed” fish, swims along the water surface with its head
partly out of water, and each of its two eyes has two pupils, one kept
above the surface, the other below
• Each pupil focuses light upon a different retina
Surfaces. Problems
2 Fundamental Problems
∕
Reconstruction
\
Segmentation
of surfaces (onto a uniform grid)
of surfaces into different
from sparse depth measurements
surface types for object
that may have outliers.
recognition & refinement of the
surface estimates.
Contours → 2-D
Surfaces → 3-D
27
Surfaces. Problems (Cont.)
• Basic methods include
– Converting point measurements into a mesh of
triangular facets,
– Segmenting range measurements into simple
surface patches,
– Fitting a smooth surface to the point
measurements, and
– Matching a surface model to the point
measurements.
28
Surfaces. Problems (Cont.)
• One of the interpolation methods is based on the use of radial
basis functions (proposed by V. Savchenko )
• Reconstruction from 3D scattered data (“Phobos” project, 1986)
29
Reconstruction and Representation
of 3D Objects with Radial Basis
Functions
Carr et al. SIGGRAPH 2001
Problem
• Given a set of points 𝑝𝑖 , 𝑖 = 1 … 𝑛
• Define an implicit function 𝑓: 𝑅3 → 𝑅 , such
that:
𝑓 𝑝𝑖 = 0, 𝑖 = 1 … 𝑛
• Surface is defined by the zero set of f
Constrains
• f(x) = 0 is a trivial solution
• Add off-surface normal points to constrain the
solution further
• Both side approach was proposed by Turk &
O’Brien in 2001
Off-Surface Normal Points
• For each point, add two more points defining
the inside and outside
Off-Surface Normal Points
• Create points
𝑝𝑖 𝑜𝑢𝑡 = 𝑝𝑖 + 𝜖𝑛
𝑝𝑖 𝑖𝑛 = 𝑝𝑖 − 𝜖𝑛
• New constrains
𝑓 𝑝𝑖 = 0
𝑓 𝑝𝑖 𝑖𝑛 = −𝛿
𝑓 𝑝𝑖 𝑜𝑢𝑡 = +𝛿
𝑖 = 1…𝑛
Choice of Parameters
•  is chosen as the distance of the new point
from 𝑝𝑖 , signed distance function
•  must be chosen such that the displacement
to the new point doesn’t intersect other parts
of the surface
Choice of

Generating Normals
• Input is a point cloud, no normals
• Generate normals as per [Hoppe 92]
– Estimation from plane fitted to neighborhood
• Additionally, use consistency and/or scanner
positions to resolve ambiguities
• If fails, don’t define off-surface normal points
Find the Implicit Function
• Given a set of distinct nodes 𝑥𝑖 ∈ 𝑅3 , 𝑖 = 1 … 𝑛
and a set of function values 𝑓𝑖 ∈ 𝑅, 𝑖 = 1 … 𝑛,
find an interpolant s, such that
𝑠 𝑥𝑖 = 𝑓𝑖 , 𝑖 = 1 … 𝑛
𝐵𝐿(2) (𝑅 3 )
• The interpolant is choose from 𝐵𝐿(2) (𝑅3 ) , the
Beppo-Levi space of distributions on 𝑅3 with
square integrable second derivatives
• Square integrable means
•
∞
𝑓 𝑥
2
𝑑𝑥 < ∞
−∞
𝐵𝐿(2) (𝑅 3 ) is the space of functions in 𝑅3
second derivatives fall off quickly
whose
Radial Basis Functions
• [Duchon 77] shown that the smoothest
intepolant in 𝐵𝐿(2) (𝑅3 ) is
N
𝑠 𝑥 =𝑝 𝑥 +
λi |x − xi |
i=1
• It is a particular example of Radial Basis
N
Functions:
𝑠 𝑥 =𝑝 𝑥 +
λi ϕ|x − xi |
i=1
Radial Basis Functions
N
𝑠 𝑥 =𝑝 𝑥 +
λi ϕ|x − xi |
i=1
•
•
•
•
p(x) is a low degree polynomial
𝜆𝑖 are real coefficients
|| is the Euclidean norm in 𝑅3
ϕ(x), the basis function, is a real function
– Thin plate spline function 𝜙 𝑟 = 𝑟 2 log r , r = |x − xi |
– Guassian 𝜙 𝑟 = 𝑒𝑥𝑝(−cr 2 )
– Biharmonic and triharmonic splines 𝜙 𝑟 = 𝑟 𝑜𝑟 𝜙 𝑟 = 𝑟 3
Radial Basis Functions
• For biharmonic spline RBF, 𝑝
𝐴
𝑃𝑇
𝑃
0
𝑥 = c1 + c2 x + c3 y + c4 z
𝑓
𝜆
𝜆
=𝐵
=
𝑐
𝑐
0
– 𝐴𝑖,𝑗 = 𝑥𝑖 − 𝑥𝑗 , 𝑖, 𝑗 = 1, … , 𝑁
𝑖 = 1, … , 𝑁
– 𝑃𝑖 = 1, 𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ,
• B is symmetric and invertible under very mild
conditions
Non-Compact Support
• Biharmonic spline RBF
• Pros
– Suitable to non-uniformly sample data
– Handle holes
• Cons
– A is not sparse, more computation and not
scalable
Fast Methods
• Approximated with Fast Multipole Method
(FMM)
– Cluster RBF centers into a hierarchy
– Approximate evaluation for cluster far away
– Direct evaluation for clusters near by
– Reduce both storage and computation
RBF Center Reduction
• Use fewer centers to achieve desired accuracy
• A greedy algorithm
–
–
–
–
–
1. Choose a subset of centers, fit a RBF
2. Evaluate the residual, 𝜖𝑖 = 𝑓𝑖 − 𝑠 𝑝𝑖 , 𝑖 = 1 … 𝑛
3. If max{ 𝜖𝑖 } < desired accuracy then stop
4. Add new centers where 𝜖𝑖 is large
5. Refit RBF and goto step 2
RBF Center Reduction
Noisy Data
• Consider both interpolation and smoothness
1
min(𝜌 𝑠 +
𝑠
𝑁
–
𝑁
(𝑠 𝑥𝑖 − 𝑓𝑖 ))
𝑖=1
𝑠 measures the smoothness, 𝜌 is the weight
• The linear system changes to
𝐴 − 8𝑁𝜋𝜌𝐼
𝑃𝑇
𝑃
0
𝑓
𝜆
=
𝑐
0
Noisy Data
RBF approximation of a noisy LIDAR data
Noisy Data
Surface Evaluation
• A marching tetrahedra variant, optimized for
surface following
– Avoid RFS evaluation at a grid of voxels
– Start from several seed points
– Wavefronts of facets spread out across surfaces
– Stop when intersect the bounding box
Surface Evaluation
Iso-surfacing a RBF from a single seed point,
wavefront marked by red.
Results – Complicate Datasets
Results – Mesh Repair
Results – Mesh Repair
Results – Non-uniform Data