Demetri Terzopoulos, et. al., "Elastically Deformable Models":
I found this paper to be extremely interesting; I was a physics
major as an undergraduate, and so much of the material Terzopoulos
discusses was familiar to me. I hadn't known that splines were actually
based in this sort of mathematics; a little more elaboration on this point
would have been nice. Their procedure itself seems rather costly in terms
of computing power, and I got a little lost in the discussion of
discretization. At the same time, this modeling method has many benefits
that tend to outweigh these costs (such as conceptual simplicity and very
physically accurate results).
Andrew Witkin, et. al., "Energy Constraints on Parametrized Models":
An interesting approach with some annoying problems. Since the
forces modeled do not necessarily match real-world physical forces, some
models defined using this method might act in ways that defy natural
physical intuition; this is a mixed blessing at best. Future developers of
this method might also wish to have the program resolve the local minima
problem itself, without human intervention. On a more technical note, I
was confused by some of Witkin's notation: are the (u,v) pairs used in the
shape-defining functions coordinates along the surface of a given object,
as I suspect, or something else?
Elastically Deformable Models
Demetri Teropoulos, John Platt, Alan Barr, Kurt Fleischer
The authors put forth using elastically deformable models for use in computer graphics to take the place of kinematic models. Kinematic models do not naturally interact with one another, nor are they affected by external "forces." The authors models are "active" meaning that they react to collisions and eachother.
They use energy functions to represent the internal forces of the models. The models all have a rest energy, a potential energy of zero. Any deformation causes the potential energy to increase, which, like a physical system, can result in kinetic energy which brings the model back to its rest state. The paper is very technical and I had some trouble understanding all of the math. I think that the simulations are a step ahead of kinematic models, and further extension would be to model the loss of heat in a physical system to further increase the accuracy of the modeling technique.
Energy Constraints On Parameterized Models
Andrew Witkin, Kurt Fleischer, Alan Barr
This paper is an approach to representing systems of "connected" models as hierarchies of energy functions. This approach seems to be applicable to any kinds of model with any kind of energy function. Basically a joint between two models may be a strong energy gradient that has a minimum when there is no pressure on the joint. As the model moves, the joint will stay intact because it would take a lot of "energy" to displace it. This seems like a novel approach, buy I wonder what the computational intensity is like for a complex connected system.
The physically based dynamics of felxible objects is incorporated into geometric models. The elasticity theory can represent the shape and motion of deformable materials .
It simulates pghisycal properties, tension, rigidity. In this way static shapes can be modelled in deformable objets. Also, by using other properties such as mass and damping , the dynamics of objects can be simulated. All this is based on the definition
of the partial differential equations that specify teh evolving shape of the objects and its motion in space and their numerical resolution.The general dynamics of deformable models is represened by the inertial forces, the damping forces, and the elastic
forces which are balanced with the external forces.
Deformation is defined using differential geometry of curves, surface and solids. Deformation should be insensitive to rigid body motion, since this does not cause deformation. Potential energy of deformation restores deformed bodies to their natural shap
e. This energy become larger as the model gets more deformed, compared with its natural shape.
In order to measure the amount of deformation it uses the strain energy, which is a norm of the difference between the natural form and the deformed body.
An implementation, is described in the paper and some simulation examples are also shown.
How to get equations for different kind of materials?, their dynamcs can be very different. Is there any way to ge it automatically?
In general I think that this technique can be very useful in simplyfing animation of complex objects, it can also facilitate the representation of non-rigid objects by using the same basic structure of the object. The animations seem to have good realism
.
It seems also that it is specially useful when objects defined using this technique can interact with other physically-based computer graphics objects. But, I think that it can represent some types of materials, but what about other types of materials w
ith a different dynamics, how do we get the right differential equations for those objects?.
An approach to expressing and solving constraints on parameterized model hierarchies is formulated here. It formulates the constraints as energy functions, they are non-negative functions. These functions get a value equal to zero at points where the cons
traints are satisfied.
All the constraints are added (i.e. we sum all the energy functions) in order to get a single scalar function with all the parameters. Then the system move through the parameter space to minimize the global energy.
By using this technique, objects can rotate and translate. Also objects can vary their internal parameters (e.g. length, radius, height, and so on).
According to its authors, by using this tech., we can achieve self assembling models, animated models, generality and additivity. The also talk about an interactive control, something that help the technique not to get stuck, it also helps solving ambigui
ties.
Talking about the implementation, they use a tree that defines the model's geometry by using a collection of mathematical functions. This is useful because this structure helps avoiding the waste of computation that comes from differentiating and solving
energy terms with respect to parameter on which they do not depend.
The solving procedure is in general to formulate energy functions that represent constraints and then move trough parameter space according to an energy gradient. The steepest descent method is mentioned in the paper, more accurate methods can also be use
d.
Some examples of constraints and how they work are showed in the paper. Also a example with 3 objects with similar characteristics, an assembling example, and a working model are shown.
I do not know how easy could it be to specify these constraints. To formulate these energy constraints, it is neccesary to construct a non-negative smooth function such that is 0 at the values of the function for which the constraint(s) are satisfied. If
this is done hierarchically I think the complexity of the problem can be partially reduced, but how good would this ensemble work?. I expected a more automatic way, although this is very helpful for some kind of applications, animations. Another point I w
anted to mention is that the fact that the object deformations are done in a very smooth and stylized way, which may be OK for some environments only.
One of the major challenges that researchers in computer graphics face is
the problem of making objects in a virtual world appear more
"natural" and "true to life". One way of doing this is by
making virtual objects interact with one another in the same manner as
their counterparts in the real-world would, by making them follow the same
laws that nature imposes on real-world objects. This involves studying the
whole gamut of dynamic physical laws and applying them to the virtual
world. Several researchers have attempted to simulate special purpose
physical models following laws of mathematical physics. The authors of
this paper study a particular class of these laws, namely, the dynamic
deformation of non-rigid, elastic solids. The virtual models they study
are subject to forces, such as gravity and friction (due to a viscous
fluid), constraints such as linkages, as well as collisions with rigid
non-deformable immovable solids. The authors provide very impressive
examples, of a solid ball resting on a supporting elastic solid cube, a
flag waving in the wind and a carpet falling off the surfaces of two rigid
stationary bodies in a gravitational field, to illustrate the potential of
their algorithm. This algorithm has the clear advantage of simplifying
animation due to the dynamic nature of the objects created. But this is
achieved at a very high computational cost of having to solve numerically
a set of partial differential equations.
The approach to physics-based animation that is presented in this paper is
more general than the one suggested by Terzopoulos et al., although the
authors concede that they do not intend the model to be physically
plausible, i.e., the virtual objects do not necessarily obey physical
laws such as gravity, etc. The main aim of this work is to create a user
friendly environment that helps in the design of mechanical parts of an
object for an industrial application. The authors address the problem of
imposing constraints on parametrized models and propound a solution in
which an energy (or cost) function is used to find the best parameter
choices for the problem. This is important because as the complexity of a
model increases, the number of parameters involved also increases and
these parameters may interact with one another in non-linear ways, making
the search for good parameter choices very difficult. The authors propose
to formulate constraints on the model as energy functions in the
parameter space of the model and use gradient descent techniques to find
the global minimum of the total energy function in this space. The
formulation of such energy functions is useful because the implementation
of new constraints does not require any knowledge of previous ones and
these can be represented as new additive terms of the energy function. We
can do this because minimizing the total energy function is equivalent to
individually minimizing all its constituent terms. Thus, new constraints
can be dynamically imposed on the system as it evolves.
The authors use a simple gradient descent algorithm to solve for the
minimum value of the energy function. Such an algorithm is very likely to
get stuck in local minima. The authors suggest no solution for this
problem other than user interaction. They claim that "local minima are
easy to interpret geometrically", so that "the user can correct the
situation by manually repositioning a part". This is clearly a very
inelegant way of handling the problem and it might be more useful to
consider other methods such as neural networks that are capable of
providing more robust solutions. Backpropagation, Hopfield networks and
genetic algorithms are techniques that spring to my mind as possible
alternatives that might make user intervention unnecessary.
Energy Constraints on Parameterized Models
Timothy Frangioso
Scott Harrison
Leslie Kuczynski
Shih-Jie Lin
Geoffry Meek
Romer Rosales
Elastically Deformable Models
This work is based on the problem of modeling the behavior of non-rigid curves, surfaces and solids in time. It uses ellasticity theory in order to construct differntial equations that can model such behavior.
Energy Constraints on Parameterized Models
Article Review
In general this work deals with the problem of imposing and solving geometric constraints on parameterized models. When modeling, the main difficulty is to set the parameters that achieve the effect that we know the combinations of objects are supposed to
have
Lavanya Viswanathan
1) D. Terzopoulos, J. Platt, A. Barr and K. Fleischer. Elastically
deformable models. In Computer Graphics Proceedings, ACM SIGGRAPH, volume
21, p 205--214, 1987.
2) A. Witkin, K. Fleischer and A. Barr. Energy constraints on
parameterized models. In Computer Graphics Proceedings, ACM SIGGRAPH,
volume 21, p 225--232, 1987.
Stan Sclaroff
Created: Jan 21, 1997
Last Modified: Feb 19, 1997