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 A journey between dimensions

Researchers were able to minimize distortion created in geometric bodies due to their flattening or stretching

Representations with geometric distortions
Representations with geometric distortions

Computer graphics is a field of computer science that studies methods for the digital creation and processing of visual content by computer means. Its applications include editing images or bodies in XNUMXD and XNUMXD, planning using a computer, computer animation, virtual reality, computer games and character design for film and television.

Prof. Ofir Weber from the Faculty of Engineering at Bar-Ilan University is a researcher in the field of computer graphics who develops algorithms and software for digital geometry processing and in the process deals with parameterization (mapping of two and three-dimensional bodies). This research may contribute, among other things, to the development of visual applications such as computer games, movies, animation and medical applications. For his developments, Prof. Weber is largely based on mathematical fields such as differential geometry and topology, conformal and harmonic mappings, optimization and linear algebra.

"My research focuses on a better understanding and representation of geometric bodies and the development of new methods for preserving them and adapting them to different applications," he says. "For example, if we want to use a form that represents a person in computer games, a movie or animation, we need to make it dynamic and changing and as close to reality as possible; And if we are interested in painting a three-dimensional body or pasting a picture on it, we need to flatten it first and make it two-dimensional because that makes the painting job very easy."

In the transition from three-dimensional to two-dimensional and in flattening the geometric bodies or stretching them, engineering problems arise. One of the main ones is geometric distortion, which cannot be prevented, but can be minimized or blocked. "In the end, the geometric bodies have physical properties and we are supposed to prevent the engineering problems that may arise in them during the changes, for example by preserving lengths, angles and areas, otherwise they will look unnatural. For example, the volume of the head of a human figure can increase excessively due to geometric distortion and will not appear believable to the viewer who is sensitive to this. And beyond the visual matter, such distortion can fail processes such as physical simulations and production processes", explains Prof. Weber.

In their latest study, which won a research grant from the National Science Foundation, Prof. Weber and his research team sought to solve engineering problems, including the minimization of geometric distortion. To this end, they used mathematical methods such as differential equations (where the missing is a function and describes dependencies between the function and its derivatives), linear algebra, complex numbers and optimization (finding the optimal value of a function under given constraints). According to him, "We came to the conclusion that in order to reduce geometric distortion we need to solve a certain optimization problem. If you minimize it, you minimize the distortion. Because, an optimization problem is energy reduction subject to the constraints that define a domain/shape (for example a deformation that should be as small as possible). If you manage to reduce this energy, the distortion is small. An optimization problem is usually solved by making an approximation, which is an inaccurate representation of the optimization problem. But this way the solution is approximate and sometimes even wrong".

In view of these insights, the researchers were able to solve the optimization problem of mapping geometric bodies represented on the computer (for example, of humans, animals and vehicles) through the use of unique conversion formulas - through which they convert an optimization problem that is difficult to calculate and solve, into an optimization problem that is easier to calculate and solve (that is, they did Computational reduction - an algorithmic method in computer science that allows you to convert a difficult problem into an easy one and thus solve it). "The secret lies in finding the appropriate conversion through which the energy can be reduced and thus obtain a geometric model with a small distortion. We tested this in imaging experiments and indeed we received representations with minimal and imperceptible distortions", Prof. Weber concludes.

Life itself:

Prof. Ofir Weber, 46 years old, lives in Ra'anana with his wife Bat-Chen, his three daughters and two sons.

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