In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
, a caustic or caustic network is the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
of
light rays which have been
reflected or
refracted
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
by a curved surface or object, or the
projection of that envelope of rays on another surface.
The caustic is a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
or
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
to which each of the light rays is
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
, defining a boundary of an envelope of rays as a curve of concentrated light.
Therefore, in the photo to the right, caustics can be seen as patches of light or their bright edges. These shapes often have
cusp singularities.
Explanation
Concentration of light, especially
sunlight
Sunlight is a portion of the electromagnetic radiation given off by the Sun, in particular infrared, visible, and ultraviolet light. On Earth, sunlight is scattered and filtered through Earth's atmosphere, and is obvious as daylight when ...
, can burn. The word ''caustic'', in fact, comes from the Greek καυστός, burnt, via the Latin ''causticus'', burning.
A common situation where caustics are visible is when light shines on a drinking glass. The glass casts a shadow, but also produces a curved region of bright light. In ideal circumstances (including perfectly parallel rays, as if from a point source at infinity), a
nephroid-shaped patch of light can be produced. Rippling caustics are commonly formed when light shines through waves on a body of water.
Another familiar caustic is the
rainbow
A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It takes the form of a multicoloured circular arc. Rainbows c ...
. Scattering of light by raindrops causes different
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
s of light to be refracted into arcs of differing radius, producing the bow.
Computer graphics
In computer graphics, most modern
rendering systems support caustics. Some of them even support volumetric caustics. This is accomplished by
raytracing the possible paths of a light beam, accounting for the refraction and reflection.
Photon mapping is one implementation of this. Volumetric caustics can also be achieved by
volumetric path tracing. Some computer graphic systems work by "forward ray tracing" wherein photons are modeled as coming from a light source and bouncing around the environment according to rules. Caustics are formed in the regions where sufficient photons strike a surface causing it to be brighter than the average area in the scene. “Backward ray tracing” works in the reverse manner beginning at the surface and determining if there is a direct path to the light source. Some examples of 3D ray-traced caustics can be foun
here
The focus of most computer graphics systems is aesthetics rather than
physical accuracy. This is especially true when it comes to real-time graphics in computer games where generic pre-calculated
textures are mostly used instead of physically correct calculations.
Caustic engineering
Caustic engineering describes the process of solving the
inverse problem
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
to
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
. That is, given a specific image, to determine a surface whose refracted or reflected light forms this image.
In the discrete version of this problem, the surface is divided into several micro-surfaces which are assumed smooth, i.e. the light reflected/refracted by each micro-surface forms a Gaussian caustic. Gaussian caustic means that each micro-surface obey
gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. The position and orientation of each of the micro-surfaces is then obtained using a combination of
Poisson integration and
simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It ...
.
There have been many different approaches to address the continuous problem. One approach uses an idea from
transportation theory called ''
optimal transport
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.G. Monge. '' ...
'' to find a mapping between incoming light rays and the target surface. After obtaining such a mapping, the surface is optimized by adapting it iteratively using
Snell's law
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
of refraction.
Optimal-transport-based caustic pattern design
Basic principle
Controlling caustic pattern is rather a challenging problem as very minor changes of the surface will significantly affect the quality of the pattern since light ray directions might be interfered by other light rays as they intersect with and refract through the material. This will lead to a scattered, discontinuous pattern. To tackle this problem, optimal-transport-based is one of the existing proposed methods to control caustic pattern by redirecting light's directions as it propagates through the surface of a certain
transparent material
In the field of optics, transparency (also called pellucidity or diaphaneity) is the physical property of allowing light to pass through the material without appreciable light scattering by particles, scattering of light. On a macroscopic scale ...
. This is done by solving an inverse optimization problem based on
optimal transport
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.G. Monge. '' ...
.
Given a reference image of an object/pattern, the target is to formulate the mathematical description of the material surface through which light refracts and converges to the similar pattern of the reference image. This is done by rearranging/recomputing the initial light intensity until the minimum of the optimization problem is reached.
Design pipeline
Here considering only refractive caustic, the objective can be determined as follows (similar principle for reflective caustic with different output):
''
Input:'' image of pattern to be obtained after propagating lights through the material, given the light source position.
''Output:'' caustic geometry on the receiver (flat solid surface, e.g.: floor, wall, etc...)
In order to achieve the target pattern, the surface where light refracts through and exits to the outer environment must be manufactured into certain shape to achieve desired pattern on the other side of the material.
As mentioned, given an input image, this process will produce the similar caustic pattern as the output. In principle, there are two core stages with each includes two sub-stages:
*
Solving Optimal Transport Problem
*# Compute Target Light Distribution
*#Compute Mapping from Initial Distribution to Target Distribution
* Optimizing Target Surface
*# Compute Normal Representation of Surface
*# Surface Refinement
Solving optimal transport problem
As the case refraction occurs through a transparent surface, for instance the patterns appearing under clear water surface, 3 main phenomena can be observed:
* Very bright (condensed light intensity) points (so-called
singularity)
* Curve-like objects that connect the points
* Regions with low light intensity
To perform computation, the following 3 quantities are being respectively introduced to describe the geometric characteristics of the pattern: point singularity
(measuring light intensity at certain highly concentrated light-point), curve singularity
(measuring light intensity at/around a light-curve), and
irradiance measure (measuring intensity in a certain poorly concentrated light-area). Putting them altogether, the following function defines the total
radiant flux measure at a certain section Ω on the target surface:
:
After this step, there are two existing measures of the radiant flux measures of the source
(uniform distribution, by initialization) and the target
(computed in previous step). What remains to compute is the mapping from the source to target. In order to do this, there are several quantities to be defined. Firstly, two light intensities evaluated by probabilities:
(light intensity evaluated by dividing
by the
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
of the union region between
and
),
(light intensity evaluated by dividing
by the
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
of the union region between
and
) are defined. Secondly, the source mesh is generated as multiple sites
, which is later being deformed. Next, a
power diagram
In computational geometry, a power diagram, also called a Laguerre–Voronoi diagram, Dirichlet cell complex, radical Voronoi tesselation or a sectional Dirichlet tesselation, is a partition of the Euclidean plane into polygonal cells defined from ...
(a set of
power cells) is defined on this set of sites
weighted by a weight vector
. Finally, the goal is to decide whether which power cells are going to be move. Considering all
vertices on the surface, finding the minimizer
of the following
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
will produce the matched power diagram for the target:
:
Optimizing target surface
After solving optimal transport problem, the vertices are achieved. However, this gives no information about what the final surface should look like. To achieve the desired target surface given the incoming light ray
, outgoing light ray
and power diagram from the step above, the surface normals representation can be computed according to
Snell's law
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
as:
:
where,
:
:
refraction coefficient
:
: target position obtained from solving above optimal transport problem
As the normal representation is obtained, surface refinement is then achieved by minimizing the following
compound energy function:
: