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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. ...
. 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 throug ...
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. 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 \Phi_ (measuring light intensity at certain highly concentrated light-point), curve singularity \Phi_ (measuring light intensity at/around a light-curve), and irradiance measure \Phi_ (measuring intensity in a certain poorly concentrated light-area). Putting them altogether, the following function defines the total radiant flux measure \Phi_T at a certain section Ω on the target surface: :\Phi_T(\Omega) = \textstyle \sum_^ \Phi_^i(\Omega) + \sum_^ \Phi_^j(\Omega) + \Phi_I(\Omega)\displaystyle After this step, there are two existing measures of the radiant flux measures of the source \Phi_ (uniform distribution, by initialization) and the target \Phi_ (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: \mu_S (light intensity evaluated by dividing \Phi_ 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 ...
of the union region between \Phi_ and \Phi_), \mu_T (light intensity evaluated by dividing \Phi_ 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 ...
of the union region between \Phi_ and \Phi_) are defined. Secondly, the source mesh is generated as multiple sites s_i \in (\Phi_S \cup \Phi_T), 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 ...
P_w (a set of C_i^w power cells) is defined on this set of sites s_i weighted by a weight vector \omega. Finally, the goal is to decide whether which power cells are going to be move. Considering all x vertices on the surface, finding the minimizer \omega_ 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 the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
will produce the matched power diagram for the target: :f(\omega) = \sum_ (\omega_i.\mu_S(C_i^0) - \int\limits_ (\, x - s_i\, ^2 - \omega_i)d\mu_T(x))


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 d_I, outgoing light ray d_O 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 throug ...
as: :n = d_I + = d_I + where, :\eta: refraction coefficient :x_R: 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: :\underset \, \omega \, \cdot E_, \, E_,\, E_, \, E_, \, E_ /math> where, : E_ = \sum_ \lVert n_o - n\rVert_2^2 is the integration energy that aligns the vertex normals n_o obtained from the Optimal Transport with the target normals n obtained from the Snell's law computation above. :E_ = \sum_ \lVert x - \textbf_(x)\rVert_2^2 as mesh generated in step Solving Optimal Transport cannot adapt to the sharp instances from the discontinuities, this energy is to penalize the vertices to not change significantly from the incoming light ray. : E_ = \sum_ \lVert \Phi_T(t) - \Phi_S(t_s)\rVert_2^2 is the energy measuring the flux over the triangle t in the mesh. : E_ = \lVert \textbf\textbf\rVert_2^2 is the energy that regularizes the shape of the triangles to maintain its well-shapedness. : E_ = \sum_ \lVert max(0, -log((1 - n_R.(x - x_R)) + d_)\rVert^2 is barrier energy to ensure that surface does not deform beyond a certain distance threshold d_.


Differentiable inverse rendering caustic pattern design


Basic principle

Inverse graphics is a method of observing the data from an image and inferring all possible properties including 3D geometry, lighting, materials, and motion in order to generate a realistic image. In conventional computer graphics, to render an image with desired appearance and effects, it is given all the relevant properties/characteristics. This could be described as the forward method. On the contrary, in caustic design, the properties and characteristics of objects (especially the material surface) are not trivial. The given constraint is the target image to obtain. Therefore, the goal is to obtain its properties and characteristics by observing and inferring the target image. This can be considered the inverse/backward method. The following is the basic
loss function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
explaining how to optimize the parameters: :L(c) = \lVert f(c)-I \rVert^2 where, :: loss function, mean square error of the rendered image and the target :: contains elements which can influence the generated image :: target image


Designed pipeline

At first, the target pattern is designed and the forward pass computed to get the synthetic pattern. It's compared to the target pattern and get the loss. The objection is to let the synthetic pattern is similar to the target pattern as much as possible. And then do the back propagation to get the optimized properties need to use in caustic manufacturing.


Elements contributing to generated image

* Appearance (A): per-pixel surface appearance is modeled as product of mipmapped texture and per-pixel brightness. * Geometry (V): assume a 3D scene to be approximated by triangles, parameterized by vertices V. * Camera (C): focal length, the point of view, the center of the camera. There could be more elements, for example
albedo Albedo (; ) is the measure of the diffuse reflection of solar radiation out of the total solar radiation and measured on a scale from 0, corresponding to a black body that absorbs all incident radiation, to 1, corresponding to a body that refle ...
and refraction coefficient.


General differentiable framework

Introduce U as an intermediate variable indicating 2D projected vertex coordinate positions. The gradient of these properties can be derived by chain rule indirectly. : \frac = \frac \times \frac : \frac = \frac \times \frac : \frac = \frac \times \frac After applying the
stochastic gradient descent Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of ...
, the optimal A, V and C could be achieved. Subsequently, these quantities are used to carve or mill the material to generate the target pattern.


Implementation

One common approach is to utilize the ability to perform differential operations in various
deep learning Deep learning (also known as deep structured learning) is part of a broader family of machine learning methods based on artificial neural networks with representation learning. Learning can be supervised, semi-supervised or unsupervised. ...
auto-differentiation frameworks/libraries such as:
Tensorflow TensorFlow is a free and open-source software library for machine learning and artificial intelligence. It can be used across a range of tasks but has a particular focus on training and inference of deep neural networks. "It is machine learnin ...
,
PyTorch PyTorch is a machine learning framework based on the Torch library, used for applications such as computer vision and natural language processing, originally developed by Meta AI and now part of the Linux Foundation umbrella. It is free and open ...
,
Theano In Greek mythology, Theano (; Ancient Greek: Θεανώ) may refer to the following personages: *Theano, wife of Metapontus, king of Icaria. Metapontus demanded that she bear him children, or leave the kingdom. She presented the children of Mel ...
. One more approach is to make use of the OpenDR framework to build a forward graphics model and to automatically obtain derivatives with respect to the model parameters for optimization. As optimization properties are obtained, the target image can be generated. OpenDR provides a local optimization method that can be incorporated into probabilistic programming frameworks. This can be used to solve the problem of caustic.


Manufacturing

Once the caustic pattern has been designed computationally, the processed data will be then sent to the manufacturing stage to get the final product. The most common approach is subtractive manufacturing (
machining Machining is a process in which a material (often metal) is cut to a desired final shape and size by a controlled material-removal process. The processes that have this common theme are collectively called subtractive manufacturing, which utilizes ...
). Various materials can be used depending on the desired quality, the effort it takes to manufacture, and the available manufacturing method. * Common refractive materials:
acrylic Acrylic may refer to: Chemicals and materials * Acrylic acid, the simplest acrylic compound * Acrylate polymer, a group of polymers (plastics) noted for transparency and elasticity * Acrylic resin, a group of related thermoplastic or thermosett ...
,
polycarbonate Polycarbonates (PC) are a group of thermoplastic polymers containing carbonate groups in their chemical structures. Polycarbonates used in engineering are strong, tough materials, and some grades are optically transparent. They are easily work ...
,
polyethylene Polyethylene or polythene (abbreviated PE; IUPAC name polyethene or poly(methylene)) is the most commonly produced plastic. It is a polymer, primarily used for packaging ( plastic bags, plastic films, geomembranes and containers including b ...
,
glass Glass is a non- crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling (quenchin ...
,
diamond Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, b ...
* Common reflective materials:
steel Steel is an alloy made up of iron with added carbon to improve its strength and fracture resistance compared to other forms of iron. Many other elements may be present or added. Stainless steels that are corrosion- and oxidation-resistan ...
,
iron Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in ...
,
aluminum Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. It ha ...
,
gold Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile ...
,
silver Silver is a chemical element with the symbol Ag (from the Latin ', derived from the Proto-Indo-European ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, white, lustrous transition metal, it exhibits the highest electrical ...
,
titanium Titanium is a chemical element with the symbol Ti and atomic number 22. Found in nature only as an oxide, it can be reduced to produce a lustrous transition metal with a silver color, low density, and high strength, resistant to corrosion i ...
,
nickel Nickel is a chemical element with symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and ductile transition metal. Pure nickel is chemically reactive but large pieces are slow t ...
Caustic pattern design has many real-world applications, for example in: * Luminaires * Jewelry * Architecture * Decorative glass production


See also

*
Focus (optics) In geometrical optics, a focus, also called an image point, is a point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blu ...
*
Circle of confusion In optics, a circle of confusion (CoC) is an optical spot caused by a cone of light rays from a lens not coming to a perfect focus when imaging a point source. It is also known as disk of confusion, circle of indistinctness, blur circle, or ...
* Caustic (mathematics)


References

* *


Further reading

* * {{DEFAULTSORT:Caustic (Optics) Geometrical optics