Path tracing is a computer graphics Monte Carlo method of rendering images of three-dimensional scenes such that the global illumination is faithful to reality. Fundamentally, the algorithm is integrating over all the illuminance arriving to a single point on the surface of an object; this illuminance is reduced by a surface reflectance function to determine how much of it will go towards the viewpoint camera. This integration procedure is repeated for every pixel in the output image; when combined with physically accurate models of surfaces, accurate models of real light sources, optically correct cameras, path tracing can produce still images that are indistinguishable from photographs. Path tracing simulates many effects that have to be added to other methods, such as soft shadows, depth of field, motion blur, ambient occlusion, indirect lighting. Implementation of a renderer including these effects is correspondingly simpler. An extended version of the algorithm is realized by volumetric path tracing, which considers the light scattering of a scene.
Due to its accuracy and unbiased nature, path tracing is used to generate reference images when testing the quality of other rendering algorithms. In order to get high quality images from path tracing, a large number of rays must be traced to avoid visible noisy artifacts; the rendering equation and its use in computer graphics was presented by James Kajiya in 1986. Path Tracing was introduced as an algorithm to find a numerical solution to the integral of the rendering equation. A decade Lafortune suggested many refinements, including bidirectional path tracing. Metropolis light transport, a method of perturbing found paths in order to increase performance for difficult scenes, was introduced in 1997 by Eric Veach and Leonidas J. Guibas. More CPUs and GPUs have become powerful enough to render images more causing more widespread interest in path tracing algorithms. Tim Purcell first presented a global illumination algorithm running on a GPU in 2002. In February 2009 Austin Robison of Nvidia demonstrated the first commercial implementation of a path tracer running on a GPU, other implementations have followed, such as that of Vladimir Koylazov in August 2009.
This was aided by the maturing of GPGPU programming toolkits such as CUDA and OpenCL and GPU ray tracing SDKs such as OptiX. Path tracing has played an important role in the film industry. Earlier films had relied in scanline renderers to produce animation. In 1998, Blue Sky Studios rendered the Academy Award-winning short film Bunny with their proprietary CGI Studio path tracing renderer, featuring soft shadows and indirect illumination effects. Sony Pictures Imageworks' Monster House was, in 2006, the first animated feature film to be rendered in a path tracer, using the commercial Arnold renderer. Walt Disney Animation Studios has been using its own optimized path tracer known as Hyperion since the production of Big Hero 6 in 2013. Pixar Animation Studios has adopted path tracing for its commercial RenderMan renderer. Kajiya's rendering equation adheres to three particular principles of optics. In the real world and surfaces are visible due to the fact that they are reflecting light; this reflected light illuminates other objects in turn.
From that simple observation, two principles follow. I. For a given indoor scene, every object in the room must contribute illumination to every other object. II. Second, there is no distinction to be made between illumination emitted from a light source and illumination reflected from a surface. Invented in 1984, a rather different method called. However, radiosity relates the total illuminance falling on a surface with a uniform luminance that leaves the surface; this forced all surfaces to be Lambertian, or "perfectly diffuse". While radiosity received a lot of attention at its invocation diffuse surfaces do not exist in the real world; the realization that scattering from a surface depends on both incoming and outgoing directions is the key principle behind the Bidirectional reflectance distribution function. This direction dependence was a focus of research resulting in the publication of important ideas throughout the 1990s, since accounting for direction always exacted a price of steep increases in calculation times on desktop computers.
Principle III follows. III; the illumination coming from surfaces must scatter in a particular direction, some function of the incoming direction of the arriving illumination, the outgoing direction being sampled. Kajiya's equation is a complete summary of these three principles, path tracing, which approximates a solution to the equation, remains faithful to them in its implementation. There are other principles of optics which are not the focus of Kajiya's equation, therefore are difficult or incorrectly simulated by the algorithm. Path Tracing is confounded by optical phenomena not contained in the three principles. For example, sharp caustics. Subsurface scattering. Chromatic aberration, iridescence; the following pseudocode is a procedure for performing naive path tracing. This function calculates a single sample of a pixel. All these samples must be averaged to obtain th
Finite element method
The finite element method, is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, electromagnetic potential; the analytical solution of these problems require the solution to boundary value problems for partial differential equations. The finite element method formulation of the problem results in a system of algebraic equations; the method approximates the unknown function over the domain. To solve the problem, it subdivides a large system into smaller, simpler parts that are called finite elements; the simple equations that model these finite elements are assembled into a larger system of equations that models the entire problem. FEM uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function. Studying or analyzing a phenomenon with FEM is referred to as finite element analysis; the subdivision of a whole domain into simpler parts has several advantages: Accurate representation of complex geometry Inclusion of dissimilar material properties Easy representation of the total solution Capture of local effects.
A typical work out of the method involves dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem, followed by systematically recombining all sets of element equations into a global system of equations for the final calculation. The global system of equations has known solution techniques, can be calculated from the initial values of the original problem to obtain a numerical answer. In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are partial differential equations. To explain the approximation in this process, FEM is introduced as a special case of Galerkin method; the process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE.
The residual is the error caused by the trial functions, the weight functions are polynomial approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with a set of algebraic equations for steady state problems, a set of ordinary differential equations for transient problems; these equation sets are the element equations. They are linear if the underlying PDE is linear, vice versa. Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler's method or the Runge-Kutta method. In step above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains' local nodes to the domain's global nodes; this spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system.
The process is carried out by FEM software using coordinate data generated from the subdomains. FEM is best understood from its practical application, known as finite element analysis. FEA as applied in engineering is a computational tool for performing engineering analysis, it includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm. In applying FEA, the complex problem is a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system. FEA is a good choice for analyzing problems over complicated domains, when the domain changes, when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations.
For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear. Another example would be in numerical weather prediction, where it is more important to have accurate predictions over developing nonlinear phenomena rather than calm areas. While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering, its development can be traced back to the work by R. Courant in the early 1940s. Another pioneer was Ioannis Argyris. In the USSR, the introduction of the practical application of the method is connected with name of Leonard Oganesyan. In China, in the 1950s and early 1960s, based on the computations of dam constructions, K. Feng proposed a systematic numerical method for solving partial differential equations; the method was called the finite difference method based on variation principle, another independent invention of the finite element met
Radiosity (computer graphics)
In 3D computer graphics, radiosity is an application of the finite element method to solving the rendering equation for scenes with surfaces that reflect light diffusely. Unlike rendering methods that use Monte Carlo algorithms, which handle all types of light paths, typical radiosity only account for paths which leave a light source and are reflected diffusely some number of times before hitting the eye. Radiosity is a global illumination algorithm in the sense that the illumination arriving on a surface comes not just directly from the light sources, but from other surfaces reflecting light. Radiosity is viewpoint independent, which increases the calculations involved, but makes them useful for all viewpoints. Radiosity methods were first developed in about 1950 in the engineering field of heat transfer, they were refined for the problem of rendering computer graphics in 1984 by researchers at Cornell University and Hiroshima University. Notable commercial radiosity engines are Enlighten by Geomerics.
The inclusion of radiosity calculations in the rendering process lends an added element of realism to the finished scene, because of the way it mimics real-world phenomena. Consider a simple room scene; the image on the left was rendered with a typical direct illumination renderer. There are three types of lighting in this scene which have been chosen and placed by the artist in an attempt to create realistic lighting: spot lighting with shadows, ambient lighting, omnidirectional lighting without shadows; the image on the right was rendered using a radiosity algorithm. There is only one source of light: an image of the sky placed outside the window; the difference is marked. The room glows with light. Soft shadows are visible on the floor, subtle lighting effects are noticeable around the room. Furthermore, the red color from the carpet has bled onto the grey walls, giving them a warm appearance. None of these effects were chosen or designed by the artist; the surfaces of the scene to be rendered are each divided up into one or more smaller surfaces.
A view factor is computed for each pair of patches. Patches that are far away from each other, or oriented at oblique angles relative to one another, will have smaller view factors. If other patches are in the way, the view factor will be reduced or zero, depending on whether the occlusion is partial or total; the view factors are used as coefficients in a linear system of rendering equations. Solving this system yields the radiosity, or brightness, of each patch, taking into account diffuse interreflections and soft shadows. Progressive radiosity solves the system iteratively with intermediate radiosity values for the patch, corresponding to bounce levels; that is, after each iteration, we know how the scene looks after one light bounce, after two passes, two bounces, so forth. This is useful for getting an interactive preview of the scene; the user can stop the iterations once the image looks good enough, rather than wait for the computation to numerically converge. Another common method for solving the radiosity equation is "shooting radiosity," which iteratively solves the radiosity equation by "shooting" light from the patch with the most energy at each step.
After the first pass, only those patches which are in direct line of sight of a light-emitting patch will be illuminated. After the second pass, more patches will become illuminated as the light begins to bounce around the scene; the scene continues to grow brighter and reaches a steady state. The basic radiosity method has its basis in the theory of thermal radiation, since radiosity relies on computing the amount of light energy transferred among surfaces. In order to simplify computations, the method assumes that all scattering is diffuse. Surfaces are discretized into quadrilateral or triangular elements over which a piecewise polynomial function is defined. After this breakdown, the amount of light energy transfer can be computed by using the known reflectivity of the reflecting patch, combined with the view factor of the two patches; this dimensionless quantity is computed from the geometric orientation of two patches, can be thought of as the fraction of the total possible emitting area of the first patch, covered by the second.
More radiosity B is the energy per unit area leaving the patch surface per discrete time interval and is the combination of emitted and reflected energy: B d A = E d A + ρ d A ∫ S B 1 π r 2 cos θ x cos θ x ′ ⋅ V i s d A ′ where: Bi dAi is the total energy leaving a small area dAi around a
In computer graphics, photon mapping is a two-pass global illumination algorithm developed by Henrik Wann Jensen that solves the rendering equation. Rays from the light source and rays from the camera are traced independently until some termination criterion is met they are connected in a second step to produce a radiance value, it is used to realistically simulate the interaction of light with different objects. It is capable of simulating the refraction of light through a transparent substance such as glass or water, diffuse interreflection between illuminated objects, the subsurface scattering of light in translucent materials, some of the effects caused by particulate matter such as smoke or water vapor, it can be extended to more accurate simulations of light such as spectral rendering. Unlike path tracing, bidirectional path tracing, volumetric path tracing and Metropolis light transport, photon mapping is a "biased" rendering algorithm, which means that averaging many renders using this method does not converge to a correct solution to the rendering equation.
However, since it is a consistent method, any desired accuracy can be achieved by increasing the number of photons. Light refracted or reflected causes patterns called caustics visible as concentrated patches of light on nearby surfaces. For example, as light rays pass through a wine glass sitting on a table, they are refracted and patterns of light are visible on the table. Photon mapping can trace the paths of individual photons to model where these concentrated patches of light will appear. Diffuse interreflection is apparent. Photon mapping is adept at handling this effect because the algorithm reflects photons from one surface to another based on that surface's bidirectional reflectance distribution function, thus light from one object striking another is a natural result of the method. Diffuse interreflection was first modeled using radiosity solutions. Photon mapping differs though in that it separates the light transport from the nature of the geometry in the scene. Color bleed is an example of diffuse interreflection.
Subsurface scattering is the effect evident when light enters a material and is scattered before being absorbed or reflected in a different direction. Subsurface scattering can be modeled using photon mapping; this was the original way. With photon mapping, light packets called photons are sent out into the scene from the light sources. Whenever a photon intersects with a surface, the intersection point and incoming direction are stored in a cache called the photon map. Two photon maps are created for a scene: one for caustics and a global one for other light. After intersecting the surface, a probability for either reflecting, absorbing, or transmitting/refracting is given by the material. A Monte Carlo method called. If the photon is absorbed, no new direction is given, tracing for that photon ends. If the photon reflects, the surface's bidirectional reflectance distribution function is used to determine the ratio of reflected radiance. If the photon is transmitting, a function for its direction is given depending upon the nature of the transmission.
Once the photon map is constructed, it is arranged in a manner, optimal for the k-nearest neighbor algorithm, as photon look-up time depends on the spatial distribution of the photons. Jensen advocates the usage of kd-trees; the photon map is stored on disk or in memory for usage. In this step of the algorithm, the photon map created in the first pass is used to estimate the radiance of every pixel of the output image. For each pixel, the scene is ray traced. At this point, the rendering equation is used to calculate the surface radiance leaving the point of intersection in the direction of the ray that struck it. To facilitate efficiency, the equation is decomposed into four separate factors: direct illumination, specular reflection and soft indirect illumination. For an accurate estimate of direct illumination, a ray is traced from the point of intersection to each light source; as long as a ray does not intersect another object, the light source is used to calculate the direct illumination.
For an approximate estimate of indirect illumination, the photon map is used to calculate the radiance contribution. Specular reflection can be, in most cases tracing procedures; the contribution to the surface radiance from caustics is calculated using the caustics photon map directly. The number of photons in this map must be sufficiently large, as the map is the only source for caustics information in the scene. For soft indirect illumination, radiance is calculated using the photon map directly; this contribution, does not need to be as accurate as the caustics contribution and thus uses the global photon map. In order to calculate surface radiance at an intersection point, one of the cached photon maps is used; the steps are: Gather the N nearest photons using the nearest neighbor search function on the photon map. Let S be the sphere that contains these N photons. For each photon, divide the amount of flux that the photon represents by the area of S and multiply by the BRDF applied to that photon.
The sum of those results for each photon represents total surface radiance returned by the surface intersecti
In optics, a caustic or caustic network is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an envelope of rays as a curve of concentrated light. Therefore, in the adjacent image, the caustics can be the patches of their bright edges; these shapes have cusp singularities. Concentration of light sunlight, can burn; the word caustic, in fact, comes from the Greek καυστός, via the Latin causticus, burning. A common situation where caustics are visible is; the glass casts a shadow, but produces a curved region of bright light. In ideal circumstances, a nephroid-shaped patch of light can be produced. Rippling caustics are formed when light shines through waves on a body of water. Another familiar caustic is the rainbow. Scattering of light by raindrops causes different wavelengths of light to be refracted into arcs of differing radius, producing the bow.
In computer graphics, most modern rendering systems support caustics. Some of them 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 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 found here; the focus of most computer graphics systems is aesthetics rather than physical accuracy. This is true when it comes to real-time graphics in computer games where generic pre-calculated textures are used instead of physically correct calculations.
Caustic engineering describes the process of solving the inverse problem to computer graphics. Given a specific shape or image one wants to find a surface such that the light refracted 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. The position and orientation of each of the micro-surfaces is obtained using a combination of Poisson-integration and so-called simulated annealing. For the continuous problem there have been many different approaches to solving it. One approach uses an idea from transportation theory called ‘optimal transport’ to find a mapping between incoming light rays and target surface. After obtaining such a mapping, the surface is optimized by adapting it iteratively using Snell’s law of refraction, but there are several other approaches using different methods. Focus Circle of confusion Caustic Born, Max.
Principles of Optics: Electromagnetic Theory of Propagation and Diffraction of Light. Cambridge University Press. ISBN 0-521-64222-1. Nye, John. Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations. CRC Press. ISBN 978-0-7503-0610-2. Ferraro, Pietro. "What a caustic!". The Physics Teacher. 34: 572. Bibcode:1996PhTea..34..572F. Doi:10.1119/1.2344572. Dachsbacher, Carsten. "Real-time volume caustics with adaptive beam tracing". Symposium on Interactive 3D Graphics and Games. ACM: 47–54
Ray tracing (graphics)
In computer graphics, ray tracing is a rendering technique for generating an image by tracing the path of light as pixels in an image plane and simulating the effects of its encounters with virtual objects. The technique is capable of producing a high degree of visual realism higher than that of typical scanline rendering methods, but at a greater computational cost; this makes ray tracing best suited for applications where taking a long time to render a frame can be tolerated, such as in still images and film and television visual effects, more poorly suited for real-time applications such as video games where speed is critical. Ray tracing is capable of simulating a wide variety of optical effects, such as reflection and refraction and dispersion phenomena. Optical ray tracing describes a method for producing visual images constructed in 3D computer graphics environments, with more photorealism than either ray casting or scanline rendering techniques, it works by tracing a path from an imaginary eye through each pixel in a virtual screen, calculating the color of the object visible through it.
Scenes in ray tracing are described mathematically by a visual artist. Scenes may incorporate data from images and models captured by means such as digital photography; each ray must be tested for intersection with some subset of all the objects in the scene. Once the nearest object has been identified, the algorithm will estimate the incoming light at the point of intersection, examine the material properties of the object, combine this information to calculate the final color of the pixel. Certain illumination algorithms and reflective or translucent materials may require more rays to be re-cast into the scene, it may at first seem counterintuitive or "backward" to send rays away from the camera, rather than into it, but doing so is many orders of magnitude more efficient. Since the overwhelming majority of light rays from a given light source do not make it directly into the viewer's eye, a "forward" simulation could waste a tremendous amount of computation on light paths that are never recorded.
Therefore, the shortcut taken in raytracing is to presuppose that a given ray intersects the view frame. After either a maximum number of reflections or a ray traveling a certain distance without intersection, the ray ceases to travel and the pixel's value is updated. On input we have: E ∈ R 3 eye position T ∈ R 3 target position θ ∈ [ 0, π ) field of view - for human we can assume ≈ π / 2 rad = 90 ∘ m, k ∈ N numbers of square pixels on viewport vertical and horizontal direction i, j ∈ N, 1 ≤ i ≤ k ∧ 1 ≤ j ≤ m numbers of actual pixel w → ∈ R 3 vertical vector which indicates where is up and down w → = - roll component which determine viewport rotation around point C The idea is to find the position of each viewport pixel center P i j which allows us to find the line going from eye E through that pixel and get the ray described by point E and vector R → i j = P i j − E. First we need to find the coordinates of the bottom left viewport pixel P 1 m and find the next pixel by making a shift along directions parallel to viewport multiplied by the size of the pixel.
Below we introduce formulas which include distance d between the eye and the viewport. However, this value will be reduced during ray normalization r → i j. Pre-calculations: let's find and normalise vector t → and vectors b →, v → which are parallel to the viewport t → = T