CN104658029B - The rendering intent of the minute surface light based on MCMC - Google Patents

Abstract

The present invention provides a rendering method based on MCMC specular light. When rendering, the pixel value of each pixel point is first determined, and then the pixel value of each pixel point is used for rendering. For any pixel point, each pixel point is determined through the following steps The pixel value of the pixel: the effective path of the current pixel is determined based on the MCMC method, and the effective path is the specular light path; according to the effective path of the current pixel, the pixel value of the current pixel is calculated based on the light propagation path integral formula, and the The current pixel to render. The rendering method of the present invention uses the MCMC method to model the effective specular light path in a high-dimensional space, and the pixel value of each pixel point can be obtained by using the effective specular light path, and then the rendering method for complex specular light is completed .

Description

Rendering method of specular light based on MCMC

technical field

The invention relates to the technical field of computer graphics, in particular to an MCMC-based rendering method for specular rays.

Background technique

Unbiased Monte Carlo rendering of some complex ray propagation paths has been a long-standing problem, especially for some, where there are paths of illumination rays that include specular or smooth surfaces, as this slows down the rate at which light converges. become very slow.

For the past 25 years, simulating the path of light rays has been the focus of researchers in the field of computer graphics; initially as a complementary method to finite element simulations, or radiosity and ray tracing. The use of Monte Carlo methods for ray tracing arose out of Kajiya's work formulating the rendering equations under global illumination, which led to the formation of a field called Monte Carlo Global Illumination. Treating each pixel in the image as a random variable, and making its expected value equal to the solution of the rendering equation, the unbiased sampling method obtained starts from the source path tracing method proposed by Kajiya to the two-way path tracing, in which the light propagation The path can be constructed partly from the light from the light source, and partly from the light seen by the eyes; in addition, the open Metropolis ray propagation algorithm proposed by Veach and Guibas in 1997 utilizes the two-way under the Markov Monte Carlo framework Path tracing method.

Many two-pass methods use a "particle tracing" pass to send the energy from the light source, which is then traced throughout the scene in the form of photons, and stored in a spatial data structure. Another pass uses ray tracing to render the image, and of course uses the stored particle density to estimate the light intensity.

Photon mapping and other two-pass methods are characterized by storing an approximate representation of the partial lighting in the scene, based on the assumption of a smooth distribution of lighting. On the one hand, this makes it difficult to render some propagation models with an unbiased approach, since the exact path the ray takes does not need to be found. Under the assumption of smoothness, the blind ray paths seen by the eye eventually end up in nearby regions. However, this smoothing assumption itself leads to image smoothing errors; that is, in Monte Carlo scenarios, the obtained results are biased. For smooth-to-smooth propagation, not having a sufficiently good diffuse surface to store photons will make it difficult to process photon maps because we have to collect a large number of photons in order to be in the direction, position space Fully sampled. Some photon mapping variants avoid this problem by treating smooth materials as specular surfaces, but this means that as the number of rough surfaces in the input scene grows, the treatment becomes more and more like path tracing.

In addition to the global illumination algorithm, there is still a lot of work on the properties of specular reflection paths. Mitchell and Hanrahan proposed a method for calculating radiance in 1992, which uses Fermat's principle interval Newton method to locate the source point of the total reflection path. In 2009, Walter et al. proposed a method to calculate the monoscatter radiance in a refraction body bounded by a triangular mesh.

Contents of the invention

Aiming at the deficiencies in the prior art, the present invention provides a rendering method based on MCMC (Markov Chain Monte Carlo) specular light, which can effectively process the specular path in the rendering scene and reduce time consumption , and easy to implement.

A method for rendering specular light based on MCMC. When rendering, the pixel value of each pixel point is first determined, and then the pixel value of each pixel point is used for rendering. For any pixel point, the pixel value of each pixel point is determined by the following steps:

(1) The method based on MCMC determines the effective path of current pixel point, and described effective path is specular ray path;

(2) According to the effective path of the current pixel point, the pixel value of the current pixel point is calculated based on the light propagation path integral formula (that is, the derivation formula of the rendering equation), and the current pixel point is rendered.

The MCMC-based rendering method for processing complex specular rays of the present invention uses Markov chain Monte Carlo technology to sample the contributing specular paths in the scene. In the rendering process, the state space is the space formed by all the paths through the scene. The state points in the space are the paths. The probability distribution of the final expected path is the contribution of each path to the rendered image (that is, the contribution to the camera) light intensity) is positively correlated. The final image is the image obtained by projecting the path distribution onto the image plane.

The step (1) determines the effective path of the current pixel point by the following method:

(1-1) For the current pixel point, determine the original path from the viewpoint to the light source of the current pixel point;

(1-2) Select the first non-mirror node from the original path as the initial node, and randomly select a ray outgoing direction as the disturbance direction for the initial node;

(1-3) A ray is emitted from the initial node along the disturbance direction, so that the ray travels along the original path:

(a1) If a non-mirror node is encountered during transmission, the propagation is stopped, and the obtained first propagation path is completed to obtain a suggested path in the disturbance direction;

(a2) If no non-mirror node is encountered during transmission, stop until it reaches the light source, and use the obtained first propagation path as the suggested path in the direction of the disturbance;

(1-4) Adding a disturbance direction for the initial node, performing step (1-3) for the newly added disturbance direction to determine a suggested path in the newly added disturbance direction, and judging the validity of the suggested path;

Steps (1-4) are executed cyclically until the number of perturbation directions reaches the preset number threshold and stop, and the effective path of the current pixel is obtained.

The number threshold is adjusted according to specific application scenarios. The number threshold in the present invention is 50-150, preferably 100.

Completing the first propagation path based on the iterative method in the step (a1) includes the following steps:

(S1) Determine whether ||x _b -x' _b ||>εL is satisfied, where x _b is the second non-mirror node in the original path, x' _b is the second non-mirror node in the propagation path, and ε is the pre- Set the error threshold, L is the maximum distance between two adjacent nodes of the original path;

If it is not satisfied, then the part after the second non-mirror node x _b in the original path is directly added to the last node in the first propagation path to obtain the suggested path;

Otherwise, proceed as follows:

(S2) Determine the outgoing direction of the light source according to the iterative step size, emit a ray along the outgoing direction, and make the light propagate along the original path until it reaches a non-mirror node and stop propagating to obtain a corresponding second propagation path;

(S3) Judging whether ||x _cn -x' _b ||<||x _b -x' _b || is satisfied, where x _cn is the last node in the second propagation path:

(b1) If satisfied, add the second propagation path to the first propagation path to obtain the corresponding suggested path, update x _b to x _cn , and update the iteration step size, and then return to the execution step (S1) for the next iteration ;

(b2) If not satisfied, the second propagation path is added to the first propagation path to obtain a corresponding suggested path, the iteration step size is updated, and then the execution step (S1) is returned for the next iteration;

Loop in this way until the number of iterations reaches the maximum number, and stop, and use the suggested path obtained last time as the suggested path in the direction of the disturbance, and the iteration step size is 1 in the first iteration.

The maximum number of iterations and the error threshold set are directly related to the accuracy and rendering efficiency of the final rendering result. Usually, the maximum number of iterations is 10 to 20. Considering the accuracy and rendering efficiency of the rendering result, as a preference, the The maximum number of iterations is 20. Further preferably, the error threshold ε=10 ^-7 .

In the step (S2), the outgoing direction of the light source is determined according to the following formula:

p=x ₂ -βT(x ₂ )P ₂ A ^-1 B _k T(x _n ) ^T (x' _n -x _n );

Among them, β is the iteration step size;

x ₂ is the second node in the original path;

x _n is the second non-mirror node in the original path;

x' _n is the second non-mirror node in the first propagation path;

A ^-1 is a matrix with a size of 8×8, and B _k is a matrix with a size of 8×1, where the matrices A and B _k satisfy Y=[B ₁ AB _k ], where Y is the constraint Jacobian matrix corresponding to the original path, B ₁ , A, B _k are block representations of the constrained Jacobian matrix Y;

T(x ₂ ) is a matrix composed of a set of orthogonal bases of the tangent plane at the second node x ₂ in the original path, with a size of 3×3;

T(x _n ) is a matrix composed of a set of orthogonal bases of the tangent plane at the second non-mirror node x _n of the original path, with a size of 3×3;

P ₂ is the projection matrix corresponding to the second node x ₂ in the original path, and its size is 4×8.

In the present invention, the iterative process is continuously more similar to the iterative step size β, so as to ensure the convergence speed. As preferably, utilize following formula in described step (b1):

β _k+1 = min{1,2β _k },

Update the iteration step size;

Utilize following formula in described step (b2):

β _k+1 = 1/2β _k ,

Update the iteration step size;

Among them, k is the number of iterations, β _k+1 is the iteration step size used in the k+1 iteration, and β _k is the iteration step size used in the k iteration.

In the step (1-4), a disturbance direction is added to the initial node by using the spherical von Mises-Fisher distribution based on the principle of uniform sampling.

In the step (1-4), for any suggested path, it is judged whether the suggested path is valid according to the transition probability between the suggested path and the corresponding original path by using the property of the steady-state distribution of the Markov chain.

There is no special description in the present invention. When determining the number of nodes or non-mirror nodes for any path, the initial node when the path is formed is the first node.

Compared with prior art, the beneficial effect of the present invention is:

(1) The present invention introduces a new transfer rule method with different properties, and utilizes the Markov chain Monte Carlo method to effectively sample the specular illumination path;

(2) The present invention uses an iterative method to process the path space. This algorithm can perform each step of path tracing according to the local linear model of the path, and then supplemented with corresponding specular environment information, a new path can be obtained;

(3) The transition rules proposed by the present invention can systematically study a large number of mirror paths.

detailed description

The present invention will be described in detail below in conjunction with specific embodiments.

In the MCMC-based rendering method for processing complex specular rays in this embodiment, the pixel value of each pixel point is first determined during rendering, and then the pixel value of each pixel point is used for rendering. For any pixel point, each pixel point is determined through the following steps The pixel value of:

(1) Using the algorithm idea of MCMC, cyclically sample the path related to the current pixel point to determine several effective paths:

(1-1) For the current pixel point, determine the original path from the viewpoint to the light source of the current pixel point;

(1-2) Select a non-mirror node (a node without specular reflection, in this embodiment, select the first non-mirror node along the original path from the viewpoint) from the original path as the initial node X _a , and for The initial node X _a is based on the principle of uniform sampling and uses the spherical Von Mises-Fisher distribution to randomly select a light emitting direction for the initial node as the disturbance direction (along the direction of the light source or along the direction of the camera);

(1-3) A ray is emitted from the initial node X _a along the disturbance direction, so that the ray travels along the original path:

(a1) If a non-mirror node is encountered during transmission, the propagation is stopped, and the obtained first propagation path is completed to obtain a suggested path in the disturbance direction;

In this embodiment, the first propagation path is completed based on an iterative method, which specifically includes the following steps:

(S1) Determine whether ||x _b -x' _b ||>εL is satisfied, where x _b is the second non-mirror node in the original path, x' _b is the second non-mirror node in the propagation path, and ε is the pre- Set the error threshold, L is the maximum distance between two adjacent nodes of the original path;

If it is not satisfied, then the part after the second non-mirror node x _b in the original path is directly added to the last node in the first propagation path to obtain the suggested path;

Otherwise, proceed as follows:

(S2) Determine the outgoing direction of the light source _Xc according to the iterative step size, emit a ray along the outgoing direction, and make the light propagate along the original path until it reaches a non-mirror node and stop propagating to obtain a corresponding second propagation path;

In this embodiment, the outgoing direction of the light source _Xc is determined according to the following method:

First determine the node p according to the following formula, and then use the direction of the line between the light source X _c and the node p (from the light source X _c to the node p) as the outgoing direction of the light source:

p=x ₂ -βT(x ₂ )P ₂ A ^-1 B _k T(x _n ) ^T (x' _n -x _n );

Among them, β is the iteration step size;

x ₂ is the second node in the original path;

x _n is the second non-mirror node in the original path;

x' _n is the second non-mirror node in the first propagation path;

A ^-1 is a matrix with a size of 8×8, and B _k is a matrix with a size of 8×1, where the matrices A and B _k satisfy Y=[B ₁ AB _k ], where Y is the constraint Jacobian matrix corresponding to the original path, B ₁ , A, B _k are block representations of the constrained Jacobian matrix Y;

T(x ₂ ) is a matrix composed of a set of orthogonal bases of the tangent plane at the second node x ₂ in the original path, with a size of 3×3;

T(x _n ) is a matrix composed of a set of orthogonal bases of the tangent plane at the second non-mirror node x _n of the original path, with a size of 3×3;

P ₂ is the projection matrix corresponding to the second node x ₂ in the original path, and its size is 4×8.

(S3) Judging whether ||x _cn -x' _b ||<||x _b -x' _b || is satisfied, where x _cn is the last node in the second propagation path:

(b1) If satisfied, add the second propagation path to the first propagation path to obtain the corresponding suggested path, update x _b to x _cn , and update the iteration step size, and then return to the execution step (S1) for the next iteration ;

(b2) If not satisfied, the second propagation path is added to the first propagation path to obtain a corresponding suggested path, the iteration step size is updated, and then the execution step (S1) is returned for the next iteration;

Loop in this way until the number of iterations reaches the maximum number, and stop, and use the suggested path obtained last time as the suggested path in the direction of the disturbance, and the iteration step size is 1 in the first iteration.

In this embodiment, the maximum number of iterations is 20, and the error threshold ε=10 ^-7 .

In the present embodiment, step (b1) utilizes the following formula:

β _k+1 = min{1,2β _k },

Update the iteration step size;

Step (b2) utilizes the following formula:

β _k+1 = 1/2β _k ,

Update the iteration step size;

Among them, k is the number of iterations, k=1, 2, ..., N, N is the maximum number of iterations, β _k+1 is the iteration step size used in the k+1 iteration, and β _k is the kth iteration The iteration step size to use.

(a2) If no non-mirror node is encountered during transmission, stop until it reaches the light source, and use the obtained first propagation path as the suggested path in the direction of the disturbance;

(1-4) Add a disturbance direction based on the uniform sampling principle to the initial node using the spherical von Mises-Fisher, perform steps (1-3) for the newly added disturbance direction to determine the suggested path in the newly added disturbance direction, and judge the the validity of the proposed route;

Steps (1-4) are executed cyclically until the number of perturbation directions reaches the preset number threshold and stop, and the effective path of the current pixel is obtained.

For any suggested path, according to the transition probability between the suggested path and the corresponding original path, the property of the steady-state distribution of the Markov chain is used to judge whether the suggested path is valid:

(S1) Determine the current suggested path The second non-specular vertex x' _b of the intersection point with the scene, calculate the area density T(X′ _b ) at the intersection point x' _b according to the following formula (T(X′ _b ) also represents the solid angle at the initial node x _a Probability of projection):

Among them, D ^⊥ (w' _o ) is calculated according to the following formula:

D ^⊥ (w' _o )＝D(w' _o )/|cos(n _a ,w _o )|,

D(w' _o ) is a spherical von Mises-Fisher distribution, n _a is the normal at the initial node x _a corresponding to the original node, and w _o is the center of the distribution;

Calculated according to the following formula:

where P _i is the current suggested path The projection matrix of the i-th node intersecting with the scene (i.e. node i), the size of the projection matrix is 4*(k-2), i=1,2,...M, M is the current suggested path The number of nodes intersecting with the scene is related to the two-dimensional plane where node i is located; A and B are the tangent space representation elements of the path space, and the tangent space It can be expressed as:

B ₁ and B _k are block matrix representations of matrix B, B ₁ and B _k are column matrices of (k-2)*1, the size of B is (k-2)*2, and each element corresponds to a The derivative of the vertex.

(S2) Calculate the transition probability Simply take the two equal in terms of quantity, that is to say:

in, for the suggested path, is the original path corresponding to the suggested path;

(S3) Using Transition Probability Calculate the forward acceptance probability of the Markov chain of the current proposed path according to the following formula:

where f _j is the contribution function, with denote the original path and the proposed path, respectively, is the suggested path corresponding to the original path transition probability between

(S4) Determine its validity according to the forward acceptance probability of the Markov chain of the current suggested path. In this embodiment, the last 3/4 of all paths are taken, that is, the total number of paths for any pixel is 40, that is, the acceptance starts from the eleventh path.

(2) According to the effective path of the current pixel point, the pixel value of the pixel point is calculated based on the (rendering equation derivation formula) light propagation path integral formula.

In this embodiment, the selected effective path is used to complete the integration operation of the pixel points with reference to the Monte Carlo rendering method proposed by Veach et al., to obtain corresponding pixel values, and perform rendering.

Based on the ray propagation path integral formula mentioned by Veach et al., the value of each pixel of the image in the rendering problem can be considered as the integral of all relevant contribution functions in a path space;

Set the pixel value of pixel j in the image to be rendered:

in is a path in the path space P, X1...Xn are the nodes in the path respectively; is the product measure derived from the area measure on the patch M of the object in the original scene corresponding to the image to be rendered; the contribution equation is the product of some elements, and the contribution function of each path as follows:

Among them, L _e (x ₁ →x ₂ ) is the emission radiance; Reflect the importance of the jth pixel (please supplement the value range); f _s (x _k-1 →x _k →x _k+1 ) is a geometric structure defined by x _k-1 , x _k and x _k+1 The BSDF (bidirectional scattering distribution function, bidirectional scattering distribution function) function value at x _k in the middle, the value of k is 2～n; is a geometric factor, defined as:

Among them, N(x) is the normal of point x on the patch, is the visibility function between point x and point y (binary function, 0 means invisible, 1 means visible).

In order to further improve the rendering efficiency, if it is determined that the current suggested path is valid in this embodiment, the validity of the current path obtained later is not judged, and only the path is obtained.

The above is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Anyone skilled in the art can easily think of changes or substitutions within the technical scope disclosed in the present invention. All should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the protection scope of the claims.

Claims (8)

1. A rendering method based on MCMC specular rays, characterized in that, when rendering, at first determine the pixel value of each pixel point, then utilize the pixel value of each pixel point to render, for any pixel point, determine each pixel point by the following steps The pixel value of the pixel point: (1) The method based on MCMC determines the effective path of current pixel point, and described effective path is specular ray path; (2) According to the effective path of the current pixel, calculate the pixel value of the current pixel based on the light propagation path integral formula, and render the current pixel; The step (1) determines the effective path of the current pixel point by the following method: (1-1) For the current pixel point, determine the original path from the viewpoint to the light source of the current pixel point; (1-2) Select the first non-mirror node from the original path as the initial node, and randomly select a light emission direction as the disturbance direction for the initial node; (1-3) Shoot a ray from the initial node along the perturbation direction, so that the ray propagates along the original path: (a1) If a non-mirror node is encountered during propagation, the propagation is stopped, and the obtained first propagation path is completed to obtain a suggested path in the disturbance direction; (a2) If no non-specular node is encountered during propagation, stop until it reaches the light source, and use the obtained first propagation path as the suggested path in the direction of the disturbance; (1-4) Adding a disturbance direction for the initial node, performing step (1-3) for the newly added disturbance direction to determine a suggested path in the newly added disturbance direction, and judging the validity of the suggested path; Steps (1-4) are executed cyclically until the number of perturbation directions reaches the preset number threshold and stop, and the effective path of the current pixel is obtained. 2. the rendering method based on the specular light of MCMC as claimed in claim 1, is characterized in that, in described step (a1), based on iterative method, the first propagation path is complemented and comprises the steps: (S1) Determine whether ||x _b -x' _b ||>εL is satisfied, where x _b is the second non-mirror node in the original path, x' _b is the second non-mirror node in the first propagation path, ε is the preset error threshold, and L is the maximum distance between two adjacent nodes of the original path; If it is not satisfied, then the part after the second non-mirror node x _b in the original path is directly added to the last node in the first propagation path to obtain the suggested path; Otherwise, proceed as follows: (S2) Determine the emission direction of the light source according to the iteration step size, emit a ray along the emission direction, and make the ray propagate along the original path until it reaches a non-mirror node and stop propagating to obtain a corresponding second propagation path; (S3) Judging whether ||x _cn -x' _b ||<||x _b -x' _b || is satisfied, where x _cn is the last node in the second propagation path: (b1) If satisfied, add the second propagation path to the first propagation path to obtain the corresponding suggested path, update x _b to x _cn , and update the iteration step size, and then return to the execution step (S1) for the next iteration ; (b2) If not satisfied, the second propagation path is added to the first propagation path to obtain a corresponding suggested path, the iteration step size is updated, and then the execution step (S1) is returned for the next iteration; Loop in this way until the number of iterations reaches the maximum number, and stop, and use the suggested path obtained last time as the suggested path in the direction of the disturbance, and the iteration step size is 1 in the first iteration. 3. The rendering method based on MCMC specular rays as claimed in claim 2, wherein the maximum number of iterations is 20. 4 . The rendering method based on MCMC specular rays according to claim 2 , wherein the preset error threshold ε=10 ⁻⁷ . 5. the rendering method of the specular light based on MCMC as claimed in claim 2, is characterized in that, in the described step (S2), determine the emission direction of light source according to the following formula: P＝x ₂ -βT(x ₂ )P ₂ A ^-1 B _k T(x _b ) ^T (x ' _b -x _b ); Among them, β is the iteration step size; x ₂ is the second node in the original path; A ^-1 is a matrix of size 8×8, B _k is a matrix of size 8×1, where matrix A and B _k satisfy Y=[B ₁ AB _k ], where Y is the constraint Jacobian matrix corresponding to the original path, B _1. A and B _k are block representations of the constrained Jacobian matrix Y; T(x ₂ ) is a matrix composed of a set of orthogonal bases of the tangent plane at the second node x ₂ in the original path, with a size of 3×3; T(x _b ) is a matrix composed of a set of orthogonal bases of the tangent plane at the second non-mirror node x _b of the original path, with a size of 3×3; P ₂ is the projection matrix corresponding to the second node x ₂ in the original path, and its size is 4×8. 6. the rendering method of the specular light based on MCMC as claimed in claim 2, is characterized in that, utilizes following formula in described step (b1): β _k+1 = min{1,2β _k }, Update the iteration step size; Utilize following formula in described step (b2): β _k+1 = 1/2β _k , Update the iteration step size; Among them, k is the number of iterations, β _k+1 is the iteration step size used in the k+1 iteration, and β _k is the iteration step size used in the k iteration. 7. The rendering method based on MCMC-based specular rays according to any one of claims 2 to 6, wherein, in the step (1-4), the spherical Von Mises-Fisher distribution is used for the initial The node adds a perturbation direction. 8. The rendering method based on MCMC-based specular rays as claimed in claim 7, characterized in that, in the step (1-4), for any one suggested path, according to the transition probability of the suggested path and the corresponding original path, the Marl The properties of the steady-state distribution of the Cove chain determine whether the proposed path is valid.