The problem of increasing of shading productivity according to widespread methods Gourand and Phong is examine

Romanyuk, O. N.

Романюк, А. Н.

Романюк, О. Н.

Repository of Vinnytsia National Technical University

УДК 621.833  METHODS AND FACILITIES FOR 3-D OBJECTS SHADING WITH USAGE LOCAL ILLUMINATION MODELS Oleksandr Romanyuk  Vinnitsa National Technical University, Vinnitsa, Ukraine  The problem of increasing of shading productivity according to widespread methods Gourand and Phong is examined  1. Introduction The most complicated and resource-intensive computing at the rendering stage takes place in the shading of 3-D objects, which have voluminosity effect created by color gradation. The color intensity and coordinates are detected for every pixel on the surface in the process of shading. Taking into consideration, that high resolution displays are used for representing realistic images, shading process takes a lot of time. Especially it reveals when illumination models are used, in particular that which represent specular constituent of the color. That’s why the question of the increasing shading production in computer graphic systems is very urgent one.  Gourand and Phong methods are the most widespread for today. This is caused by their relative easiness, satisfactory quality and hardware supply ability. This work proposes new approaches of their realization that allows considerable increase production of them.  2. The methods of increasing of shading productivity This is theoretically proved [1],  that during the shading process according Gourand growth of colour intensities ГIΔ  and  correspondingly along horizontal and vertical rasterization lines are constant values and this eliminates necessity of their computing for every rasterization line. Let’s find out interconnection between growth of colour intensities along horizontal and vertical rasterization lines. It’s proved that growth of intensitiesBIΔHIΔ , VIΔ and DIΔ  in horizontal, vertical and diagonal directions correspondingly are constants and are equal  ACBCBCACACCBBCCAГ YXYXY)II(Y)II(I ΔΔΔΔΔΔΔ ⋅−⋅⋅−−⋅−= ,    ACBCBCACACCABCCBВ YXYXY)II(Y)II(I ΔΔΔΔΔΔΔ ⋅−⋅⋅−−⋅−= , ВГД ІII ΔΔΔ += ,  95where -colour intensities in triangle vertexes (Figure 1).  CBA I,I,IThe received property allows to propose sequence of approaches Gourand paralleling rendering [2].  In the first method first order Serpinsky triangulation is used. It lies in dividing initial triangle to 4 parts by drawing centerlines. With considering the fact that colour intensities growth is permanent value an approach was theoretically founded. It lies in shading only one of rectangular triangles and transformation of rendering results etc. During the shading rasterization of top triangle 1 only is performed and elementary step growth are to be repeated for all the other triangles with considering that triangles 1, 2 and 3 have identical shading directions and triangle 4  is shaded in the oncoming direction (Figure 2). Before rasterization coordinates of all the rectangular triangle vertexes are to be defined.           Figure 2.  Shading directions of rectangular triangles АС BHE G1243Figure 1. Initial triangle  Analogously during detection of colour intensities of internal points belonging to the top left rectangular triangle all the actions of code interpolation are to be duplicated for all the other triangles, but according to their vertexes. Inasmuch initial triangle triangulation is performed by the bisection its edges and growth of initial triangle sides are not always even numbers the task of dividing triangle without non-rasterized points appears. Different divisions’ variants were examined and optimal ones were defined. The second method of Gourand shading production increase is based on the independent forming of even and odd points in the rasterization line. In the preparation cycle colour intensities  ,  are to be found according to the first and next points in the rasterization line. The intensities 1I 2IКI  of the first points in scanlines are to be defined by means of  96code interpolation of colour intensities along basic edges, and  - with help of expression:  2IГК III Δ+=2 . P24 I2II Δ+= ,  P46 I2II Δ+= , ..., P2w2w2 I2II Δ+= − , ...; P13 I2II Δ+= , PIII Δ+= 235 , P57 I2II Δ+= ,..., P1w21w2 I2II Δ+= −+ ,... . One of the possible approach for Gourand rendering improvement lies in the detection of highlights in triangle borders and their further forming [3] . Suppose normal vectors 210 ,, NNNrrr are given in the triangle , ,  edges. In the graphic images forming the most common case [2], is when light source and observer are situated in the infinity, h.i. vector ),( 00 yx ),( 11 yx ),( 22 yxLr has identical direction for all the triangle points. It was proved [6], that in this case maximal values of colour intensity are at the such edge points that satisfies equations  )cos)(cos1(coscoscoscos1 γ+β−ϕβ−ϕγ=t , ,)cos)(cos1(coscoscoscos2 γ+λ−φλ−φγ=t )cos)(cos1(coscoscoscost3 βλϑλϑβ+−−= .  Conformity of angles between normal vectors is represented in the Figure 3.  Figure 3. Normal vectors of triangle ABC  Variables  are parametric ones and has change interval from 0 to 1 Initial triangle can be divided to several depending on the result of threshold valuations comparing with colour intensities in the points .The triangles received are further shaded with Gourand method. Such an approach increases Gourand rendering quality due to highlights forming in case of their intersection by one or several triangle edges. In 32,1 , ttt32,1 , ttt 97the classic Gourand rasterization realization this shading is absent and highlight is eliminated, that is essential artifact.  In the computer graphics systems Blinn illumination model is widespread, according to what intensity of specular colour constituent is with following formula:  γ= nss kII cos , where I  - intensity of external light source;  - coefficient of specular reflection; skγ  - angle between normal vector Nr  and surface normal vector Hr, that is received by the adding of light direction vector Lr and observing vector Vr;  - coefficient of surface specularity.  nThe maximal amount of computing in the process of specular colour constituent detection takes place for BRDF computing ,γncos 1000,1n = . A new BRDF [4]  was proposed that has considerably lower power in comparison with such BRDF like . It’s expressed with formula γncos)k/n(cosk γ⋅ , where k  –  coefficient, that is detected depending on surface specularity coefficient   (n nk << ). The usage of various powers k  for different diapasons of values was proposed for guaranteeing of necessary approximation accuracy. For instance in approximation of  by function nγncos )k/n(cosk γ⋅  with relative error 3% it’s enough to take only to values of k: 1, 3. Modification of well known Schlick BRDF was proposed. It lies in using different values of powers depending on specular coefficient. This is the following formula [5] ( )γ+γ−γ⎥⎦⎥⎢⎣⎢−coscos/cos22log2nnn. A problem of BRDF approximation with function of 2nd and 3rd degree [6, 7] and its hardware realization was examined.  New BRDF approximation, being proposed in this work, represents highlights on the surface of graphic objects with rather high accuracy, at the same time approximating function operates with cosine of angle between vector of normal N and middle vector H. In comparison with Schlick BRDF considerable reduction of approximation error is reached both in highlight epicenter and fading area representation. With the purpose of increasing the shading productivity it is offered to use in one computing process combined and Phong methods. [8] So, for example, it is possible to divide scanline into digital segments of length, divisible by degree of a two. The values of colour intensities according to the Phong method are determined in endpoints of a segment, and in  98intermediate – according to the a Gourand method. The length of the digital segment is able for adaptive changing and is determined by curvature of a surface and direction of vector H. During the computing specular and diffusive colour constituent for points on 3D surface normalization of normal vectors is required. This is explained by the fact, that cosines of angles that are used in shading function can easily be found by the inner product of unit length vectors. Taking into consideration that normalization is performed for observer’s vectors, light source’s vector and vector of normal, the problem of computational complexity decreasing for this resource-intensive procedure is urgent. The method of adaptive normal vectors normalization is proposed [9]. The normalization is performed only in cases of considerable influence upon rsult of colour intensity determination.  On the Figure 4 an example of determination of intermediate normal vector between vectors aNr and bNr, that are situated at the scanline endpoints is provided. It’s shown, that  4dd)2/1()2/1(1ii ≈+. The following correlation is received for error detection : )2/1( id2z21di1i)2/1()2/1(+−=+ . It is proposed at the error value lower than certain threshold valuation to use linear interpolation between vectors of normal. )2/1( id  Figure 4. Determination of intermediate vectors of normal  The normal vectors are received by the bisection of angle between given vectors. So, for example,   99,zNN)cos1(2NNN)21(baba)21(rrrrr +=++= ψ  where )cos1(2z )21( ψ+= . For the further division following formulas are proposed ,z2NNN)2/1()21(a)21(nn1n ++=+rrr)2/1()2/1( n1nz2z +=+ . The problem of spherical angular interpolation for the Phong rendering is examined. Recursive correlations are found, according to which normal vector for current point could be computed with help of normal vectors for two previous points, i.e.  )1t(Ncos)t(N2)1t(N −−=+ rrr ϕ , where φ angle between start and final vector in the scanline. Analogous expression takes place for the determination of diffusive colour constituent  ( ) ( )ddd 1tIcostI2)1t(I −−=+ ϕ  The usage of found recursive correlations allows bringing the computation of diffusive colour constituent and inner product to two multiplication operations: two subtraction operations and two shift operations that allows increase productivity to 40%. The new approach for accelerated diffusive colour constituent determination according to the Phong method is proposed in 3D figures shading. The peculiarity of this approach lies in the synchronous hardware determination of two points at the scanline. For this approach new approximation formula is used for normalization of normal vectors, and new properties of Phong rendering. The method of accelerated definition of color intensities at shading of 3D graphic objects is proposed. With usage of quadratic interpolation an equation of a surface and color intensities for orthonormal triangle are found according to values of intensities at apexes and at those points on the edges, where color intensities take their extreme values. The obtained results are transferred on a real triangle with usage of affine transformations. The adaptive method of shading [10], in which two different illumination models are used depending on availability of highlight within the triangle. The increase of productivity is reached due to using in case of availability of highlight within the triangle the illumination model that considers specular colour constituent, and in all the others – simple model, that considerably decreases computation complicity. The method for detecting fact intersection of edge and highlight or its identification within triangle is proposed.   100Considering, that highlight on the surface of graphic object occupies a tiny area adaptive approach allows essential increase productivity of the shading. Not only problems dealt with adaptive usage of illumination models are highlighted but the methods of shading either. At the same time surface curvature is considered and level of specular and diffusive light.  3. Conclusion The methods being proposed allow considerable increasing of shading productivity, and can be used for designing software and hardware for computer graphics.  4. References [1] Romanyuk A.N. One of the ways of computation process simplification during the shading of 3D objects according to the Gourand method. //Вимірювальна та обчислювальна техніка в технологічних процесах, №2, 2004, стр. 72-75. [2] Romanyuk O.N., Chornij A.V. Effective algorithms of rendering procedure parallelization for realistic image forming // Proceedings of the 6th international conference on Development and Application System. - Suceava. – 2002, - pp. 357 - 362. [3] Romanyuk A.N. The new approach of increasing the quality of 3D objects shading according to the Gourand method // Інформаційні технології та комп'ютерна інженерія, 2005, №3, c. 106-110.  [4] Romanyuk A.N., Chornij A.V. The new approach of specular colour constituent determination // Оптико – електронні інформаційно – енергетичні технології. - 2004.– №2. - С. 85-92. [5] Romanyuk A.N.., Lyashenko Y.L., Melnikov A.M., Savitska L.A. The new formula for computation BRDF //Вестник Херсонського национального технического университета, №25, 2006. –С.412-416. [6] Romanyuk O.N., Lyashenko Y.L. Acceleration of painting with guadratic BRDF interpolation // XIII Міжнародна конференція з автоматичного управління (Автоматика -2006). Тези доповідей тринадцятої міжнародної науково-технічної конференції.-Вінниця, УНІВЕРСУМ-Вінниця, 2006. –С. 364. [7] Romanyuk O.N., Lyashenko Y.L. BRDF approximation with 3rd degree polynomial // Збірник матеріалів п’ятої міжнародної конференції конференції «Інтернет-Освіта-Наука-2006. Том 2.- Вінниця: УНІВЕСУМ-Вінниця, 2006.-С.506-509.  101[8]. Romanyuk O.N. The combined usage of binary and code linear interpolation for normalization of normal vectors during the 3D objects shading // Вестник Херсонського национального технического университета, №25, 2006. . –С.408-411. [9] Romanyuk O.N., Melnikov A.M Adaptive normalization of normal vectors during determination of diffusive and specular colour constituent // Реєстрація, зберігання і обробка даних, 2006, Т. 8, № 3, -С.11-19. [10] Romanyuk O.N., Chornij A.V. “Methods for Specular Color Component Accelerate Calculation”. IEEE Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications. September 2005,Sofia, Bulgaria, pp. 615-619.   Получено 20.05.07  102

Methods and facilities for 3-D objects shading with usage local illumination models

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