Fast Phong Shading

 Fast Phong Shading. p 103-105
 Gary Bishop
 Room 4E-626
 David M Weimer
 Room 4F-637
 AT&T Bell Laboratories
 Crawfords Corner Road
 Holmdel, NJ 07733

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 SIGGRAPH @1986 - ACM 0-89791-196-2/86/008/0103


 Computer image generation systems often represent curved surfaces as a mesh
 of polygons that are shaded to restore a smooth appearance. Phong shading
 is a well known algorithm for producing a realistic shading but it has not
 been used by real-time systems because of the 3 additions, 1 division and 1
 square root required per pixel for its evaluation. We describe a new
 formulation for Phong shading that reduces the amount of computation per
 pixel to only 2 additions for simple Lambertian reflection and 5 additions
 and 1 memory reference for Phong's complete reflection model. We also show
 how to extend our method to compute the specular component with the eye at
 a finite distance from the scene rather than at infinity as is usually
 assumed. The method can be implemented in hardware for real-time
 applications or in software to speed image generation for almost any


 Most computer image generation (CIG) systems represent curved surfaces as a
 mesh of planar polygons because polygons can be transformed and rendered
 quickly with well known algorithms. Since the polygonal representation is
 an artifact of the image generation method and is usually of no interest to
 viewers (see figure 1), these systems attempt to restore a smooth
 appearance to surfaces by varying the intensity across polygons. The
 efficiency of this shading operation is critical to the performance of a
 CIG system because it must be performed for one million or more pixels per
 image. Although algorithms for realistic shading of polygons are well
 known, real-time CIG systems have not used them because of the large amount
 of computation they require per pixel. This paper describes a new
 formulation of Phong shading that drastically reduces the amount of
 computation compared to previously known formulations. While the new
 formulation is well suited to implementation with special hardware for
 real-time display, it is also appropriate for implementation with software
 for a variety of display systems. Readers who are unfamiliar with surface
 representation or shading are referred to one of the standard graphics
 texts for more information, [Newman and Sproul, 1979; Foley and Van Dam,


 Assume, for simplicity, that the input to the shader consists of triangles,
 in screen coordinates, with normals, N, to the original curved surface
 specified at each vertex. The formulations can be extended to polygons with
 four or more sides.

 The shading algorithms must determine the intensity of each point within a
 triangle from the surface normals, and a vector to the light source, L. The
 light source is assumed to be at infinity so that L is independent of the
 point on which the light shines. We will start with diffuse reflection,

                    L * N
     I        = -------------,
      diffuse     |L| * |N|

 and later show how to extend the results to include Phong's highlight model
 and an extended highlight model.

 GOURAUD SHADING: the most commonly used shading method in real-time
 systems. Gouraud shading [Gouraud, 1971], computes the intensity at each
 point by linear interpolation of the intensities at the vertices. The
 intensities are determined using the reflection equation above with normals
 given at the vertices. The method is popular for real-time systems because
 it produces images of acceptable quality with only 1 addition per pixel,
 but the shading has several disturbing characteristics. (1) Moving objects
 tend to "flatten out" at certain viewing angles, (2) surfaces appear dull
 or chalky, and (3) images show pronounced Mach bands, exaggerations of
 intensity change at discontinuities. Figure 2 is a Gouraud shaded chess

 PHONG SHADING: Phong shading, [Phong, 1973], eliminates "flattening out"
 and dull surfaces and reduces Mach bands but has not, to my knowledge, been
 used in any real-time system because of the large amount of computation
 required for its usual formulation. Phong's method determines the intensity
 of each point using an approximate surface normal that is linearly
 interpolated from the true surface normals specified at the vertices,

   N(x,y) = Ax + By + C

 where A, B and C are chosen to interpolate the normal across the polygon.
 Interpolation for successive values of x and y and evaluating the
 illumination model requires 7 additions, 6 multiplications, 1 division and
 1 square root per pixel. Phong proposed a complicated circuit for
 evaluating this function but it has not, to my knowledge, been implemented.
 Figure 3 is a Phong shaded chess piece.

 Tom Duff has shown, in [Duff, T.,1979, "Smoothly Shaded Renderings of
 Polyhedral Objects on Raster Displays", ACM Computer Graphics, vol 13, no
 2, pp 270-275]], that Phong shading can be implemented more efficiently by
 combining the interpolation and reflection equations.

                         L        Ax + By + C
        I       (x,y) = --- * -------------------
         diffuse        |L|    |L| |Ax + By + C|

 Which can be rewritten as

                           L*Ax + L*By + L*C
        I       (x,y) = ------------------------
         diffuse        |L| |L*Ax + L*By + L*C|

 Performing the indicated dot products and expanding the vector magnitude

                                     ax + by + c
        I       (x,y) = --------------------------------------
         diffuse        SQRT( dx^2 + exy + fy^2 + gx + hy + i)


              L * A
          a = ----- ,

              L * B
          b = ----- ,

              L * C
          c = ----- ,

          d = A * A,

          e = 2A * B,

          f = B * B,

          g = 2A * C,

          h = 2B * C, and

          i = C * C.

 Using forward differences, this form can be evaluated for successive values
 of x and y with 3 additions, 1 division and 1 square root per pixel. This
 is a substantial savings over Phong's formulation but the division and
 square root are still too expensive for real-time use.

 A New Approach:

 For display purposes we need not evaluate the reflection equation exactly;
 a good approximation will suffice. The "error" introduced by the
 approximation will be of no consuquence since Phong's normal interpolation
 and the reflection equation are already approximations. the one-dimensional
 form of Taylor's series is widely used for approximation functions, such as
 sine and log, with polynomials. The less widely used, two-dimensional form
 will serve to approximate the reflection equation.

 Taylor's series for a function of two variables is:

 f(a+h, b+k) = f(a,b) + (h ... + k ...)*f(a,b) + ....... [look it up!]

 Shifting the triangle so that (0,0) lies at its center, and expanding in a
 Taylor series to the second degree produces

        I       (x,y) = T x^2 + T xy + T y^2 + T x + T y + T
         diffuse         5       4      3       2     1     0


         3ig^2 - 4cdi - 4agi
   T  = ---------------------,
    5     8 * i^2 * SQRT(i)

         3cgh - 2cei - 2bgi - 2ahi
   T  = ---------------------------,
    4        4 * i^2 * SQRT(i)

         3ch^2 - 4cfi - 4bhi
   T  = ---------------------,
    3     8 * i^2 * SQRT(i)

            2ai - cg
   T  = -----------------,
    2    2 * i * SQRT(i)

            2bi - ch
   T  = -----------------, and
    1    2 * i * SQRT(i)

   T  = ---------
    0    SQRT(i)

 Using this expansion and forward differences, the intensity can be
 evaluated with only 2 additions per pixel.

 The method can be extended to handle Phong's specular reflection model,

 I        = (N * H)

 by using Taylor's series to evaluate the dot product and table lookup to do
 the exponentiation (A table of 1K bytes will allow exponentiation to powers
 up to 20 with less than 1 percent error and a table of 8K bytes allows
 powers up to 164.) H is a vector in the direction of maximum highlight
 which is halfway between the light direction, L, and the eye direction, E,

                  E * L
             H = -------,
                 |E + L|

 The eye, like the light source, is assumed for the present to be at
 infinity. Phong's reflection equation,

 I = I       + I       +  I
      ambient   diffuse    specular

 can now be evaluated with only 5 additions and 1 memory access per pixel
 (the ambient component can be rolled into the series for the diffuse
 component and the series for the specular component can be scaled to allow
 direct addressing of the exponentiation table). It should be simple to
 configure hardware to do these operations in less than 100 nanoseconds per

 The additional computation required to determine the Ti for our new method
 is offset by the greatly reduced computation per pixel for polygons with 10
 or more pixels. This estimate is based on the following assumptions: (a)
 the algorithms are implemented on sequential processors of conventional
 design, (b) with modern hardware, the time for floating point addition,
 multiplication, and memory references are all about equal, (c) the
 computation of 1 / SQRT(x) requires about 10 operations, and (d) common
 subexpressions have been removed from the formulas for Ti. Timings and
 analysis of the method as implementd in the raster testbed [Whitted and
 Weimer, 1982] also support a 10 pixel break-even point. The break-even
 point would be much lower if some simple hardware were added to do the
 additions and table lookup for our method in parallel. Of course, special
 hardware could be built for the other methods but it would be much more
 complicated and probably not as fast.

 The images we have produced with this approximation are indistinguishable
 from those produced with Phong's method. For polygons with more than about
 60 degrees of curvature, this new expansion will produce shadings that are
 noticeably different from Phong shading but curvatures this large should
 never be represented by a single polygon.

 Going Further:

 Since the form of the polynomial we have to evaluate at each pixel is
 independent of the original reflection equation, we can use more elaborate
 models. One useful extension is to provide for finite eye distance, thus
 requiring that the vector to the eye, E, vary across the scene. The
 resulting variation in the specular component produces more natural looking
 illumination for scenes rendered with perspective. This is most obvious
 when there are planar surfaces in the scene because Phong shading with the
 eye at infinity, like flat shading, renders all parallel surfaces with the
 same intensity. Figure 4 is a comparision of Phong shading with the eye at
 infinity and with the eye at a true perspective distance. The reflection
 equation for the specular component with the eye at a finite distance now

                 N(x,y) * H(x,y)
   I        = ---------------------,
    specular   |N(x,y)| * |H(x,y)|

 where H(x,y) = E(x,y) + L and E(x,y) interpolates the eye vector across the
 triangle. This can be expanded just as before

                 (Ax + By + C) * (Dx + Ey + F)
   I        = -----------------------------------
    specular   |(Ax + By + C)| * |(Dx + Ey + F)|

 After performing the dot products and expanding the vector magnitude as

                        ax^2 + bxy + cy^2 + dx + ey + f
 I =--------------------------------------------------------------------------
  s  SQRT((gx^2 + hxy + iy^2 + jx + ky + l)*(mx^2 + nxy + oy^2 + px + qy + r))


   a = A * D,

   b = A * E + B * D,

   c = B * E,

   d = A * F + C * D,

   e = B * F + C * E,

   f = C * F,

   g = A * A,

   h = 2A * B,

   i = B * B,

   j = 2A * C,

   k = 2B * C,

   l = C * C,

   m = E * E,

   n = D * D,

   o = 2D * E,

   p = 2D * F,

   q = 2E * F, and

   r = F * F.

 And expanding in Taylor's series about (0,0).

   I(x,y) = T x^2 + T xy + T y^2 + T x + T y + T
             5       4      3       2     1     0


      8al^2r^2 - 4djlr^2 - 4fglr^2 + 3fj^2r^2 - 4dl^2pr + 2fjlpr - 4fl^2mr
 T = ----------------------------------------------------------------------
  5                           8l^2r^2 * SQRT(l*r)

           + 3fl^2p^2
     + ---------------------,
        8l^2r^2 * SQRT(l*r)

      4bl^2r^2 - 2dklr^2 - 2ejlr^2 - 2fhlr^2 + 3fjkr^2 - 2dl^2qr
 T = ------------------------------------------------------------
  4                       4l^2r^2 * SQRT(l*r)

         + fjlqr - 2el^2pr + fklpr - 2fl^2nr + 3fl^2pq
     + ------------------------------------------------,
                       4l^2r^2 * SQRT(l*r)

      8cl^2r^2 - 4eklr^2 - 4filr^2 + 3fk^2r^2 - 4el^2qr + 2fklqr - 4fl^2or
 T = ----------------------------------------------------------------------
  3                           8l^2r^2 * SQRT(l*r)

           + 3fl^2q^2
     + ---------------------,
        8l^2r^2 * SQRT(l*r)

       2dlr - fjr - flp
 T = -------------------,
  2    2lr * SQRT(l*r)

       2elr - fkr - flq
 T = -------------------,
  1    2lr * SQRT(l*r)

 T = -----------
  0   SQRT(l*r)


 We have shown that computer image generation systems can use Phong shading
 with only a little more computation per pixel than is required for Gouraud
 shading. This result should allow real-time systems to generate more
 realistic scenes and conventional systems to produce images more rapidly.
 To test the method, we have implemented it as a shader subroutine for the
 raster testbed. The cpu time for rendering figure 4 with 14620 polygons at
 2048x2048 resolutions on a DEC microVAX-II are: Shading method CPU time
 Notes Gouraud 406s Taylor (infinite) 767s incl 119s for Taylor coeffs
 Taylor (finite) 850s incl 202s for Taylor coeffs Duff 2597s incl 1420 in
 sqrt Phong 3900s incl 1405 in sqrt


 Tom Duff and the reviewers provided many helpful suggestions.

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Date this article was posted to 7/16/1999
(Note that this date does not necessarily correspond to the date the article was written)

See Also:
Lighting and Shading

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