Real-time Shading with Free-form Planar Area Lights using Linearly Transformed Cosines (JCGT & I3D 2022)

Presentation slides for the research, "Real-time Shading with Free-form Planar Area Lights using Linearly Transformed Cosines" by Kuge et al., which is presented at I3D 2022 (invited from JCGT).

JCGT article link: http://jcgt.org/published/0011/01/01/

Transcript

1. Real-time Shading with Free-form Planar Area Lights using
   Linearly Transformed Cosines
   Takahiro Kuge1 ○Tatsuya Yatagawa2 Shigeo Morishima1
   1Waseda University 2The University of Tokyo
   

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2. Pioneering work by Baum et al. [1989]
   l Lambertian surface illuminated by polygonal light
   2
   Real-time rendering of flat area lights
   𝐼 = #
   !
   𝐿 cos 𝜃 d𝜔
   ＝ Cosine is integrated over a solid angle
   ＝ The integral equals to area of projected
   figure [Lambert 1760]
   Light
   source
   Solid angle: 𝛀
   Projected figure
   Shading point
   * A figure is projected to “local” shading plane, and
   the shaded object should not be planar.
   

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3. Area of projected figure is analytic when light shape is polygon
   3
   Baum’s formula
   ＝ － －
   Sum of areas of projected “fans”
   𝐼 = +
   #
   acos 𝐩# ⋅ 𝐩$
   𝐩#×𝐩$
   𝐩#
   ×𝐩$
   ⋅ 𝐞%
   𝑗 = 𝑖 + 1 mod 𝑁, 𝐞𝐳
   is unit vector along z axis.
   Light
   source
   Solid angle: Ω
   Projected figure
   Shading point
   

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4. l Surfaces with Phong BRDF [Arvo 1995]
   l Surfaces with Microfacet BRDF (GGX) [Heitz+ 2016]
   l Special light shapes, e.g., ellipses and lines [Heitz & Hill 2017]
   l Polygon-clipped spherical harmonics [Wang+ 2018; Belcour+ 2018]
   4
   Extensions of Baum’s approach
   [Heitz & Hill 2017] [Wang et al. 2018]
   Free-form area lights hasn’t been well investigated!
   

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5. Real-time shading with free-form area lights bounded by 3rd-order Bezier
   l Approximate boundary curve with polyline, then apply Baum’s method
   → Adaptive polygonization based on magnitude of light contribution
   l Extend [Heitz+ 2016] to handle Microfacet BRDF
   → Efficient clipping operation (shown later) for polynomial curve
   Research objectives
   5
   Real-time shading with free-form area lights bounded by 3rd-order Bezier
   l Approximate boundary curve with polyline, then apply Baum’s method
   → Adaptive polygonization based on magnitude of light contribution
   l Extend [Heitz+ 2016] to handle Microfacet BRDF
   → Efficient clipping operation (shown later) for polynomial curve
   

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6. Requirement: Area of projected figure is solved analytically
   Assume polynomial curve 𝐶 𝑡 = 𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 is projected to (
   𝐶(𝑡)
   6
   Why difficult to handle polynomial curves?
   𝐴 = #
   !
   "($)
   $
   𝑦 d$
   𝑥 = # $
   𝑦 𝑡
   d$
   𝑥(𝑡)
   d𝑡
   d𝑡
   +
   𝐶 𝑡 = $
   𝑥 𝑡 , $
   𝑦 𝑡 , ̃
   𝑧 𝑡 =
   𝑥 𝑡
   𝐶 𝑡
   ,
   𝑦 𝑡
   𝐶 𝑡
   ,
   𝑧(𝑡)
   𝐶(𝑡)
   = #
   𝑦 𝑡
   𝑥& 𝑡 + 𝑦& 𝑡 + 𝑧&(𝑡)
   d
   d𝑡
   𝑥(𝑡)
   𝑥& 𝑡 + 𝑦& 𝑡 + 𝑧&(𝑡)
   d𝑡
   In general, fractional function does not have analytic infinite integral
   Projected figure
   Light source
   Shading point
   = #
   𝑃(3𝐷 − 1)
   𝑃(4𝐷)
   d𝑡
   z
   y x
   𝐶(𝑡)
   %
   𝐶(𝑡)
   (𝑃 𝐷 : 𝐷th-order polynomial, 𝐷: order of polynomial curve)
   projection on
   hemisphere
   

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7. Ramer-Douglas-Peucker algorithm
   Originally, a method for approximating curve based on “distance” (see figure below)
   Adaptive boundary curve polygonization
   7
   Extension for light contribution:
   l Compute increase/decrease of light contribution when new point added
   l Subdivide a curve until increase/decrease is sufficiently small
   

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8. For each partial curve, add a midpoint and calculate light contribution from a triangle
   8
   Adaptive boundary curve polygonization
   continue
   much
   contribution
   terminate
   little
   contribution
   terminate
   Contribution of each triangle is calculated using Baum’s formula
   Partial curve is always
   divided by a midpoint
   No re-computation
   required
   little
   contribution
   

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9. 9
   Threshold for traverse termination
   Experiments with constant threshold for different roughness
   l Intuition: small threshold slows down shading computation.
   Low roughness
   Time: not affected
   Quality: affected
   → small thres. can be used
   High roughness
   Time: affected
   Quality: not affected
   → large thres. will be fine
   

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10. 10
    Threshold for traverse termination
    Accordingly, we can use a threshold proportional to roughness
    With adaptive thresholding, we obtain both speed and accuracy
    

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11. Real-time shading with free-form area lights bounded by 3rd-order Bezier
    l Approximate boundary curve with polyline, then apply Baum’s method
    → Adaptive polygonization based on magnitude of light contribution
    l Extend [Heitz+ 2016] to handle Microfacet BRDF
    → Efficient clipping operation (shown later) for polynomial curve
    Research objectives
    11
    

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12. Microfacet BRDF is approximated by linearly transforming cosine distribution
    12
    Linearly Transformed Cosines [Heitz et al. 2016]
    Linear transformation
    Cosine distribution BRDF
    By linear transformation, Baum’s formula is available for microfacet BRDF!
    Light shape deformation
    

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13. A part can be hidden under the ground after deformed by inverse transformation
    l Baum’s formula enforce area light is on upper hemisphere
    → We need to clip (= exclude) the part under the ground
    13
    Clipping operation
    For polygon, each edge is checked intersection with the ground plane
    Unfortunately, adaptive polygonization results in considerably many edges
    Cosine distribution BRDF
    Linear transformation
    Light shape deformation
    This part must
    be clipped!
    

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14. l Points of intersection between boundary curve and shading plane
    → Those with 𝑧 = 0 on the boundary curve
    → Solving cubic equation for z coordinates is sufficient (3rd-order Bezier)
    l Owing to affine invariance of Bezier, we can efficiently evaluate transformed
    curve by transforming only control points
    → We can instantly obtain the cubic equation
    Algebraic clipping
    14
    𝑎 = −𝑝&,(
    + 3 𝑝),(
    − 𝑝*,(
    + 𝑝+,(
    𝑏 = 3(𝑝&,(
    − 2𝑝),(
    + 𝑝*,(
    )
    𝑐 = 3(−𝑝&,(
    + 𝑝),(
    )
    𝑑 = 𝑝&,(
    Specifically, we solve the equation below
    𝑎𝑡+ + 𝑏𝑡* + 𝑐𝑡 + 𝑑 = 0,
    We can algebraically solve polynomial equation whose order is less than five.
    𝑡 ∈ 0, 1
    

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15. What we need are only real-number solutions
    l [Blinn 2007] shows an efficient solution for cubic equation
    15
    Algebraic solution for cubic equation
    Reference: http://momentsingraphics.de/CubicRoots.html#_Blinn07b
    

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16. Computer:
    CPU: Intel Xeon 5119 Gold
    GPU: NVIDIA GeForce 1080 (8GB Memory)
    RAM: 64GB
    Implementation:
    l We use the method for both diffuse and specular (shading is computed twice).
    l We assume light source itself can intersect with a shading surface (clipping is
    performed twice)
    Reference pictures:
    Rendered by Blender Cycles, Unidirectional Path Tracing (GPU enabled)
    16
    Experiment
    

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17. Currently, our method spends 1ms for each interval of Bezier curve
    17
    Rendering result: Concave shape with cavity
    

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18. Currently, our method spends 1ms for each interval of Bezier curve
    18
    Rendering result: Character
    

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19. Currently, our method spends 1ms for each interval of Bezier curve
    19
    Rendering result: Textured light
    

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20. Comparison method
    l Divide curve domain into a fixed number of intervals
    l Then, apply [Heitz+ 2016] with clipping operation to resulting polygon
    Comparison with [Heitz+ 2016]
    20
    For same-quality image,
    our method is twice faster!
    equal time equal quality
    

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21. Order of computation is a dominant factor.
    21
    Why ours is faster?
    [Heitz et al. 2016]
    1. Divide entire boundary curve
    into line segments
    2. For every line segment,
    ・Shading computation
    ・Clipping operation
    Our method
    1. Clip boundary curve
    → Reduce # of curves
    2. For only remained curves,
    ・Adaptive polygonization
    ・Shading computation
    First clipping operation reduce the number of curves processed
    ＋
    Adaptive polygonization concentrates more resource on more important parts
    

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22. Our method shares the same limitations with [Heitz+ 2016]
    22
    Limitations
    Low accuracy for textured lights Highly rough surface illuminated
    from small grating angle
    Light is entirely hidden
    under the ground
    

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23. New real-time shading method for area lights contoured by 3rd-order Bezier.
    l Contrib. #1: Adaptive boundary curve polygonization based on light contribution
    l Contrib. #2: Efficient clipping operation for polynomial curve (algebraic clipping)
    → More efficient than straightforward application of previous method
    Conclusion
    23
    The same idea will be available for rational Bezier and B-splines!
    

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24. 24
    Thank you for listening
    GitHub: https://github.com/Paul180297/BezierLightLTC.git
    

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