Deep Shadow Maps

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Deep Shadow Maps

• Tom Lokovic Eric Veach
• Pixar Animation Studios
• Presented by Tom Lechner

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Outline

• Traditional Shadow Maps (TSMs)
• Other Shadowing Techniques
• Deep Shadow Maps (DSMs)
• Generation
• Sampling
• The Transmittance Function
• The Visibility Function
• Compression
• Lookups
• Comparing DSMs to TSMs
• Implementations
• Examples

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So why are Shadow Maps Important?
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Traditional Shadow Maps (TSMs)

• Generation
• Shadow Camera (SC)
• Rectangular array of depths to the closest
  surface of a given pixel
• Sampling
• Transform P into SC coordinate system
• Compare point depth to shadow depth
• Higher quality images require
• Percentage closer filtering - examines depth
  samples within a given filter region and computes
  the fraction that are closer than a given depth z
• Stratified sampling in both the original shadow
  map and filtering subset

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Traditional Shadow Maps (contd)

• Advantages
• Renders large objects well
• Stores only one depth value per pixel
• Disadvantages
• Poorly Renders highly detailed geometries (e.g.
  fur, hair)
• Produces artifacts, esp. in animation (e.g.
  sparkling)
• Rendering time and memory allocation for detailed
  renders rapidly increase due to super sampling

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Other Shadowing Techniques

• Ray Casting
• Fuzzy objects (potentially millions of hairs) too
  expensive!
• No soft shadows (unless using an expensive area
  light source)
• Smoke and fog require ray marching for each hair
• 3D Texturing
• Has had some success for volume datasets (clouds,
  fog, medical imaging)
• Relatively course resolution
• Low accuracy in z (creates bias problems)
• Become prohibitively large as detail increases
• Multi-layer Z-buffers
• Renders opaque surfaces from differing viewpoints
• Similar drawbacks to TSMs

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Deep Shadow Maps (DSMs)

• Generation
• Rectangular array of pixels in which every pixel
  stores a visibility function
• Every value is a function of transmittance - the
  fraction of light that penetrates to a given
  depth z
• The original beam can be shaped and filtered to
  any particular filter
• A visibility function is calculated by filtering
  the nearby transmittance functions and
  re-sampling at the pixel center

transmittance function
desired band limiting pixel filter (centered
around the origin)
filter radius
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Deep Shadow Maps (contd)

• Generation (contd)
• Visibility functions are closely related to alpha
  channels
• Thus a DSM is equivalent to computing the
  approximate value 1 - ? at all depths, and
  storing result as a function of z
• Contains the combined attenuation and coverage
  information for every depth

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Deep Shadow Maps (contd)

• Sampling
• Select a set of sample points across the shadow
  cameras image plane
• For each sample point, determine the
  corresponding transmittance function
• Given an image point (x,y) compute the surfaces
  and volume elements intersected by the
  corresponding primary ray
• surface transmission function volume
  transmission function transmission function of
  (x,y)
• For each pixel, compute its visibility function
  by taking a weighted combination of the
  transmittance function at nearby sample points.

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Deep Shadow Maps (contd)

• The Transmittance Function
• Surface Transmittance
• Each surface hit has a depth value, zis, and an
  opacity, Oi
• Start with a transparency of 1 and then multiply
  by 1 Oi at every surface hit (resulting in
  piecewise constant function Ts)
• Volumetric Transmittance
• Sample atmospheric density at regular intervals
  along primary ray
• Each volume sample has a depth value, ziv, and an
  extinction coefficient, ki, that measures light
  falloff per unit distance
• Linearly interpolate between the samples to get
  the extinction function, k
• Since not piecewise linear, approximate by
  evaluating transmittance at each vertex of the
  extinction function and linearly interpolating
• Composite like surface transparencies to find Tv,
  except this time interpolate between vertices
  rather than forcing discrete steps

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Deep Shadow Maps (contd)

• The Transmittance Function (contd)
• Merge surface and volume transmission functions
• Since result is not piecewise linear, evaluate it
  at the combined vertices of Ts and Tv,
  interpolating linearly between them

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Deep Shadow Maps (contd)

• The Visibility Function
• At each depth z, the nearby transmittance
  functions are filtered like ordinary image
  samples
• Results in a piecewise linear function with
  approx. n times as many vertices as the
  transmittance functions
• Takes into account the fractional coverage of
  semitransparent surfaces and fog as well as light
  attenuation

number of transmittance functions within the
filter radius around (i ½, j ½)
normalized filter weight for each corresponding
sample point (xk, yk)
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Deep Shadow Maps (contd)

• Compression
• Visibility functions can have a large number of
  vertices depending on filter size and number of
  samples per shadow pixel
• Fortunately, functions tend to be very smooth
• Compression must preserve z values of important
  features, since simple errors in z can cause
  self-shadowing artifacts

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Deep Shadow Maps (contd)

• Compression (contd)

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Deep Shadow Maps (contd)

• Lookups
• Apply a reconstruction and re-sampling filter to
  a rectangular array of pixels (similar to
  textures)
• Given a point (x,y,z) at which to perform a
  lookup and a 2D filter kernel f, the filtered
  shadow value is
• Evaluating each visibility function requires
  searching through its data points to determine
  which segment contains the z-value
• May be implemented as a binary or linear search

filter weight for pixel (i, j)
sum over all pixels in filter radius
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Comparing DSMs to TSMs

• Support prefiltering
• Faster lookups
• Much smaller in comparison to an equivalent high
  resolution depth map (dependant upon compression)
• Fortunately, at any sampling rate there is an
  error tolerance that allows significant
  compression without compromising quality
• Shadows of detailed geometry have an expected
  error of about O(N-½)
• This error is a measure of the noise inherent in
  the sampled visibility function
• Actually implemented with a tolerance of
  0.25(N-½) half of the maximum expected noise
  magnitude
• TSM uses O(N) storage, where DSM uses O(N-½),
  approaches O(N-¼) when functions are piecewise
  linear

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Comparing DSMs to TSMs (contd)

• Significantly more expensive to compute than a
  regular shadow map at the same pixel resolution
• Bias artifacts possible, due to constant z depths
• Exacerbated by encouragement of large filter
  widths
• Might be useful, as it provides an extra degree
  of freedom
• Shadow resolution should be chosen according to
  minimum filter width desired (shadow detail)
• Number of samples per pixel should be determined
  by maximum acceptable noise
• Note Bias artifacts and computational expense of
  DSMs are no worse off than TSMs that are
  equivalent in terms of pixel resolution or sample
  size

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Comparing DSMs to TSMs (contd)
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Comparing DSMs to TSMs (contd)
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Implementations

• Incremental Updates
• Can be optimized to proceed in O(nlogn)
• Colored Shadows
• At expense of twice the storage
• Allows for some compression for gray shadows
• Mip-Mapping
• Can dramatically reduce lookup costs when objects
  are viewed over a wide range of scales
• Tiling and Caching
• Stored similar to textures and share in some of
  their advantages
• Tile directory
• Motion Blur

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Examples

• With and without DSMs

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Examples (contd)
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Examples (contd)
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Questions?

• E.G.
• Was there anything you didnt understand about my
  explanation of the algorithms?
• Do you see the correlation between DSMs and Light
  Fields?
• Do you need clarification on the implementations?

Thanks!