Antialiasing Miniaturized Textures CSS 552 – Prof. Sung Daniel R. Lewis June 3, 2013.

Antialiasing Miniaturized Textures

CSS 552 – Prof. SungDaniel R. Lewis

June 3, 2013

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Texels and Pixels (1)Textures are composed of discrete texels:

Pixels are projected onto texture:

TexturedGeometry

Eye

ImagePlane

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Texels and Pixels (2)

Image credit: Heckbert

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Texture Magnification

Image resolution higher than texture resolution Pixel grid “denser” than texel grid

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Texture Minification

Image resolution lower than texture resolution Texel grid “denser” than pixel grid

TexelGrid

Pixel Footprint

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Point Sampling

Color comes from texel that happens to be at the intersection position

Poor representation of pixel area Temporal aliasing e.g. “blinking”

At right: How would the pixel coloration change as the camera or geometry moved?

Improved by supersampling to a degree—becomes expensive as the texel:pixel ratio increases

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Image credit:Angel

Aliasing from point sampling

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Prefiltered Texel Averaging

How many? What shape? Reminder: pixel coverage is view-

and scene-dependent (1a, 1b) Pixels cover various numbers of

texels even within the same frame (2)

Cannot know a priori what to average

All methods prefilter the texture to generate approximations of all possible pixel footprints

(1a)

(1b)

camera moves back

(2)

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Mipmapping Invented by

Lance Williams, seminal paper published in 1983

MIP an acronym for multum in parvo, meaning “many things in a small place”

Well-supported in hardware

point sampling

mipmapping

detail

detail

Image credit: Williams

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Mipmapping: How It Works (1)

Start with 2n×2n texture Create n downsampled

texture maps, where each map M

i is 25% of the size of

map Mi-1

Average four pixels into one How should pixels be

“averaged”? Hold that thought...

Mipmaps require only 33.3% of the memory of the original texture

Image credit: Wikipedia(a)

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Mipmapping: How It Works (2)

Pixel color derived from (u, v, d) triple, where d is the level of detail

Image credit: Akenine-Moller et al.

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Finding d

(a) d || longest_pixel_edge ||∝ lpe = || longest_pixel_edge || d = 1 - ( lpe / texture_width )

(b) dx = sqrt((Δu / Δx)2 + (Δv / Δx)2) dy = sqrt((Δu / Δy)2 + (Δv / Δy)2) d max(dx, dy)∝ e.g.: d = (1 / sqrt(2)) × max(dx, dy, sqrt(2))

Challenge: in ray tracing, all we have is a point–not enough to reconstruct the red quadrilateral or find dx/dy

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Using d

d

Derive two maps from d Sample both at (u, v) and interpolate between

the samples Common to add/subtract a level of detail bias

from d to tweak results. Specified by user.

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Filtering Algorithms

When downsampling, want to eliminate high frequency information that cannot be reproduced, before it causes aliasing Low-pass filter (LPF)

Pixel averaging, a.k.a. box filtering, is a bad LPF

Ideal LPF is sinc sin(πx)/(πx) Infinitely wide

Approximate sinc by multiplying it by a windowing function and cutting it off

Purple: Ideal filterYellow: Box filterGreen: Kaiser filterBlue: Point sampling

Image credit: Blow(a)

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Brightness and Gamma Correction (1)

Human eye is more sensitive to darker shades than brighter ones

Linear encoding of color wastes space on bright shades that are indistinguishable

Hence, images store color non-linearly, which is then scaled exponentially by γ to become linear

Clinear

= (Cstore

)γ

Typically, 1.8 ≤ γ ≤ 2.8

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Filtering should not reduce brightness (ideally) However:

A = tc1γ + tc

2γ + tc

3γ + tc

4γ

B = ((tc1 + tc

2 + tc

3 + tc

4) / 4)γ

A > B Fixed by converting texel colors to linear space

(decode), filter that, and then convert back (encode)


Image credit:Blow(b)

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direct filtering linear filtering

Image credit: Blow(b)

Detail view of mipmapped textures

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Achilles’ Heel of Mipmapping

Mipmapping weakness: all pixel projections that do not vaguely approximate squares

Below: color of pixel (red outline) comes from all shaded texels—mostly error!

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Image credit:Angel

Aliasing from mipmapping

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Ripmapping

Extension of mipmapping generating rectangular maps

Requires 300% more memory than original texture

Rarely used today

Image credit: Wikipedia(b)

d

v

d

u

d

u

d

u

d

v

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Summed Area Tables

Invented by Frank Crow, seminal paper published in 1984

Based on rectangles like ripmapping, but with a dissimilar algorithm

Limited hardware support

Top: Point samplingMiddle: MipmappingBottom: Summed area tables

Image credit: Akenine-Moller et al.

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Generating the SAT

n×m SAT for n×m texture Each slot (i, j) holds the sum of

all texels in rectangle between origin (0, 0) and (i, j) Requires more bits per channel

(16–32) than original texture color (8)

(0,0)

(i, j)

m

n

Efficiently computed in a single pass: I(i,j) = tc(i,j) + I(i−1,j) + I(i,j−1) − I(i−1,j−1)

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Finding Rectangle Color R

s = I(il,jb) − I(ir,jt) − I(il,jb) + I(il,jt)

Ra = R

s / ((ir − il)(jb − jt))

(0,0)

il ir

jt

jb

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Ra = R

s / ((ir − il)(jb − jt))

(0,0)

il ir

jt

jb

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Ra = R

s / ((ir − il)(jb − jt))

(0,0)

il ir

jt

jb

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Ra = R

s / ((ir − il)(jb − jt))

(0,0)

il ir

jt

jb

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Ra = R

s / ((ir − il)(jb − jt))

(0,0)

il ir

jt

jb

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Finding the Bounding Box u

min = min(u

a, u

b, u

c, u

d)

umax

= max(ua, u

b, u

c, u

d)

vmin

= min(va, v

b, v

c, v

d)

vmax

= max(va, v

b, v

c, v

d)

Rectangle: Vertice[0] = (u

min, v

min)

Vertice[1] = (umin

, vmax

)

Vertice[2] = (umax

, vmax

)

Vertice[3] = (umax

, vmin

)

(ua,

va)

(ub, v

b)

(uc, v

c)

(ud, v

d)

v

u

v

uv0

v1

v3

v2

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Interpolating SAT Rectangles

Instead of rounding bounding box vertices to integers, allow the corners to fall between texel boundaries

Find color of all boxes at neighboring texels and interpolate between them based on distance

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Achilles’ Heel ofRipmapping and SATs

Weakness of rectangle-based methods is pixel projections that cut diagonally across the texture

Below: color of pixel (red outline) comes from the texels in the shaded bounding box—mostly error!

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Unconstrained Anisotropic Filtering

Built on top of mipmapping Find (approximate) the major axis of the

pixel footprint Sample the mipmap (trilinear

interpolation) at least twice along the axis, more if the footprint is long

Unlike normal mipmapping, use the length of the shorter side to compute the d value

Supported by PC graphics cards since early 2000s

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Not Herein Considered...

Non-color textures: most of the antialiasing techniques can be extended for textures of other types (e.g., bump maps)

Curved surfaces Literal edge cases

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Implementation Plan (1)

Focus on implementing basic mipmapping “Color” textures only Time allowing, will implement more

Use RayTracer framework, with MP4 as the starting point

Risk: do not have enough information to compute the size or shape of the pixel footprint Substantial modifications required

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Implementation Plan (2)

Must have additional intersections (either world space or uv space) down within GetTexile() Deep stack; might be easiest to stuff extra

information into the IntersectionRecord ComputeImage()

ComputeShading()

GetDiffuse()

AttributeLookup()

TextureLookup()

GetTexile()

Naïve implementation: ray trace four extra intersections at the pixel corners

Better: shifting “box” of four intersection records No more expensive, except edges

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Image Credits

Akenine-Moller, Tomas; Eric Haines; Naty Hoffman. Real-Time Rendering (3rd ed.).

Angel, Edward. Interactive Computer Graphics.

Found at http://vip.cs.utsa.edu/classes/cs5113s2007/lectures/materials/TextureInterpolationExample.html

Blow, Jonathan (a). “Mipmapping, Part 1.” Game Developer Magazine Dec. 2001.

Blow, Jonathan (b). “Mipmapping, Part 2.” Game Developer Magazine Jan. 2002.

Heckbert, Paul. “Fundamentals of Texture Mapping and Image Warping.” Thesis. University of California, Berkeley, 1989.

Wikipedia (a). <http://en.wikipedia.org/wiki/Mipmap>

Wikipedia (b). <http://en.wikipedia.org/wiki/Anisotropic_filtering>

Williams, Lance. “Pyramidal Parametrics.” Computer Graphics 13.3 (1983).

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Questions?

Antialiasing Miniaturized Textures CSS 552 – Prof. Sung Daniel R. Lewis June 3, 2013.

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