© 2011 Frank Nielsen

INF555

Fundamentals of 3D

Lecture 7: ColorsRandomized algorithms

RANSAC/MINIBALL

Frank Nielsennielsen@lix.polytechnique.fr

2nd November 2011

© 2011 Frank Nielsen

Supersampling: Averaging

Uniform versus non-uniform sampling(raytracing software: Povray, Blender)

The essence of pixels: Integration !

© 2011 Frank Nielsen

Color and perceptionCommission Internationale del’ ´ Eclairage in French, CIE, http:// www.cie.co.at/

© 2011 Frank Nielsen

A candela unit is defined as the luminous intensity ofa light source emitting a monochromatic light of frequency 540 × 10^12 Hz that has a radiant intensity of 1 683 watt per steradian.

A luminous flux is measured in lumen units (lm).A lumen unit is defined as the amount of light that falls on a unit area at unit distance from a light source of one candela.

The lux (lx) represents the luminance and defined as one lumen per square meter.

Color and measurements

© 2011 Frank Nielsen

Cone responses in human eyes

Cones (color) and rods (B&W) in human eyes

© 2011 Frank Nielsen

Additive and subtractive color modes

Display/printing industries

Color checking in printing industry

© 2011 Frank Nielsen

Choosing colors: Colors pick-up

© 2011 Frank Nielsen

Pseudo-coloring

Volume rendering and transfer function

© 2011 Frank Nielsen

Color spaces

Normalizing coefficient

converts the spectrum absorption of light L to XYZ colors

© 2011 Frank Nielsen

The basic response functions for CIE XYZ hypothetical colors.Observe the artificial response curve x(·) of color X.

The basic response functions for CIE XYZ hypothetical colors.Observe the artificial response curve x(·) of color X.

© 2011 Frank Nielsen

Color cube

© 2011 Frank Nielsen

HLS/HSV color spaces

(hue, lightness, saturation (hue, saturation, value)

© 2011 Frank Nielsen

© 2011 Frank Nielsen

CMYK color space (for printing)

© 2011 Frank Nielsen

CIE L*a*b

Perceptual difference between two colors:

CIE L*a*b (where Euclidean distance make sense)

© 2011 Frank Nielsen

Many standards....

© 2011 Frank Nielsen

Gamma color correction:

Bayer tile

© 2011 Frank Nielsen

Half-toning and dithering

t=127 Random t

© 2011 Frank Nielsen

Dithering: Trade space for grey level perception

© 2011 Frank Nielsen

Error-diffusion process: Floyd-Steinberg

© 2011 Frank Nielsen

Low dynamic range images (LDRs) versusHigh dynamic range images (HDRs)

© 2011 Frank Nielsen

Low dynamic range imageHigh dynamic range imageTone mapping

© 2011 Frank Nielsen

Tone mapping

© 2011 Frank Nielsen

Quicksort:

O(n log n) expectedO(n^2) worst-caseO(n) best case

© 2011 Frank Nielsen

Randomization: A Powerful principle

Analysis:

© 2011 Frank Nielsen

RANSAC on Harris-Stephens points.Epipolar points intersect on their epipoles

RANSAC: Random Sample Consensus

© 2011 Frank Nielsen

© 2011 Frank Nielsen

Probability of failure of picking two inliers

Macthing points: Inliers/outliers

© 2011 Frank Nielsen

RANSAC: Bounding failure probability

Probability of failure after one round

Probability of failure after k independent rounds

© 2011 Frank Nielsen

© 2011 Frank Nielsen

Estimating homographies with RANSAC

© 2011 Frank Nielsen

Adaptive RANSAC: Guessing the ratio of inliers

© 2011 Frank Nielsen

Demo of autostich:

© 2011 Frank Nielsen

Bounding sphere hierarchy

© 2011 Frank Nielsen

MINIBALL: Smallest enclosing ballUNIQUE

http://www.inf.ethz.ch/personal/gaertner/miniball.html

© 2011 Frank Nielsen

LP-type problem in optimization

© 2011 Frank Nielsen

MINIBALL: Solving for basis of 2 and 3 points

Intersection of bisectors(use cross-product on homogeneous coordinates)

Intersection of bisector/segmentlinking the two points:Circumcenter=half-point