Index

F -measures, see image classificationF1-measure, see image classificationFβ-measure, see image classificationPb, see probability of boundaryχ-squared kernel, see classifier0-1 loss, 459, 5112 12-dimensional sketch, 210

aberrations, 10–12astigmatism, 11chromatic aberration, 12circle of least confusion, 11coma, 11distortion, 11

radial, 27field curvature, 11primary aberrations, 11spherical aberration, 10

absoluteconic, 249dual quadric, 250

absolute reconstruction, 611accumulator array, see fittingactions from the web, see datasetsactive markers, see peopleactive sensors, 422active vision, 218, 221activity recognition, see peopleaffine, 230affine geometry

affine shape, 232affine transformation, 231

affine motion model, see optical flowaffine shape, 232affine structure from motion

ambiguity, 232definition, 230Euclidean upgrade, 238–240

multiple views, 239–240from binocular correspondences, 233–

236

from multiple imagesfactorization, 237–238

affine transformation, 15affinity, see clusteringaffordances, see object recognitionAIC, see model selectionairlight, see color sourcesalbedo, 34

low dynamic range, 90reflectance, 76spectral albedo, 76spectral reflectance, 76

algebraic surface, 454aliasing, see samplingalignment, 377all-vs-all, see classifieralpha expansion, 214, 680ambient illumination, 35ambiguity

affine, see affine structure from motionEuclidean, 223, see structure from mo-

tion, 246projective, see projective structure from

motionaperture problem, see cornerapparent curvature, 402, 406apparent motion, 315arc length, 400, 404area source, 36area sources

shadows, 36, 37aspect, 416, 531, 547aspect graph, 392, 416–417astigmatism, 11asymptotic

bitangent, 415curve, 409–410

red and blue, 409directions, 399, 403–405, 425spherical image, 412spherical map, 407, 410–411

737

INDEX 738

tangent, 409, 412, 414atlas, see medical imagingattached shadow, 403attributes, 550, see object recognitionAUC, 466average pooling, see poolingaverage precision, see image classification

background subtraction, 261–263, 314backward variable, see peoplebarycentric coordinates, 379baseline, 199basis pursuit, 184Bayes classifier, see classifierBayes information criterion, see model se-

lectionBayes risk, see classifierbeak-to-beak, 412beat frequency, 424bed-of-nails function, 125best bin first, see nearest neighborsbetween-class variance, 495between-class variation, see image classifi-

cationBhattacharyya coefficient, 336bias, 515binary terms, see tree-structured modelsbinocular fusion, 197binormal

vector, 403birds, see image classificationbitangent, 413

line, 414asymptotic, 415limiting, 415

plane, 415, 417ray manifold, 407, 413–414

black body, see color sourcescolor, physical terminology, 75

blind spot, 14blurring, see smoothingBM3D, 183–184boosting, 475bootstrapping, see classifierbottles, see datasetsbrightness, 32, 44, see color spacesbrowsing, 643browsing images, 627

desiderata for systems, 643BSDS, see datasetsBuffy, see datasets

bundle adjustment, 245, see for mosaics

Caltech 101, see image classificationCaltech 256, see image classificationCaltech-101, see datasetsCaltech-256, see datasetscamels, see datasetscamera

obscura, 3pinhole, see pinhole camera

camera calibrationlinear, 23–26

camera parameters, 25–26degeneracies, 24–25projection matrix, 23–24

photogrammetry, 27–29weak, see weak calibration

camera consistency, 367camera model

affine, 20–22extrinsic parameters, 22intrinsic parameters, 22projection equation, 6, 7, 22

perspective, 14–20extrinsic parameters, 18–19intrinsic parameters, 16–18projection equation, 18

weak perspective, 20–22canonical variates, see classifiercapacity, see graphsCART, 450carved visual hull, 572–573cascade, see detectioncast shadow, 403catadioptric optical systems, 8category, see object recognitioncenter of curvature, 395central limit theorem, 430central perspective, see projection model →

pinhole perspectivechaff, 349Chamfer matching, 388chromatic aberration, 12CIE, see color spacesCIE LAB, see color spacesCIE u’v’ space, see color spacesCIE xy, see color spacesCIE xy color space, see color spacesCIE XYZ, see color spacesCIE XYZ color space, see color spacescircle of curvature, 395

INDEX 739

circle of least confusion, 11circuit, see graphclass-confusion matrix, see classifierclassifier, 457, see also image classification

Bayes classifier, 459, 460Bayes risk, 459boosting, 475

decision stump, 475decision stump, training, 476discrete adaboost, 477fast face detection by, 524real adaboost, 478weak learner, 475

class confusion matrix, 465class-confusion matrix, 464correlated image keywords, 649, 650cost of misclassification, 457cross-validation, 464decision boundary, 458

chosen to minimize total risk, 458,459

definition, 457error rate, 463estimating performance, 459, 464

detection rate, 466leave-one-out cross-validation, 464receiver operating characteristic curve,

466receiver operating curve, 466, 501true positive rate, 466

examplesfinding faces using a neural network,

522, 523false negative, 457false positive, 457feature selection

by canonical variates, 494, 497–499by principal component analysis or

PCA, 493–495problems with principal component

analysis, 496from class histograms

skin finding examples, 466, 500, 501general methods for building

directly determining decision bound-aries, 467

from explicit probability models, 467kernel machine, 473

kernel, 474kernel machinesχ-squared kernel, 488

histogram intersection kernel, 489spatial pyramid kernel, 489

linking faces to names, 651, 652logistic regression, 461

logistic loss, 461loss, 457Mahalanobis distance, 468multiclass

all-vs-all, 479one-vs-all, 479

naive Bayes, 469nearest neighbor classifier

k-nearest neighbor classifier, 469nearest neighbors, 469, 470neural network

finding faces using, 522, 523plug-in classifier

example, 467, 468practical tips, 475

bootstrapping, 477hard negative mining, 478manipulating training data, 477

predicting keywords for images, 646–648, 654, 656

regularization, 460choosing weight with cross-validation,

463for numerical reasons, 463possible norms, 463to improve generalization, 463

risk function, 458software, 480

LIBSVM, 480multiple kernel learning, 480PEGASOS, 480SVMLight, 480

support vector machine, 526for linearly separable datasets, 470–

472for non-separable data, 472hinge loss, 472

total risk, 458closing point, 564closure, see segmentationclustering, 268

as a missing data problem, see missingdata problems

complete-link clustering, 270dendrogram, 270graph theoretic, 277

agglomerative, 280, 281

INDEX 740

divisive, 281–283normalized cuts, 284, 285

group average clustering, 270grouping and agglomeration, 269image pixels using agglomerative or di-

visive methods, 270image pixels using K-means, 272, 273image pixels, as a missing data prob-

lem, 308normalized cut, 284partitioning and division, 269single-link clustering, 270using hierarchical k-means, 640using K-means, 172

clutter, 377CMU Quality of Life dataset, see datasetsCMY space, see color spacescoarse-to-fine matching, 137coding, see image classificationcollections of images, browsing

MDS for layout, 644, 645color bleeding, 59, 101color constancy, 95

finite-dimensional linear model, 96recovering surface color by gamut map-

ping, 98recovering surface color from aver-

age reflectance, 98recovering surface color from gamut,

99recovering surface color from specu-

lar reflections, 98human color constancy, 95, 96lightness computation, 44

color matching functions, see color spacescolor perception, 68

cone, 72, 73Grassman’s laws, 70lightness

computing lightness, 44photometry does not explain, 95, 96primaries, 68principle of univariance, 71rod, 72subtractive matching, 69surface color, 96surface color perception, 95test light, 68trichromacy, 69

color sourcesairlight, 73

black body, 75color temperature, 75

daylight, 73fluorescent light, 74incandescent light, 74mercury arc lamps, 75Mie scattering, 74Rayleigh scattering, 73skylight, 73sodium arc lamps, 75

color spaces, 77brightness, 83by matching experiments, 78CIE, 79CIE LAB, 85CIE u’ v’ space, 87CIE u’v’ space, 85CIE xy color space, 80CIE XYZ color space, 79, 81

CIE xy, 80, 82, 83CMY space, 81

mixing rules, 81use of four inks, 82

color differences, 86color matching functions, 78

negative values, 78obtaining weights from, 78

cyan, 81HSV space, 84, 85hue, 83imaginary primaries, 78just noticeable differences, 84lightness, 83Macadam ellipses, 84, 86magenta, 81opponent color space, 80RGB color space, 80RGB cube, 80saturation, 83subtractive matching, required, 78uniform color space, 85, 87uniform color spaces, 86value, 83yellow, 81

color temperature, see color sourcescolor, modeling image, 87

color constancy using model, 96color depends on surface and on illu-

minant, 89complete equation, 88diffuse component, 89

INDEX 741

finding specularities using model, 92finite-dimensional linear model, 96, 97

computing receptor responses, 98recovering surface color by gamut map-

ping, 98recovering surface color from aver-

age reflectance, 98recovering surface color from gamut,

99recovering surface color from specu-

lar reflections, 98illuminant color, 88, 89

color, physical terminologyspectral energy density, 68

coma, 11comb function, 125combinatorial optimization, 663common fate, see segmentationcommon region, see segmentationcompact support, 116complete data log-likelihood, see missing data

problemscomposition across the body, see peoplecompound lenses, 12computational molecules, 429computed tomography imaging, see medical

imagingconcave point, 405

plane curve, 397surface, 399

cone strip, 561, 567cones, 13conjugate directions, 403, 405connected, see graphconnected components, see graphconnected graph, see graphconsecutive, see graphcontent based image retrieval, see digital li-

brariescontinuity, see segmentationcontour

image, see image contouroccluding, see occluding contour

convex pointplanar curve, 397surface, 399

convolution, 107associative, 117commutative, 117continuous, 115, 117

1D derivation, 115

2D derivation, 117impulse response, 117point spread function, 117properties, 117

discrete, 108convention about sums, 108effects of finite input datasets, 118

examplesfinite differences, 110, 112, 141, 142,

144, 145local average, 107ringing, 108, 109smoothing, see smoothingweighted local average, see smooth-

inggives response of shift invariant linear

systemdiscrete 1D derivation, 113discrete 2D derivation, 114

kernel, 107, 117like a dot product, 131–133notation, 114, 115

convolution theorem, 120convolutional nets, 216coordinate system

local, see differential geometrycopyright, see near duplicate detectioncorner

aperture problem, 148covariant windows, 156detection, 149estimating scale and orientation, 151estimating scale with Laplacian of Gaus-

sian, 151, 153finding pattern elements, 154, 155Harris corner detector, 150interest point, 148self-similar, 149

correspondence problem, 197cosine similarity, 633cost-to-go function, see tree-structured mod-

elscovariance, 154cross-validation, see model selection, see clas-

sifiercrowdsourcing, 515crust algorithm, 454curvature, 396, 402, 404

curve, 395, 397, 428apparent, 402sign, 397

INDEX 742

surfaceGaussian, 401, 404, 406normal, 398, 401, 405principal, 41, 399, 401

cusp, 391, 394, 399, 402, 407–409, 411–413,416

crossing, 415, 416of Gauss, 408, 410, 411of the first kind, 394of the second kind, 394point, 391, 404

cut, see graphcyan, see color spacescyclopean retina, 205cylindrical panorama, 371

data association, 349Data-driven texture representations, 164datasets

activityactions from the web, 625CMU Quality of Life dataset, 626IXMAS dataset, 626KTH action dataset, 625movies and scripts datasets, 626MSR action dataset, 625Olympic sports dataset, 626PMI dataset, 626UCF activity datasets, 626VIRAT dataset, 626Weizmann activity, 625

BSDS, 287crowdsourcing, 515dataset bias, 515face detection, 539general object detection, 539human interactions, 624human signers, 625image classification, 513

bottles, 514Caltech-101, 513Caltech-256, 513camels, 514flowers, 514Graz-02, 514Imagenet, 514LabelMe, 513Lotus Hill data, 514materials, 515Pascal challenge, 514repositories, 515

scenes, 515LHI, 287near duplicate detection, 640parsing

Buffy, 624HumanEva dataset, 626

pedestrian detection, 538people, 624pose

Buffy, 624segmentation, 287

daylight, see color sourcesdecision boundary, see classifierdecision stump, see classifierdecision tree, 448Decision trees, 448–450decoding, see object recognitiondefinite patch neighbor, 577definitely visible patch, 576deformable objects

detection, 530, 533, 534software, 535

deformation, 531degree, see graphDelaunay triangulation, 454delta function, see convolutiondendrogram, see clusteringdense depth map, 47depth map, 47, 422depth of field, 9depth of focus, 9derivative of Gaussian filters, see gradient,

estimatingderivatives

using finite differences, 430derivatives, estimating

differentiating and smoothing with oneconvolution, 142

using finite differences, 110noise, 112smoothing, 141, 142, 144, 145

detectioncascade, 524deformable objects

model, 530, 533, 534software, 535

evaluationoverlap test, 535

face detection, 520, 524non-maximum suppression, 519occluding boundaries, 527, 529

INDEX 743

evaluation, 530method, 531

pedestrian detection, 524–526, 528evaluation, 527, 537

sliding windows, 520state of the art, 536

detection rate, see classifierdictionary, 183, 184dielectric surfaces, 90differential geometry, 391, 441

analytical, 424–426descriptive, 393–401plane curves, 393–397

concave point, 397convex point, 397curvature, 395, 396, 428cusp of the first kind, 394cusp of the second kind, 394Gauss map, 396Gaussian image, 396inflection, 394, 396local coordinate system, 393normal line, 393normal vector, 393regular point, 394singular point, 394tangent line, 393tangent vector, 393

space curves, 397curvature, 397Gauss map, 397normal plane, 397osculating plane, 397parametric curve, 420principal normal line, 397tangent line, 397

surfaces, 397–401asymptotic directions, 399concave point, 399conjugate directions, 403, 405convex point, 399differential of the Gauss map, 400,

425elliptic point, 399, 405first fundamental form, 425Gauss map, 399Gaussian curvature, 401, 404Gaussian image, 399hyperbolic point, 399local coordinate system, 399Monge patch, 425

normal curvature, 398normal line, 398normal section, 398normal vector, 398parabolic point, 399parametric surface, 41principal curvatures, 399principal directions, 398, 403second fundamental form, 400, 425second-order model, 399tangent plane, 398

differential of the Gauss map, 400, 425diffraction, 8diffuse reflection, 33diffusion equation, 138digital libraries, 627

browsing, 627desiderata for systems, 643MDS for layout, 644, 645

correlated image keywords, 649, 650evaluation, 629

news search example, 506patent search example, 506precision, 506recall, 506web filtering example, 506

example applications, 628image browsing, 629image search, 628near duplicate detection, 628trademark evaluation, 628

image search by keyword, 645linking faces to names, 651, 652predicting keywords for images, 646–

648, 654, 656user behavior, 630, 631user needs, 629

dilation, 499, 500diopter, 13directed graph, see graphdiscrete Adaboost, 475discrete cosine transform, 184discrete wavelet transform, 184disparity, 197distance minimization

geometric, 225distance transformation, 388distant point light source, 34distortion, 11distributional semantics, see information re-

trieval

INDEX 744

dual problem, 473dynamic programming, 210, see hiddenMarkov

modelsdynamics, 327

E-step, see missing data problemsecologically valid, 260edge

detection, 424roof edges, 428–429step edges, 427–428

edge detection, 144gradient based, 145

examples, 148finding maxima of gradient magni-

tude, 145, 146hysteresis, 146nonmaximum suppression, 147

poor behavior at corners, 149rectifying, see corner

roof edges, 161step edges, 161

edge points, 145edge-preserving smoothing, 138edges, 141, 145egg-shaped, see differential geometry→ sur-

faces → ellipticegomotion, 315eigenvalue problem, 666

generalized, 666eight-point algorithm, see weak calibrationelastic net, 675elliptic point, 399, 405EM, see missing data problemsempirical cost function, 673entropy, 94, 387entropy coding, 586envelope, 410epipolar

geometry, 198–199line, 198, 199, 202, 210plane, 198

epipolar constraint, 199affine fundamental matrix, 233–235

parameterization, 235essential matrix

characterization, 201parameterization, 200

fundamental matrix, 201characterization, 201

Longuet–Higgins relation

uncalibrated, 201Longuet-Higgins relation

calibrated, 200instantaneous, 218

epipoles, 199erosion, 500, 501error rate, see classifieressential matrix, see epipolar constraintestimating scale with Laplacian of Gaussian

cornerLaplacian, 151

Euclidean geometryEuclidean shape, 223, 246rotation matrix

exponential representation, 219quaternions, 434Rodrigues formula, 455

Euclidean structure from motionambiguity, 224definition, 222from binocular correspondences, 224–

228Euclidean upgrade

from affine structure, 238–240multiple views, 239–240

from projective structure, 246–248partially-calibrated cameras, 246–248

Euler angles, 15exemplars, 609EXIF tags, 17exitance, 55expectation-maximization, see EMexterior orientation, 27extremal points, 564extrinsic parameters, 18

affine camera, 22perspective camera, 18–19

f-number, 9, 206false negative, see classifierfalse positive, see classifierfalse positive rate, 466familiar configuration, see segmentation, see

segmentationfeature tracking, 137feature vectors, 457field curvature, 11field of view

camera, 9human eye, 13

figure-ground, see segmentation

INDEX 745

filtering, see linear filteringfinite difference, 110finite differences, 112, 142

choice of smoothing, 145derivative of Gaussian filters, 142, 144,

145differentiating and smoothing with one

convolution, 142smoothing, 141, 142, 144

finite-dimensional linear model, see color,modeling image

first fundamental form, 425first-order geometric optics, see paraxial →

geometric opticsfitting, see segmentation, 290

curves, 297parametric curves, distance from a

point to, 299Hough transform, 290

accumulator array, 292for lines, 290–292practical issues, 292

identifying outliers with EM, 312, 313layered motion models with EM, 313,

318–320least squares, 294, 295

outliers, 299sensitivity to outliers, 299, 300

linesby least squares, 294, 295by total least squares, 294, 295fitting multiple lines, 296identifying outliers with EM, 312, 313incremental line fitting, 296k-means, 296

outliers, see robustnessplanes, 295robust, see robustnesstokens, 293total least squares, 294, 295using k-means, 296

flecnodalcurve, 411, 412, 414, 417point, 411, 412

flow, see graphflow-based matching, 334flowers, see datasetsFM beat, 424focal points, 9focus of expansion, see optical flowfold

of the Gauss mapplane curve, 396surface, 400, 404

point, 391, 404footskate, see peoplefor mosaics

bundle adjustment, 374forest, see graphforward selection, 673forward variable, see peopleFourier transform, 118, 119

as change of basis, 118, 119basis elements as sinusoids, 119definition for 2D signal, 119inverse, 120is linear, 120of a sampled signal, 126pairs, 120phase and magnitude, 120

magnitude spectrum of image unin-formative, 121, 122

sampling, see samplingfovea, 13frame-bearing group, 369free-form surface, 441Frobenius norm, 247, 669frontier point, 561, 564fronto-parallel plane, 6fundamental matrix, see epipolar constraint

gamut, 81gate, 350Gauss map

differential of the, 400, 425fold, 396, 400, 404plane curve, 396space curve, 397surface, 399

Gauss sphere, 399, 439Gauss–Newton, see nonlinear least squaresGaussian curvature, 401, 404, 406, 425Gaussian image

curve, 396, 404surface, 399, 401

Gaussian kernel, 474Gaussian smoothing, 426generalized

eigenvalue problem, 666eigenvector, 666

generalized eigenvalue problem, 496generalized polyhedron, 561

INDEX 746

generalizing badly, 460generative model, 306generic

surface, 409geoconsistency, 572geometric consistency

in pedestrian detection, 528geometrical modes, 65geometry

differential, see differential geometrygestalt, see segmentationGIST features, see image classificationglobal shading model, 52

color bleeding, 59comparing black and white rooms, 56governing equation, 56interreflections, 35reflexes, 59smoothing effect of interreflection, 57

gradient, estimatingdifferentiating and smoothing with one

convolution, 142using derivative of Gaussian filters, 142,

145using finite differences, 110

noise, 112smoothing, 141, 144

graph, 277agglomerative clustering using, 280, 281circuit, 278connected, 278connected components, 279connected graph, 278consecutive, 278cut, 279degree, 277directed graph, 278divisive clustering using, 281–283flow, 279forest, 279min-cut, 283path, 278self-loop, 278spanning tree, 279tree, 278undirected graph, 278weighted graph, 278

graph cuts, see min-cut/max-flow problemsand combinatorial optimization

graphscapacity, 279

Grassman’s laws, see color perceptionGraz-02, see datasetsgrouping, see segmentation, see fittinggutterpoint, 408gzip, 586

half-angle direction, 40hard negative mining, see classifier, 535hard thresholding, 183Harris corner detector, see cornerheight map, 47Hessian, 671hidden Markov models, see also matching

on relations, see also people, seealso tracking, see also tree struc-tured energy models

dynamic programming, 594homogeneous Markov chain, 591Markov chain, 591state transition matrix, 591stationary distribution, 592Viterbi algorithm, 594

hierarchical k-means, see clusteringhigh dynamic range image, 39highlights, see specularhinge loss, see classifierhistogram equalization, 521histogram intersection kernel, see classifierHOG feature, 155, 157, 159, 524–526

software, 160HOG features

difficulties with, 546homogeneous

projection matrices, 19homogeneous Markov chain, see hiddenMarkov

modelshomography, see projective transformation,

372, 589homotopy continuation, 419horopter, 204Hough transform, see fittingHSV space, see color spaceshue, see color spaceshuman

eye, 12–14blind spot, 14cones, 13fovea, 13Helmoltz’s schematic eye, 13macula lutea, 13rods, 13

INDEX 747

stereopsis, 203–205, 217cyclopean retina, 205horopter, 204monocular hyperacuity threshold, 204random dot stereogram, 204

human parser, see peopleHumanEva dataset, see datasetshyperbolic point, 391, 399, 405hypernyms, 511hyponyms, 511hysteresis, see edge detection

ICP algorithm, see iterated closest-point al-gorithm

illumination cone, 65illusory contour, 260illusory contours, see segmentationimage browsing, 629image classification, see also classifier

as information retrieval, 639between-class variation, 482classifying images of single objects, 504

evaluation, 505general points, 505

correlated image keywords, 649, 650dataset bias, 515datasets, 512, 513

birds, 515bottles, 514Caltech 101, 510Caltech 256, 510Caltech-101, 513Caltech-256, 513camels, 514crowdsourcing, 515flowers, 514Graz-02, 514Imagenet, 514LabelMe, 513Lotus Hill data, 514materials, 515Pascal challenge, 509, 514repositories, 515scenes, 515

evaluationF -measures, 506F1-measure, 506Fβ-measure, 506average precision, 507news search example, 506patent search example, 506

precision, 506recall, 506web filtering example, 506

example applicationsexplicit images, 482, 498material classification, 483, 502scene classification, 484, 502

featurescoding, 544contour features, 546general points, 484geometric representations, 548GIST features, 486histogram equalization, 521HOG feature, 524–526pooling, 545preclustering, 546shading features, 547spatial pyramid kernel, 489visual words, 488, 639

linking faces to names, 651, 652output

affordances, 549attributes, 550–553

predicting keywords for images, 646–648, 654, 656

software, 512color descriptor, 513course software, 513GIST, 513link repository, 513pyramid match, 513VLFeat, 513

specialized problems, 511state of art

number of classes, 509performance on fixed classes, 508

within-class variation, 482image completion, 176

by matching, 180by texture synthesis, 181methods, 179state of the art, 182

image contour, 391convexities and concavities, 405curvature, 404cusp, 391, 403, 404inflection, 403, 404Koenderink’s theorem, 404–407T-junction, 391

image denoising, 182–186

INDEX 748

BM3D, 183learned sparse coding, 184non-local means, 183results, 186

image hole filling, 176by matching, 180by texture synthesis, 181methods, 179state of the art, 182

image irradiance equation, 59image mosaic, 370image plane, 4image pyramid, 134, see also scale, 135

coarse scale, 135Gaussian pyramid, 135

analysis, 135, 136applications, 136, 137

image rectification, 202–203, 205image search, see digital libraries

correlated image keywords, 649, 650linking faces to names, 651, 652predicting keywords for images, 646–

648, 654, 656using keywords, 645

image stabilization, 330image-based modeling and rendering, 221,

559PMVS, 573visual hulls, 559

Imagenet, see datasetsimpulse response, 114, see convolutionincremental fitting, see fittingindex of refraction, 9, 11inertia

axis of least, 667second moments, 667

inflection, 394, 396, 399, 408, 412, 413information retrieval, 632

distributional semantics, 634latent semantic analysis, 634latent semantic indexing, 634pagerank, 638

for image layout, 642query expansion, 639ranking by importance, 638stop words, 632strategies applied to image classifica-

tion, 639tf-idf weighting, 633word counts, 632

smoothing, 633

integrability, 50in lightness computation, 45in photometric stereo, 50

integral image, 522interactive segmentation, see segmentationinterest point, see cornerinterior orientation, 27interior-point methods, 677interpretation tree, 438, 454interreflections, see global shading modelintrinsic parameters, 17

affine camera, 22perspective camera, 16–18

intrinsic representations, 43invariant image, see shadow removalinverted index, 632isotropy, see textureiterated closest points, 369iterated closest-point algorithm, 434–436IXMAS dataset, see datasets

Jacobian, 670joint angles, see peoplejoint positions, see people

k-d tree, see nearest neighbors, 637software, 638

k-d trees, 435k-means, see clusteringk-nearest neighbor classifier, see classifierKalman filter, 345Kalman filtering, see trackingkernel, see convolution, see convolution, see

classifierkernel profile, 274, 336kernel smoother, 336kernel smoothing, 273key frame, see shot boundary detectionKinect, 446–453Koenderink’s theorem, 404–407KTH action dataset, see datasetsKy-Fan norm, 649

LabelMe, see datasetsLagrange multiplier, 673Lambert’s cosine law, 34Lambertian+specular model

image color model, 91, 92lambertian+specular model, 36Laplacian, see estimating scale with Lapla-

cian of Gaussian

INDEX 749

Lasso, 184latent semantic analysis, see information re-

trievallatent semantic indexing, see information

retrievallatent variable, 531layered motion, 318, see also fitting, see also

optical flowsupport maps, 319

learned sparse coding, 184–185least squares, see fitting, 663

linear, 201homogeneous, 436non-homogeneous, 441

nonlinear, 202least-angle regression, 673leave-one-out cross-validation, 322, see clas-

sifierlenses

depth of field, 9f number, 9

level set, 436Levenberg–Marquardt, see nonlinear least

squaresLHI, see datasets287LIBSVM, see classifierlight field, 559, 584–586light slab, 585lightness, 44, see color spaceslightness computation, 44

algorithm, 43, 45assumptions and model, 44constant of integration, 45

lightness constancy, 44limiting bitangent developable, 415line space, 291linear, see properties, 113linear filtering, see convolution, see linear

systems, shift invariantlinear least squares, 663–669

homogeneous, 665–669eigenvalue problem, 666generalized eigenvalue problem, 666

nonhomogeneous, 664–665normal equations, 664pseudoinverse, 665

linear systems, shift invariantconvolution like a dot product, 131–

133filtering as output of linear system, 107

filters respond strongly to signals theylook like, 131

impulse response, 117point spread function, 117properties, 112

scaling, 113superposition, 113

response given by convolution, 1151D derivation, 1152D derivation, 117discrete 1D derivation, 113discrete 2D derivation, 114

lines of curvature, 421lip, 412local shading model, 36Local texture representations, 164local visual events, 407, 412–413locality sensitive hashing, see nearest neigh-

bors, 636software, 638

logistic loss, see classifierlogistic regression, see classifierLonguet–Higgins relation, see epipolar con-

straint, 200loss, see classifierLotus Hill data, see datasetsluminaires, 33

M-estimator, see robustnessM-step, see missing data problemsmacula lutea, 13magenta, see color spacesmagnetic resonance imaging, see medical imag-

ingmagnification, 6magnitude spectrum, see Fourier transformMahalanobis distance, see classifiermanifold, 414MAP inference, 213marching cubes, 437markerless motion capture, 589Markov chain, 339, see hiddenMarkov mod-

elsMarkov models, hidden, see hidden Markov

models, see also matching on re-lations, see also people, see alsotracking

matching on relationshidden Markov models, 590

backward variable, 598dynamic programming, 594

INDEX 750

dynamic programming algorithm, 595dynamic programming figure, 596example of Markov chain, 592fitting a model with EM, 595, 597,

598forward variable, 598node value, 594trellis, 592, 593Viterbi algorithm, 594, 595

pictorial structure models, 602, 604, 605,608, 610

tree-structured energy models, 600material properties, 164materials, see datasetsmatrix

nullspace, 24range, 664rank, 664

matte, 266max pooling, see poolingMDL, see model selectionMDS, see multidimensional scalingMean average precision, 508mean shift

tracking with, 335medical imaging

applications of registration, 383, 387atlas, 384imaging techniques, 384

computed tomography imaging, 385magnetic resonance imaging, 385nuclear medical imaging, 385ultra-sound imaging, 385

metric shape, see Euclidean shapeMeusnier’s theorem, 401Mie scattering, see color sourcesmin-cut/max-flow, 214min-cut/max-flow problems and combina-

torial optimization, 675–682min-cuts, 663minimum description length, see model se-

lectionmissing data problem, 307missing data problems, 307

EM algorithm, 310, 311background subtraction example, 314complete data log-likelihood, 309E-step, 311fitting HMM with, 595, 597, 598incomplete data log-likelihood increases

at each step, 311

M-step, 311motion segmentation example, 313,

318–320outlier example, 312, 313practical difficulties, 312soft weights, 311

image segmentation, 308iterative reestimation strategy, 310layered motion, 318mixture model, 309

mixing weights, 309mixture, 309

mixing weights, see missing data problemsmixture, see missing data problemsmixture model, see missing data problemsmobile robot

navigation, 197, 221model selection, 319

AIC, 321Bayes information criterion, 321Bayesian, 321BIC, 321cross-validation, 322MDL, 321minimum description length, 321overfit, 321selection bias, 320test set, 320training set, 320

model-based visionalignment, 377application in medical imaging, 383, 384,

387verification, 377

by edge proximity and orientation,378

by edge proximity is unreliable, 378,379

Mondrian worlds, 96Monge patch, 47, 425motion capture, 451, see peoplemotion field, 218motion graph, see peoplemotion primitives, see peoplemovies and scripts datasets, see datasetsMSR action dataset, see datasetsMuller-Lyer illusion, 256multidimensional scaling, 644

for image layout, 645multilocal visual events, 407, 414–416multiple kernel learning, 475, see classifier

INDEX 751

multiple-view stereo, see stereopsis → mul-tiple views

Munsell chips, 100mutual information, 387

N-cut, see clusteringnaive Bayes, see classifiernarrow-baseline stereopsis, 217, 573near duplicate detection, 628, 639

using hierarchical k-means, 640copyright, 628trademark, 628using LSH, 640using visual words, 639

nearest neighbor classifier, see classifiernearest neighbors, 350

approximate algorithms, 635, 637best bin first, 637k-d tree, 636locality sensitive hashing, 635software, 638

correlated image keywords, 649, 650linking faces to names, 651, 652near duplicate detection, 640predicting keywords for images, 647,

648neural net, 522Newton’s method

convergence rate, 670nonlinear equations, 670nonlinear least squares, 670–671

node value, see matching on relationsnoise

additive stationary Gaussian noise, 142,143

choice of smoothing filtereffect of scale, 145

smoothing to improve finite differenceestimates, 141, 142, 144

non-local means, 183non-maximum suppression, see detectionnon-square pixels, 124nonlinear least squares, 669–672

Gauss–Newton, 671–672convergence rate, 672

Levenberg–Marquardt, 672Newton, see Newton’s method

nonmaximum suppression, see edge detec-tion

normalcurvature, 398

line, 393, 398plane, 397principal, 397section, 398vector, 393, 398

normal equations, 665normal section, 399normalized correlation, 133, 206, 445, 576normalized cut, see clustering, 285normalized image plane, 16nuclear medical imaging, see medical imag-

ingNyquist’s theorem, 126

object model acquisitionfrom range images, 436–438

object recognitionaffordances, 549aspect, 547attributes, 550–553categorization, 542–543current strategies, 542desirable properties, 540–541from range images, 438–446geometric representations, 548part representations, 553poselets, 553, 554selection, 544visual phrases, 554, 555

decoding, 555, 556observations, 327occluding boundaries

detecting, 527, 529evaluation, 530method, 531

occluding contour, 141, 391cusp point, 391, 404fold point, 391, 404

Olympic sports dataset, see datasetsone-vs-all, see classifierOpenCV, 30opening point, 564opponent color space, see color spacesoptical axis, 8optical flow, 313, 589

focus of expansion, 315layered motion, 318parametric models of

affine motion model, 316more general, 317

segmentation by, 316, 318–320

INDEX 752

yields time to contact, 315ordering constraint, 210orientation, 147orientations, 144, 147

affected by scale, 150differ for different textures, 151do not depend on intensity, 149in HOG features, 159in SIFT descriptors, 157

orthogonal matching pursuit, 673osculating plane, 397outliers, see robustnessoutline, see image contourovercomplete dictionaries, 672overfit, see model selectionoverfitting, 460overlap test, see detectionoversegmentation, 268

Pagerank, 639pagerank, see information retrieval

for image layout, 642parabolic

curve, 407–409, 411, 417point, 410, 412

parabolic point, 391, 399, 404, 405paraboloid, 399parallelism, see segmentationparametric

curve, 420surface, 41

parametric models ofoptical flow, 316

parametric surface, 424part representations, see object recognitionparts, 532, 551Pascal challenge, see image classification,

see datasetspassive markers, see peoplepatch-based multi-view stereopsis, 573–584path, see graphpattern elements

describing neighborhoods, 155, 157finding with corner detector, 154finding with Laplacian of Gaussian, 155shape context, 613software, 160yield covariant windows, 156

PCA, see classifierPEGASOS, see classifierpenumbra, 37

people3D from 2D, 611, 612, 614

ambiguities, 613snippets, 616

activity is compositional, 619composition across the body, 620motion primitives, 619

activity recognition, 617, 619by characteristic poses, 618by poselets, 618, 619by spacetime features, 621–623from compositional models, 621, 624,

625datasets, see datasetsdetecting, 525, 526, 528

evaluation, 527, 537hidden Markov models, 590

backward variable, 598dynamic programming, 594dynamic programming algorithm, 595dynamic programming figure, 596example of Markov chain, 592fitting a model with EM, 595, 597,

598forward variable, 598trellis, 592, 593Viterbi algorithm, 594, 595

human parser, 602pictorial structure models, 602, 604,

605motion capture, 617

active markers, 617computed edges, 620footskate, 617joint angles, 617joint positions, 617motion graph, 620passive markers, 617skeleton, 617

pictorial structure models, 608, 610software, see softwaretracking, 606

by appearance, 608, 610by templates, 609is hard, 606

tree-structured energy models, 600perceptual organization, see segmentationperspective

camera, see camera model → perspec-tive

effects, 5

INDEX 753

projection matrix, see projection ma-trix → pinhole perspective

phase spectrum, see Fourier transformphotoconsistency, 572photogrammetry, 27, 197Photometric stereo, 47photometric stereo

depth from normals, 49formulation, 49integrability, 45, 50normal and albedo in one vector, 48recovering albedo, 49recovering normals, 49

pictorial structure models, see peoplepinhole, 3

camera, 4–6pinhole perspective, 4planes, representing orientation of

slant, 188, 189tilt, 188, 189tilt direction, 188, 189

plenoptic function, 584PMI dataset, see datasetsPMVS, see patch-based multi-view stereop-

sis, 574point spread function, see convolutionpooled texture representations, 164pooling, see image classification

average pooling, 545max pooling, 545

pose, 367pose consistency, 367poselet, 552poselets, see object recognition

for activity recognition, 618, 619posterior, 340potential patch neighbor, 577potentially visible patch, 576pragmatics, 657precision, see image classificationpreclustering, see image classificationpredictive density, 340primaries, see color perceptionprimary aberrations, 11principal

curvatures, 399, 426directions, 398, 403, 425

principal component analysis, see classifierprincipal points, 10principle of univariance, see color percep-

tion

prior, 339probabilistic data association, 350probability distributions

normal distributionimportant in tracking linear dynamic

models, 344sampled representations, 351

probability of boundaryPb, 528

probability, formal models ofexpectation

computed using sampled representa-tions, 351, 352

integration by samplingsampling distribution, 351

representation of probability distribu-tions by sampled representations,352

marginalizing a sampled representa-tion, 353

prior to posterior by weights, 354,355

projection equationaffine, 22

orthographic, 7weak-perspective, 6

pinhole perspective, 6points, 18

projection matrixaffine, 22

characterization, 22weak-perspective, 22

pinhole perspectivecharacterization, 19–20explicit parameterization, 19general form, 18

projection modelaffine, 6–7

orthographic, 7paraperspective, 21weak-perspective, 6

pinhole perspectiveplanar, 4–6

weak perspective, 20–22projective, 230projective coordinate system, 242projective geometry

projective shape, 242projective transformation, 241

projective projection matrix, 241projective shape, 241

INDEX 754

projective structure from motionambiguity, 241definition, 241Euclidean upgrade, 246–248

partially-calibrated cameras, 246–248from multiple images

bilinear method, 244–245bundle adjustment, 245factorization, 244

from the fundamental matrix, 242–243projective transformations, 15properties

linear systems, shift invariantlinear, 107shift invariant, 107, 113

proximity, see segmentationpseudoinverse, 665pulse time delay, 424pyramid kernel, see image classification

QR decomposition, 665, 666quaternions, 433–434, 436, 440, 454query expansion, see information retrievalQuickTime VR, 588

radialcurvature, 405curve, 404distortion, 27

radiancedefinition, 52units, 53

radiometric calibration, 38radiosity, 54

of a surface whose radiance is known,54

definition and units, 54radius of curvature, 395random dot stereogram, 204random forest, 448Random forests, 448–450random forests, 446range finders, 422–424

acoustico-optical, 424time of flight, 423triangulation, 422

range image, 422ranking, see PagerankRANSAC, see robustnessratio, 232Rayleigh scattering, see color sources

recall, see image classificationreceiver operating characteristic curve, see

classifierestimating performance

ROC, 466reciprocity, 38reflectance, see albedo

color, physical terminology, 76reflexes, 58region, 164region growing, 431regional properties, 58regions, 256registration

from planes, 439–441from points, 434–436

regular point, 394regularization, 213, see classifierregularization term, 672regularizer, 463relative reconstruction, 611render, 377repetition, 164repositories, see datasetsrest positions, 619retargeting, 451Retinex, 63RGB color space, see color spacesRGB cube, see color spacesrigid transformation, 15rigid transformations and homogeneous co-

ordinates, 14–16rim, see occluding contourringing, see convolutionrisk, see classifierrisk function, see classifierrobustness, 299

identifying outliers with EM, 312, 313M-estimator, 300, 303, 304

influence function, 301M-estimators, 301

scale, 302outliers

causes, 299sensitivity of least squares to, 299,

300RANSAC, 302

how many points need to agree?, 305how many tries?, 303how near should it be?, 305searching for good data, 302

INDEX 755

ROC, see receiver operating characteristiccurve

Rodrigues formula, 455rods, 13roof, 426roof edge, 161root, 532root coordinate system, 611rotation matrices, 15Rotoscoping, 266ruled surface, 411

saddle-shaped, see differential geometry →surfaces → hyperbolic

sample impoverishment, 357sampling, 121, 122

aliasing, 123, 125–130formal model, 122, 123, 125Fourier transform of sampled signal, 126illustration, 124non-square pixels, 124Nyquist’s theorem, 126poorly causes loss of information, 123

sampling distribution, see probability, for-mal models of

saturation, see color spacesscale, 134, see smoothing

anisotropic diffusion or edge preservingsmoothing, 138

applications, 136coarse scale, 135effects of choice of scale, 145of an M-estimator, 302

scale ambiguity, see ambiguity → Euclideanscale space, 426scaled orthography, see weak perspectivescaling, see linear systems, shift invariantscan conversion, 443scene classification, see scenesscenes, 483, see datasets

scene classification, 484searching for images, see digital librariessecant, 393second fundamental form, 400, 425segmentation, 164, 255, see also clustering,

see also fitting, 424as a missing data problem, 308, see

missing data problemsby clustering, general recipe, 268example applications, 261

background subtraction, 261–263

shot boundary detection, 264gestalt, 257human, 256

closure, 258common fate, 258common region, 258continuity, 258examples, 258–261factors that predispose to grouping,

257–261familiar configuration, 258, 260figure and ground, 256, 257gestalt quality or gestaltqualitat, 256,

257illusory contours, 260parallelism, 258proximity, 257similarity, 257symmetry, 258

in humansexamples, 255

interactive segmentation, 261range data, 424–432trimaps, 281watershed, 271

selection bias, see also model selection, 460self-calibration, 250self-loop, see graphself-similar, see cornerself-similarities, 183semi-local surface representation, 441SFM, 221shading, 33shading primitives, 65shadow, 35shadow removal, 92

color temperature direction, 94estimating color temperature direction,

94examples, 95general procedure, 93invariant image, 94

shadowsarea sources, 36, 37penumbra, 37umbra, 36

shapeaffine, see affine geometryEuclidean, see Euclidean geometryprojective, see projective geometry

shape context, see pattern elements

INDEX 756

shape from shading, 59–61shape from texture

for curved surfaces, 190repetition of elements yields lighting,

190shift invariant, see propertiesshift invariant linear system, see linear sys-

tems, shift invariantshot boundary detection, 261, 264

key frame, 264shots, 264

shots, see shot boundary detectionshrinkage, 183SIFT descriptor, 155, 157–159

software, 160SIFT descriptors

difficulties with, 546silhouette, see image contoursimilarity, 223, see segmentationsimplex method, 576simulated annealing, 683singular point, 394singular value decomposition, 237, 244, 666–

669skeleton, 563, see peopleskinning, 451skylight, see color sourcesslack variables, 472slant, 188, 189

normal is ambiguous given, 189, 191smoothing, 108

as high pass filtering, 126, 128–130Gaussian kernel, 109

discrete approximation, 110Gaussian smoothing, 108, 109

avoids ringing, 108, 109discrete kernel, 110effects of scale, 110, 145standard deviation, 109suppresses independent stationary ad-

ditive noise, 111scale, 143to reduce aliasing, 126, 128–130weighted average, 107word counts, see information retrieval

Snell’s law, 8snippets, see peoplesoft thresholding, 183soft weights, see missing data problemssoftware

active appearance models, 383

approximate nearest neighbors, 638classifier

LIBSVM, 480multiple kernel learning, 480PEGASOS, 480SVMLight, 480

deformable object detection, 535deformable registration, 383face detection, 539FLANN, 638general object detection, 539homography estimation, 373image classification

color descriptor, 513course software, 513GIST, 513link repository, 513pyramid match, 513VLFeat, 513

image segmenters, 285pattern elements, 160

color descriptors, 160HOG feature, 160PCA-SIFT, 160toolbox, 160VLFEAT, 160

people, 624sources

source colors, 88, 89space carving, 587spanning tree, see graphsparse coding, 663, 672sparse coding and dictionary learning, 672–

675dictionary learning, 673–675sparse coding, 672–673supervised dictionary learning, 675

sparse model, 183spatial frequency

see Fourier transform, 118spatial frequency components, 119spectral albedo, see albedo

color, physical terminology, 76spectral colors, 68spectral energy density, see color, physical

terminologyspectral locus, 83spectral reflectance, see albedo

color, physical terminology, 76specular

dielectric surfaces, 90

INDEX 757

metal surfaces, 90specularities, 90specularity

finding, 90, 91specular albedo, 34specular direction, 34specular reflection, 34specularities, see specularspecularity, 34spherical

aberration, 10spherical panorama, 371spin

coordinates, 442images, 441–446map, 442

SSD, see sum-of-squared differencesssd, see sum of squared differencesstandard deviation, see smoothingstate, 327state transition matrix, see hidden Markov

modelsstationary distribution, see hidden Markov

modelsstep, 426step edges, 161stereolithography, 438stereopsis

binocular fusioncombinatorial optimization, 211–214dynamic programming, 210–211global methods, 210–214local methods, 205–210multi-scale matching, 207–210normalized correlation, 205–207

constraintsepipolar, 198ordering, 210

disparity, 197, 202, 203multiple views, 214–215random dot stereogram, 204reconstruction, 201–203rectification, 202–203robot navigation, 215–216trinocular fusion, 214

stop words, see information retrievalstructured light, 422stuff, 658submodularity, 213, 663subtractive matching, see color perceptionsum of absolute difference, 207

sum of squared differences, 177sum-of-squared differences, 330

SSD, 330summary matching, 334superpixels, 268superposition, see linear systems, shift in-

variantsuperquadrics, 454support maps, see layered motionsurface color, see color perceptionSVMLight, see classifierswallowtail, 412symmetric Gaussian kernel, see smoothingsymmetry, see segmentationsystem, see linear systems, shift invariantsystem identification, 362

T-junction, 391, 404, 407, 412–416tangent

crossing, 415line, 393plane, 398vector, 393

template matchingfilters as templates, 131

test error, 459test set, see model selectiontexton, 166texture

examples, 164, 165isotropy, 188local representations, 166–171pooled representations, 171, 173–175representing with filter outputs, 166

algorithm, 169example, 169–171published codes, 170scheme, 167typical filters, 168

representing with vector quantization,172

algorithm, 172example, 174, 175scheme, 173

scale, 164shape from texture, 187

for planes, 187–189synthesis, 176, 178

algorithm, 177example, 178, 179

texton, 166

INDEX 758

texture mapping, 559, 569, 585texture synthesis

for image hole filling, 181tf-idf weighting, see information retrievalthick lenses, 10

principal points, 10thin lenses, 9

equation, 9focal points, 9

tilt, 188, 189tilt direction, 188topics, 634total least squares, see fittingtotal risk, see classifiertracking

applicationsmotion capture, 326recognition, 326surveillance, 326targeting, 326

as inference, 339definition, 326hidden Markov models

backward variable, 598dynamic programming, 594dynamic programming algorithm, 595dynamic programming figure, 596example of Markov chain, 592fitting a model with EM, 595, 597,

598forward variable, 598trellis, 592, 593Viterbi algorithm, 594, 595

Kalman filters, 344example of tracking a point on a line,

344, 345forward-backward smoothing, 345, 347,

348linear dynamic models

all conditional probabilities normal,344

are tracked using a Kalman filter,qv, 344

constant acceleration, 342, 343constant velocity, 341, 342drift, 341periodic motion, 343

main problemscorrection, 340data association, 340prediction, 339

measurementmeasurement matrix, 341observability, 341

particle filtering, 350practical issues, 360sampled representations, 351–355simplest, 355simplest, algorithm, 356simplest, correction step, 356simplest, difficulties with, 357simplest, prediction step, 355working, 358–361working, by resampling, 358

smoothing, 345, 347, 348tracking by detection, 327tracking by matching, 327

tracking by detection, see trackingtracking by matching, see trackingtrademark, see near duplicate detectiontrademark evaluation, 628Training error, 459training set, see model selectiontransformation groups

affine transformations, 231projective transformations, 241similarities, 223

tree, see graphtree-structured models

binary terms, 600cost-to-go function, 601unary terms, 600

trellis, see matching on relationstrichromacy, see color perceptiontrimaps, see segmentationtrinocular fusion, see stereopsis → trinocu-

lar fusiontriple point, 415, 416tritangent, 415true positive rate, see classifiertwisted cubic, 25twisted curves, see differential geometry →

space curves

UCF activity datasets, see datasetsultra-sound imaging, see medical imagingumbra, 36unary terms, see tree-structured modelsundirected graph, see graphundulation, 410unode, 416

value, see color spaces

INDEX 759

Vector quantization, 172vector quantization, 586vergence, 204, 217viewing

cone, 391cylinder, 391

viewing sphere, 416viewpoint

general, 404vignetting, 12VIRAT dataset, see datasetsvirtual image, 4visual events, 392, 411

curves, 411equations, 421local, 412–413

beak-to-beak, 412lip, 412swallowtail, 412

multilocal, 414–416cusp crossing, 415, 416tangent crossing, 415triple point, 415, 416

visual hull, 417, 559–573visual phrases, see object recognitionvisual potential, see aspect graphvisual words, 487, see image classification,

639recovering suppressed detail, 640

Viterbi algorithm, see hidden Markov mod-els

see hidden Markov models, 594voxel, 437

watershed, see segmentationwavelet shrinkage, 183weak calibration, 224–226

eight-point algorithmminimal, 225normalized, 225overconstrained, 225

nonlinear, 225weak learner, see classifierweak perspective, 7

projection matrix, see projection ma-trix → affine → weak-perspective

weighted graph, see graphWeizmann activity, see datasetswide-baseline, 215wide-baseline stereopsis, 217, 574window, see chaff

within-class variance, 495within-class variation, see image classifica-

tionword counts, see information retrieval

yellow, see color spaces

zero-skew projection matrix, 19zippered polygonal mesh, 454

List of Algorithms

2.1 Determining the Lightness of Image Patches. . . . . . . . . . . . . . . . . 452.2 Photometric Stereo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1 Subsampling an Image by a Factor of Two. . . . . . . . . . . . . . . . . . 1294.2 Forming a Gaussian Pyramid. . . . . . . . . . . . . . . . . . . . . . . . . . 1365.1 Gradient-Based Edge Detection. . . . . . . . . . . . . . . . . . . . . . . . . 1465.2 Obtaining Location, Radius and Orientation of Pattern Elements Using a

Corner Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.3 Obtaining Location, Radius, and Orientation of Pattern Elements Using

the Laplacian of Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.4 Computing a SIFT Descriptor in a Patch Using Location, Orientation and

Scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.5 Computing a Weighted q Element Histogram for a SIFT Feature. . . . . . 1596.1 Local Texture Representation Using Filters. . . . . . . . . . . . . . . . . . 1726.2 Texture Representation Using Vector Quantization. . . . . . . . . . . . . . 1736.3 Clustering by K-Means. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.4 Non-parametric Texture Synthesis. . . . . . . . . . . . . . . . . . . . . . . 1777.1 The Marr–Poggio (1979) Multi-Scale Binocular Fusion Algorithm. . . . . . 2087.2 A Dynamic-Programming Algorithm for Establishing Stereo Correspon-

dences Between Two Corresponding Scanlines. . . . . . . . . . . . . . . . . 2128.1 The Longuet-Higgins Eight-Point Algorithm for Euclidean Structure and

Motion from Two Views. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2288.2 The Tomasi–Kanade Factorization Algorithm for Affine Shape from Motion. 2389.1 Background Subtraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639.2 Shot Boundary Detection Using Interframe Differences. . . . . . . . . . . . 2649.3 Agglomerative Clustering or Clustering by Merging. . . . . . . . . . . . . . 2699.4 Divisive Clustering, or Clustering by Splitting. . . . . . . . . . . . . . . . . 2699.5 Finding a Mode with Mean Shift. . . . . . . . . . . . . . . . . . . . . . . . 2759.6 Mean Shift Clustering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2769.7 Mean Shift Segmentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2779.8 Agglomerative Clustering with Graphs. . . . . . . . . . . . . . . . . . . . 28010.1 Incremental Line Fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29610.2 K-means Line Fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29710.3 Using an M-Estimator to Fit a Least Squares Model. . . . . . . . . . . . . 30210.4 RANSAC: Fitting Structures Using Random Sample Consensus. . . . . . . 30511.1 Tracking by Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33011.2 Tracking with the Mean Shift Algorithm. . . . . . . . . . . . . . . . . . . . 33511.3 The Kalman Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34611.4 Forward-Backward Smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . 34711.5 Obtaining a Sampled Representation of a Probability Distribution. . . . . 35211.6 Computing an Expectation Using a Set of Samples. . . . . . . . . . . . . . 35211.7 Obtaining a Sampled Representation of a Posterior from a Prior. . . . . . 35511.8 A Practical Particle Filter Resamples the Posterior. . . . . . . . . . . . . . 35911.9 An Alternative Practical Particle Filter. . . . . . . . . . . . . . . . . . . . 36014.1 The Model-Based Edge-Detection Algorithm of Ponce and Brady (1987). . 42714.2 The Iterative Closest-Point Algorithm of Besl and McKay (1992). . . . . . . 43514.3 The Plane-Matching Algorithm of Faugeras and Hebert (1986). . . . . . . 440

760

LIST OF ALGORITHMS 761

14.4 Pointwise Matching of Free-Form Surfaces Using Spin Images, after Johnsonand Hebert (1998, 1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

14.5 Training a Decision Tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44914.6 Training a Random Forest. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45015.1 The Bayes Classifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45915.2 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46415.3 Multi-class Classification Assuming Class-Conditional Densities are Normal 46815.4 (k, l) Nearest Neighbor Classification . . . . . . . . . . . . . . . . . . . . . 47015.5 Training a Two-Class Decision Stump . . . . . . . . . . . . . . . . . . . . . 47615.6 Discrete Adaboost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47715.7 Real Adaboost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47816.1 Principal Components Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 49516.2 Canonical Variates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49916.3 Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49916.4 Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50117.1 Sliding Window Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52019.1 Visual Hull Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56419.2 A Curve-Tracing Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 56519.3 The Strip Triangulation Algorithm. . . . . . . . . . . . . . . . . . . . . . . 56719.4 The PMVS Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57519.5 The Feature-Matching Algorithm of PMVS. . . . . . . . . . . . . . . . . . 57919.6 The Patch-Expansion Algorithm of PMVS. . . . . . . . . . . . . . . . . . . 58120.1 The Viterbi Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59520.2 Fitting Hidden Markov Models with EM . . . . . . . . . . . . . . . . . . . 59520.3 Computing the Forward Variable for Fitting an HMM . . . . . . . . . . . . 59720.4 Computing the Backward Variable for Fitting an HMM . . . . . . . . . . . 59720.5 Updating Parameters for Fitting an HMM . . . . . . . . . . . . . . . . . . 59821.1 Nearest Neighbor Tagging. . . . . . . . . . . . . . . . . . . . . . . . . . . 64721.2 Greedy Labeling Using Kernel Similarity Comparisons. . . . . . . . . . . 64822.1 The Alpha Expansion Algorithm of Boykov et al. (2001). . . . . . . . . . . 681