1 Spectral Texture Features - 國立中正大學資工系

Wei-Ta Chu

2010/10/21

Spectral Texture Features1

Multimedia Content Analysis, CSIE, CCU

Gabor Texture2

The Gabor representation has been shown to beoptimal in the sense of minimizing the joint two-dimensional uncertainty in space and frequency.

These filters can be considered as orientation andscale tunable edge and line (bar) detectors.

The statistics of these microfeatures in a givenregion are often used to characterize theunderlying texture information.

B.S. Manjunathand W.Y. Ma, “Texture features for browsing and retrieval of image data,” IEEE Trans. on PAMI, vol. 18, no. 8, 1996, pp. 837-842.

Gabor Texture


3

Fourier coefficients depend on the entire image (Global) →we lose spatial information

Objective: local spatial frequency analysis Gabor kernels: looks like Fourier basis multiplied by a

Gaussian Gabor filters come in pairs: symmetric and anti-symmetric

We need to apply a number of Gabor filters at differentscales, orientations, and spatial frequencies

Symmetric kernel

Anti-symmetric kernel

Gabor Texture


4

Image I(x,y) convoluted with Gabor filters hmn (totally M x N)

Using first and 2nd moments for each scale and orientations

Features: e.g., 4 scales, 6 orientations→ 48 dimensions

evenodd

Gabor Texture


5

Arranging the mean energy in a 2D form structured: localized pattern oriented (or directional): column pattern granular: row pattern random: random pattern

scale

orientation

Homogeneous Texture Descriptor6

Frequency plane partition is uniform along the angular direction (30º), non-uniform alongthe radial direction (on an octave scale)


Gabor Function7

On the top of the feature channel, the following 2D Gaborfunction (modulated Gaussian) is applied to each individualchannels.

Equivalent to weighting the Fourier transform coefficients of theimage with a Gaussian centered at the frequency channels asdefined above

Each channel filters a specific type of texture

Homogeneous Texture Descriptor


8

Partition the frequency domain into 30 channels(modeled by a 2D Gabor function)

Computing the energy and energy deviation foreach channel

Computing the mean and standard deviation offrequency coefficients

HTD = {fDC, fSD, e1,e2,…,e30,d1,d2,…,d30}

fDC and fSD are the mean and standard deviation of the imageei and di are the mean energy and energy deviation of the corresponding ith channel

Distance Measure9

Resources: http://vision.ece.ucsb.edu/texture/feature.htmlOn-line demo: http://vision.ece.ucsb.edu/texture/mpeg7/index.html


Example: Browsing Satellite Images


10

Find a vegetation patch that looks like this region


Example: Browsing Satellite Images


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(b) parts of highway (c) region containing some buildings (center of the image

toward the left) (d) a number marked on the image (lower left corner)

Wavelet Features12

Wavelet transforms refer to the decomposition of a signal witha family of basis functions with recursive filtering andsubsampling

At each level, it decomposes a 2D signal into four subbands,which are often referred to as LL, LH, HL, HH (L=low, H=high)

LL2 HL2HL1

LH2 HH2

LH1 HH1

Wavelet Features13

Using the mean and standard deviation of the energydistribution in each subband at each level.

PWT (Pyramid-structured wavelet transform) Recursively decompose the LL band Results in 30-dimensional feature vector (3x3x2+2=30)

TWT (Tree-structured wavelet transform) Some information appears in the middle frequency channels–

decomposition is not restricted to the LL band Results in 40x2 = 80 dimensional feature vector

Original image PWT TWT

T. Chang and C.C.J. Kuo, “Texture analysis and classification with tree-structure wavelet transform,” IEEE Trans. On Image Processing, vol. 2, no. 4, 1993, pp. 429-441.

Wei-Ta Chu

2010/10/21

Edge Histogram Descriptor14


Park, et al. “Efficient use of local edge histogram descriptor,” Proc. of ACM International Workshop on Standards, Interoperability and Practices, pp. 51-54, 2000.

Introduction15

Spatial distribution of edges Edge histogram descriptor (EHD)

Dividing the image into 4x4 subimages, and generatethe edge histogram based on the edges in thesubimages. Edges are categorized into five types: vertical, horizontal,

45º diagonal, 135º diagonal, and nondirectional edges. A total of 5x16=80 histogram bins

Local Edge Histogram16

Global, Semi-global, and LocalHistograms


17

Global-edge histogram Accumulate five types of edge distributions for all subimages

Semiglobal-edge histogram

Image Matching


18

Combining the local, the semiglobal, and global histogramtogether.

Total of 150 bins 80 bins (local) + 5 bins (global) + 65 bins (13x5, semiglobal)

The L1 distance measure D(A,B) can be:

This feature is one of the MPEG-7 texture descriptors.

Performance Comparison19

Retrieval performance of different texture features for the Corel photo databases.

L1 distance is used to computing the dissimilarity between images.

For the MRSAR, Mahalanobis distance is used.

MRSAR (M)

GaborTWTPWT

MRSAR

Tamura (improved)

Coarseness histogramDirectionalityEdge histogramTamura (traditional)

#relevant images

#top matches considered

Manjunath and Ma, Chapter12 of Image Database:Search and Retrieval of DigitalImagery, edited by V. Castelliand L.D. Bergman, John Wiley& Sons, 2002.

Performance Comparison20

Retrieval performance of different texture featuresfor the Brodatz texture image set.

MRSAR (M)Gabor

TWTPWT

MRSARTamura (improved)

Coarseness histogramDirectionalityEdge histogram

Tamura (traditional)

#top matches considered

Percentage ofretrieving allcorrect patterns

Wei-Ta Chu

2010/10/21

Shape for CBIR21


Shape Features


22

MPEG-7 provides contour-based shape and region-based shape tools.

contour-basedsimilarity

region-basedsimilarity

Bober, “MPEG-7 visual shapedescriptors”, IEEE Trans. On CSVT, vol. 11, no. 6, pp. 716-719, 2001.

Region-Based Shape Descriptor


23

The region-based SD expressed pixel distributionwithin a 2D object or region.

It can describe complex objects consisting ofmultiple disconnected regions.

2D Angular Radial Transformation (ART)Gives a compact and efficient way of describing

multiple disjoint regionsRobust to segmentation noise

Angular Radical Transform (ART)


24

For each image, a set of ART coefficients Fnm is extracted:

•The MPEG-7 Visual Part of the XM 4.0, ISO/IECMPEG99/W3068, Dec. 1999.•W.-Y. Kim and Y.-S. Kim, “A New Region-BasedShape Descriptor,” ISO/IEC MPEG99/M5472, Maui, Hawaii, Dec. 1999.

Contour-Based Shape Descriptor


25

The contour SD is based on theCurvature Scale-Space (CSS)representation of the contour. Distinguish between shapes that have similar

region-based shape (b) Support search for shapes that are

semantically similar, even significant intra-class variability (c)

Robust to significant nonrigid deformations (d) and to perspective transformation (e)

Curvature Scale-Space (CSS)


26

When comparing shapes, humans tend todecompose shape contours into concave and convexsections.Features: How prominent they are, their length relative

to the contour length, and their position and order onthe contour

CSS representation decomposes the contour into convexand concave sections by determining the reflectionpoints (points at which curvature is zero)

Curvature Scale-Space (CSS)


27

CSS image shows how the inflection points change whenfiltering is applied to the contour X-axis corresponds to the position on the contour (clockwise, starting

from any arbitrary point) Y-axis corresponds to the values of a shape smooth parameter (when y-

values increase, amount of smoothing increases) Any black point in the CSS image signifies that at the corresponding

position and at the corresponding scale, there is an inflection point.

Curvature Scale-Space (CSS)28

The smoothing is performed iteratively and for each level, the zero crossings of thecurvature function are computed.

The CSS image is obtained by plotting all zero-crossing points on a plane

Mokhtarian and Mackworth, “A theory of multiscale, curvature-basedshape representation for planar curves,” IEEE Trans. on PAMI, vol. 14, no. 8, pp. 789-805, 1992.

Shape Descriptor


29

Based on CSS images, the descriptor consists of Eccentricity (偏移量) and circularity (環狀) values of the

original and filtered contour Number of peaks The magnitude (height) of the largest peak The x and y positions on the remaining peaks

Chapter 15 of Introduction to MPEG-7: Multimedia ContentDescription Interface. Edited by Manjunath, et al., John Wiley & Sons,2002.

Example: The QBIC System30

Example: The QBIC System


31

ColorColor histogram

TextureCoarseness, contrast, directionality

ShapeArea, circularity, eccentricity, major-axis direction

Fusion of multiple types of features often givesbetter performance.

References


32

Tamura, et al. "Textural feature corresponding to visualperception,"IEEE Trans. on Systems, Man, and Cybernetics, vol.SMC-8, no. 6, pp. 460-473, 1978.

Park, et al. “Efficient use of local edge histogram descriptor,” Proc. of ACM International Workshop on Standards,Interoperability and Practices, pp. 51-54, 2000.

Manjunath and Ma, Chapter 12 of Image Database: Searchand Retrieval of Digital Imagery, edited by V. Castelli and L.D.Bergman, John Wiley & Sons, 2002.

Bober, “MPEG-7 visual shape descriptors”, IEEE Trans. on CSVT, vol. 11, no. 6, pp. 716-719, 2001.

Wei-Ta Chu

2010/10/21

Multidimensional IndexingTechniques

33


Types of Content-Based Query


34

Range search Find all images where feature 1 is within rang r1, and feature 2 is

within range r2, …, and feature n is within range rn

K-Nearest neighbor search Find the k most similar images to the template

Within-distance (α-cut) Find all images with a similarity score better than αwith respect to a

template

V. Castelli, “Multidimensional indexing structures for content-based retrieval,” IBM Research Report, 2001.

Curse of Dimensionality


35

In two dimensions a circle is well approximated by the minimumbounding square The ratio of the square to the circle area is 4/π

In three dimensions, the ratio is 6/π In 100 dimensions, the ratio is 4.2 x 1039

Indexing schemes that rely on properties of low-dimensionalityspaces do not perform well in high-dimensional spaces

In a high-dimensional space, most data points appear to bealmost the same distance from the query sample Difficult for k-nearest neighbor or α-cut approach

Curse of Dimensionality36

The features of each vector independently distributed as standardGaussian random variable.

A large Gaussian sample in a 3-dim space looks like a tight and wellconcentrated cloud. But it’s not so in a 100-dim space.

<12.5, return 5.3% of the database<13, return 14% of the database

Dimensionality Reduction


37

The feature space often has a local structureQuery images have close neighbors and therefore

nearest-neighbor and α-cut can be meaningful The features used to represent the images are

usually not independentThe feature vectors in the database can be wellapproximated by their “projections” onto a lower-dimensionality space

Example38

An artificial data set constructed by taking one of the off-linedigits, represented by a 64 x 64 pixel grey-level image, andembedding it in a larger image of size 100x100.

Each of the resulting images is represented by a point in the100x100 = 10000-dimensional data space.

However, there are only three degrees of freedom: verticaland horizontal translations and the rotations–intrinsicdimensionality is three.

C.M. Bishop, Chapter 12 of Pattern Recognition and Machine Learning, Springer, 2006.

Variable-Subset Selection


39

Retaining some of the dimensions of the feature spaceand discarding the remaining ones

Goal: minimize the error induced by approximatingthe original vectors with their lower-dimensionalityprojections–by linear transformation of the featurespace

Variable-Subset Selection


40

Methods: Karhunen-Loeve transform (KLT), singularvalue decomposition (SVD), principle componentanalysis (PCA)

They are data-dependent transformations and arecomputationally expensive.Poorly suited for dynamic databases

Multidimensional Scaling41

Non-linear methods to reduce the dimensionality of thefeature space.

No precise definition E.g. remapping the space Rn into Rm (m<n) using m

transformations each of which is a combination ofappropriate radial basis functions.

E.g. metric version of multidimensional scaling Generally, multidimensional scaling algorithms can

provide better reduction than linear methods. Much more expensive Data-dependent–poorly suited for dynamic databases

Beatty and Manjunath, “Dimensionality reduction using multi-dimensional scalingfor content-based image retrieval,” Proc. of ICIP, vol. 2, pp. 835-838, 1997.

Wei-Ta Chu

2010/10/21

Dimension Reduction42


1.1 Principal Component Analysis (PCA)43

Widely used in dimensionality reduction, lossydata compression, feature extraction, anddata visualization

Also known as Karhunen-Loeve transform Two commonly-used definitions

Orthogonal projection of the data onto a lowerdimensional linear space such that the variance ofthe projected data is maximized.

Linear projection that minimizes the averageprojection cost

C.M. Bishop, Chapter 12 of Pattern Recognition and Machine Learning, Springer, 2006.

Maximum Variance Formulation


44

Data set of observation {xn} with dimensionality D. Goal: project the data onto a space having

dimensionality M < D with maximizing the varianceof the projected data. Assume the value of M is given.

Begin with M=1. Data are projected onto a line in aD-dimensional space. The direction of the line isdenoted by a D-dimensional vector u1.

Each data point xn is then projected onto a scalarvalue u1

Txn.

LA Recap: Orthogonal Projection


45

cosproj

)toorthogonalofcomponent(vectorproj

)alongofcomponent(vectorproj

2

2

ua

auu

auaa

auuuu

auaa

auu

a

a

a



46

The mean of the projected data is

The variance of the projected data is given by

Where S is the covariance matrix defined by



47

Maximize the projected variance with respect to u1

Introduce a Lagrange multiplier denoted by λ1

By setting the derivative with respect to u1 equal to zero, wesee that this quantity will have a stationary point when

u1 must be an eigenvector of S The variance will be a maximum when we set u1 equal to the

eigenvector having the largest eigenvalue λ1



48

The optimal linear projection for which the variance of theprojected data is maximized is now defined by the Meigenvectors u1, …, uM of the data covariance matrix Scorresponding to the M largest eigenvalues λ1,…,λM

Principal component analysis involves evaluating the meanand the covariance matrix of the data set and then finding theM eigenvectors of S corresponding the M largest eigenvalues.

Covariance


49

High variance, low covariance High variance, high covariance→ No inter-dimension dependency → inter-dimension dependency

Minimum Error Formulation


50

Each data point can be represented by a linearcombination of the basis vectors

Our goal is to approximate this data point using arepresentation involving a restricted number M < D ofvariables corresponding to a projection onto a lower-dimensional subspace.

M-dim projection

Minimum Error Formulation


51

Minimize approximation error

Obtaining the minimum value of J by selecting eigenvectorsto those having the D-M smallest eigenvalues, and hence theeigenvectors defining the principal subspace are thosecorresponding to the M largest eigenvalues.

L.I. Smith, “A tutorial on Principal Component Analysis,” http://csnet.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf

J. Shlens, “A tutorial on Principal Component Analysis,” http://www.cs.cmu.edu/~elaw/papers/pca.pdf

Applications of PCA


52

Mean vector and the first four PCAeigenvectors for the off-line digits data set

Eigenvalue spectrum and the sum of thediscard eigenvalues

An original example together with its PCAreconstructions obtained by retaining Mprincipal components

Eigenfaces53

Eigenfaces for face recognition is a famous application of PCA Eigenfaces capture the majority of variance in face data Project a face on those eigenfaces to represent face features

M. Turk and A.P. Pentland, “Face recognition using eigenfaces,” Proc. of CVPR, pp. 586-591, 1991.

1.2 Singular Value Decomposition (SVD)54

SVD works directly on data PCA works on covariance matrix of data The SVD technique examines the entire set of data and rotates the axis

to maximize variance along the first few dimensions.

Problem: #1: Find concepts in text #2: Reduce dimensionality

http://www.cs.cmu.edu/~guestrin/Class/10701-S06/Handouts/recitations/recitation-pca_svd.ppt

SVD - Definition

A[n x m] = U[n x r] Λ [ r x r] (V[m x r])T

A: n x m matrix (e.g., n documents, m terms) U: n x r matrix (n documents, r concepts) Λ: r x r diagonal matrix (strength of each‘concept’) (r: rank of the matrix)

V: m x r matrix (m terms, r concepts)

55

SVD - Properties

‘spectral decomposition’ of the matrix:

1 1 1 0 02 2 2 0 01 1 1 0 05 5 5 0 00 0 0 2 20 0 0 3 30 0 0 1 1

= x xu1 u2

λ1

λ2

v1

v2

56

SVD - Interpretation

‘documents’, ‘terms’ and ‘concepts’: U: document-to-concept similarity matrix V: term-to-concept similarity matrix Λ: its diagonal elements: ‘strength’ of each concept

Projection: best axis to project on: (‘best’ = min sum of squares

of projection errors)

57

SVD - Example

A = U Λ VT - example:

1 1 1 0 02 2 2 0 01 1 1 0 05 5 5 0 00 0 0 2 20 0 0 3 30 0 0 1 1

datainf.

retrievalbrain lung

0.18 00.36 00.18 00.90 00 0.530 0.800 0.27

=CS

MD

9.64 00 5.29

x

0.58 0.58 0.58 0 00 0 0 0.71 0.71

x

CS-conceptMD-concept

doc-to-conceptsimilarity matrix

58

SVD - Example


1 1 1 0 02 2 2 0 01 1 1 0 05 5 5 0 00 0 0 2 20 0 0 3 30 0 0 1 1

datainf.

retrievalbrain lung

0.18 00.36 00.18 00.90 00 0.530 0.800 0.27

=CS

MD

9.64 00 5.29

x

0.58 0.58 0.58 0 00 0 0 0.71 0.71

x

‘strength’ of CS-concept

59

SVD - Example


1 1 1 0 02 2 2 0 01 1 1 0 05 5 5 0 00 0 0 2 20 0 0 3 30 0 0 1 1

datainf.

retrievalbrain lung

0.18 00.36 00.18 00.90 00 0.530 0.800 0.27

=CS

MD

9.64 00 5.29

x

0.58 0.58 0.58 0 00 0 0 0.71 0.71

x

term-to-conceptsimilarity matrix

CS-concept

60

SVD–Dimensionality reduction

Q: how exactly is dim. reduction done? A: set the smallest singular values to zero:

1 1 1 0 02 2 2 0 01 1 1 0 05 5 5 0 00 0 0 2 20 0 0 3 30 0 0 1 1

0.18 00.36 00.18 00.90 00 0.530 0.800 0.27

=9.64 00 5.29

x

0.58 0.58 0.58 0 00 0 0 0.71 0.71

x

61

SVD - Dimensionality reduction

1 1 1 0 02 2 2 0 01 1 1 0 05 5 5 0 00 0 0 2 20 0 0 3 30 0 0 1 1

0.180.360.180.90000

~9.64

x

0.58 0.58 0.58 0 0

x

62

SVD - Dimensionality reduction

1 1 1 0 02 2 2 0 01 1 1 0 05 5 5 0 00 0 0 2 20 0 0 3 30 0 0 1 1

~

1 1 1 0 02 2 2 0 01 1 1 0 05 5 5 0 00 0 0 0 00 0 0 0 00 0 0 0 0

63

2.1 Multidimensional Scaling (MDS)64

Goal: represent data points in some lower-dimensional space such that the distances betweenpoints in that space correspond to the distancebetween points in the original space

http://www.analytictech.com/networks/mds.htm

Multidimensional Scaling (MDS)


65

What MDS does is to find a set of vectors in p-dimensionalspace such that the matrix of Euclidean distances among themcorresponds as closely as possible to some function of the inputmatrix according to a criterion function called stress.

Stress: the degree of correspondence between the distancesamong points implied by MDS map and the input matrix.

dij refers to the distance between points i and j in the original spacezij refers to the distance between points i and j on the map

Multidimensional Scaling (MDS)66

The true dimensionality of the data will be revealed by therate of decline of stress as dimensionality increases.

Multidimensional Scaling (MDS)


67

Algorithm Assign points to arbitrary coordinates in p-dimensional space Compute Euclidean distances among all pairs of points to form a

matrix Compare the matrix with the input matrix by evaluating the stress

function. The smaller the value, the greater the correspondence betweenthe two.

Adjust coordinates of each point in the direction that best maximallystress

Repeat steps 2 through 4 until stress won’t get any lower

T.F. Cox and M.A.A. Cox, Multidimensional Scaling, Chapman & Hall/CRC; 2 edition, 2000

2.2 Isometric Feature Mapping (Isomap)68

Examples

J.B. Tenenbaum, V. de Silva, and J.C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” Science, vol. 290, pp. 2319-2323, 2000.

Isometric Feature Mapping (Isomap)


69

Estimate the geodesic distance between far awaypoints, given only input-space distances.Adding up a sequence of “short hops” between neighboring

points

Isometric Feature Mapping (Isomap)


70

Algorithm Step 1: construct neighborhood graphDetermines which points are neighbors on the manifoldConnect each point to all points within some fixed radius ε, or to

its K nearest neighbors

Step 2: compute shortest pathsEstimate the geodesic distance between all pairs of points on the

manifold by computing their shortest path in the graph

Step 3: construct d-dimensional embeddingApply MDS to the matrix of graph distances constructing an

embedding of the data

Isometric Feature Mapping (Isomap)71

2.3 Locally Linear Embedding (LLE)72

Eliminate the need to estimate pairwise distances between widelyseparated data points. LLE recovers global nonlinear structure from locallylinear fits.

S.T. Roweis and L.K. Saul,“Nonlinear dimensionality reduction by locally linearembedding,” Science, vol. 290, pp. 2323-2326, 2000http://www.cs.toronto.edu/~roweis/lle/publications.html

Locally Linear Embedding (LLE)73

Characterize the local geometry bylinear coefficients that reconstructeach data point from its neighbors.

Minimize the reconstruction errors

Choosing d-dimensional coordinate Yito minimize the embedding costfunction

Example74

The bottom images correspondto points along the top-rightpath, illustrating one particularmode of variability in pose andexpression.

Wei-Ta Chu

2010/10/21

Indexing Structures75


Indexing


76

After feature selection and dimensionality reduction,the third step is to the selection of appropriateindexing structure.

Vector space index methods Index feature vectors directly

Metric space index methods Index pairwise distances between objects

1. Vector Space Methods


77

Non-hierarchical methodsBrute-force, maps a d-dimensional space onto the real

line, partition the space into non-overlapping cells, …

Recursive partitioning methodsQuadtree, k-d tree, R-tree, …

Projection-based methodsSupporting fixed-radius nearest-neighbor searches,

supporting 1+εnearest-neighbor searches

1.1 Quadtree


78

Quadtrees are trees of degree 2d, where d is the dimension ofthe sample space.

Each step of the decomposition consists of identifying dsplitting points (one along each dimension), and partitioningthe space by means of (d-1)-dimensional hyperplanes passingthrough the splitting point and orthogonal to the splitting pointcoordinate axis.

Splitting a node of d-quadtree consists of dividing eachdimension into two parts, thus defining 2d hyperrectangles.

1.1 Quadtree79

Variations Region quadtree: decompose the space into squares Point quadtree: adaptive decomposition where the splitting points

depending on the data distribution

1.1 Quadtree


80

Extremely popular in geographic information systemapplications

Drawbacks: Each split node always has 2d children–the quadtree is in general

very sparse, that is, most of its nodes are empty Quadtrees are inefficient for exact α–cut and nearest-neighbor

queries, since hyperspheres are not well approximated byhyperrectangles.

Poor performance in high-dimensional spaces

Example81

Quadtree is used to describe a class ofhierarchical data structures whosecommon property is that they are basedon recursive decomposition of space.

Samet, “The quadtree and related hierarchical datastructures” ACM Computer Surveys, vol. 16, no. 2, pp. 187-260, 1984.

Example82

1.2 K-D (K-Dimensional) Tree83

The k-d tree is a binary search tree that represents a recursivesubdivision of the universe into subspaces by means of (d-1)-dimensional hyperplanes. E.g. for d=3, splitting hyperplanes are alternately perpendicular to

the x-, y-, z-axes. Vertical splitting crossing c3, then horizontal splitting crossing p10 and

c7

V. Gaede and O. Gunther, “Multidimensional access methods,” ACM Computing Surveys, vol. 30, no. 2, pp. 170-231, 1998.

1.2 K-D (K-Dimensional) Tree84

Disadvantage: The structure is sensitive to the order in which the points are inserted Data points are scattered all over the tree

Adaptive k-d tree Choosing a split such that about the same number of points on both sides Split points are not part of the input data; all data points are stored in

the leaves

1.3 R-Tree


85

An R-tree corresponds to a hierarchy of nested d-dimensionalintervals (boxes).

Each node v of the R-tree corresponds to an interval

1.3 R-Tree86

R-Trees : represent spatial objects byintervals in several dimensions

Guttman, “R-trees: a dynamic index structure forspatial searching” Proc. of SIGMOD, 1984.

1.4 Fixed-Radius Nearest-NeighborSearches


87

Projects data points onto the individual coordinatesaxes, and produces d sorted lists, on per dimension.

In response to a query, the algorithm retrieves fromeach list the points whose coordinate lie within r ofthe corresponding coordinate of the query point.

The candidate data points are exhaustivelysearched.

Friedman, et al. “An algorithm for finding nearest neighbors,” IEEE Trans. on Computer, pp. 1000-1006, 1975.

2. Metric Space Methods


88

Indexing the metric structure of a spaceVoronoi regions

Vantage-point methodsvp-tree

http://groups.csail.mit.edu/graphics/classes/6.838/S98/meetings/m25/m25.html

References


89

V. Castelli, “Multidimensional indexing structures for content-based retrieval,” IBM Research Report, 2001.

V. Gaede and O. Gunther, “Multidimensional access methods,” ACM Computing Surveys, vol. 30, no. 2, pp. 170-231, 1998.

L.I. Smith, A tutorial on Principal Component Analysis,http://csnet.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf

J. Shlens, “A tutorial on Principal Component Analysis,” http://www.cs.cmu.edu/~elaw/papers/pca.pdf

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1 Spectral Texture Features - 國立中正大學資工系

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