Multimedia DBs

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Multimedia DBs
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Time Series Data
A time series is a collection of observations
made sequentially in time.
25.1750 25.1750 25.2250 25.2500
25.2500 25.2750 25.3250 25.3500
25.3500 25.4000 25.4000 25.3250
25.2250 25.2000 25.1750 .. ..
24.6250 24.6750 24.6750 24.6250
24.6250 24.6250 24.6750 24.7500
value axis
time axis
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PAA and APCA

• Feature extraction for GEMINI
• Fourier
• Wavelets
• Another approach segment the time series into
  equal parts, store the average value for each
  part.
• Use an index to store the averages and the
  segment end points

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Feature Spaces
Korn, Jagadish, Faloutsos 1997
Chan Fu 1999
Agrawal, Faloutsos, Swami 1993
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Piecewise Aggregate Approximation (PAA)
Original time series (n-dimensional
vector) Ss1, s2, , sn
n-segment PAA representation (n-d vector) S
sv1 , sv2, , svn
PAA representation satisfies the lower bounding
lemma (Keogh, Chakrabarti, Mehrotra and Pazzani,
2000 Yi and Faloutsos 2000)
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Can we improve upon PAA?
n-segment PAA representation (n-d vector) S
sv1 , sv2, , svN
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APCA approximates original signal better than PAA
Improvement factor
3.77
1.69
1.21
1.03
3.02
1.75
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APCA Representation can be computed efficiently

• Near-optimal representation can be computed in
  O(nlog(n)) time
• Optimal representation can be computed in O(n2M)
  (Koudas et al.)

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Distance Measure
Lower bounding distance DLB(Q,S)
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Index on 2M-dimensional APCA space
Any feature-based index structure can used (e.g.,
R-tree, X-tree, Hybrid Tree)
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k-nearest neighbor Algorithm
For any node U of the index structure with MBR R,
MINDIST(Q,R) D(Q,S) for any data item S under U
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Index Modification for MINDIST Computation
APCA point S sv1, sr1, sv2, sr2, , svM, srM
R1
S2
S5
sv3
R3
S3
S1
S6
S4
sv1
R2
S8
R4
sv2
S9
sv4
S7
sr2
sr3
sr1
sr4
APCA rectangle S (L,H) where L smin1, sr1,
smin2, sr2, , sminM, srM and H smax1, sr1,
smax2, sr2, , smaxM, srM
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MBR Representation in time-value space
We can view the MBR R(L,H) of any node U as two
APCA representations L l1, l2, , l(N-1), lN
and H h1, h2, , h(N-1), hN
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Regions
M regions associated with each MBR boundaries of
ith region
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Regions

• ith region is active at time instant t if it
  spans across t
• The value st of any time series S under node U at
  time instant t must lie in one of the regions
  active at t (Lemma 2)

REGION 2
h3
value axis
l3
h1
REGION 3
h5
l1
l5
REGION 1
l2
l4
h4
h6
h2
l6
time axis
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MINDIST Computation
For time instant t, MINDIST(Q, R, t) minregion
G active at t MINDIST(Q,G,t)
MINDIST(Q,R,t1) min(MINDIST(Q, Region1, t1),
MINDIST(Q, Region2, t1)) min((qt1 - h1)2 ,
(qt1 - h3)2 ) (qt1 - h1)2
REGION 2
h3
l3
h1
REGION 3
h5
l1
l5
REGION 1
l2
l4
h4
h6
h2
l6
Lemma3 MINDIST(Q,R) D(Q,C) for any time series
C under node U
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Approximate Search

• A simpler definition of the distance in the
  feature space is the following
• But there is one problem what?

DLB(Q,S)
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Multimedia dbs

• A multimedia database stores also images
• Again similarity queries (content based
  retrieval)
• Extract features, index in feature space, answer
  similarity queries using GEMINI
• Again, average values help!

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Images - color
what is an image? A 2-d array
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Images - color
Color histograms, and distance function
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Images - color
Mathematically, the distance function is
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Images - color

• Problem cross-talk
• Features are not orthogonal -gt
• SAMs will not work properly
• Q what to do?
• A feature-extraction question

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Images - color

• possible answers
• avg red, avg green, avg blue
• it turns out that this lower-bounds the histogram
  distance -gt
• no cross-talk
• SAMs are applicable

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Images - color
time
performance
seq scan
w/ avg RGB
selectivity
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Images - shapes

• distance function Euclidean, on the area,
  perimeter, and 20 moments
• (Q how to normalize them?

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Images - shapes

• distance function Euclidean, on the area,
  perimeter, and 20 moments
• (Q how to normalize them?
• A divide by standard deviation)

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Images - shapes

• distance function Euclidean, on the area,
  perimeter, and 20 moments
• (Q other features / distance functions?

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Images - shapes

• distance function Euclidean, on the area,
  perimeter, and 20 moments
• (Q other features / distance functions?
• A1 turning angle
• A2 dilations/erosions
• A3 ... )

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Images - shapes

• distance function Euclidean, on the area,
  perimeter, and 20 moments
• Q how to do dim. reduction?

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Images - shapes

• distance function Euclidean, on the area,
  perimeter, and 20 moments
• Q how to do dim. reduction?
• A Karhunen-Loeve ( centered PCA/SVD)

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Images shapes

• Performance 10x faster

log( of I/Os)
all kept
of features kept
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Is d(u,v) sqrt ((u-v)TA(u-v) ) a metric?

• xTAx S xixjAij S ?ixi2
• ?i is the ith eigenvalue
• xi is the projection of x along the ith
  eigenvector
• d(u,v) sqrt ((u-v)TA(u-v) ) sqrt (S
  ?i(ui-vi)2 )
• d(u,v) gt 0, d(u,u) 0, d(u,v) d(v,u)
• d(u,w) lt d(u,v) d(v,w), provided
• sqrt (S ?i(ui-wi)2 ) lt sqrt (S ?i(ui-vi)2 )
  sqrt(S ?i(vi-wi)2 )
• sqrt(S (v?i ui- v?iwi)2 ) lt sqrt(S (v?iui-
  v?ivi)2 ) sqrt(S(v?ivi- v?iwi)2 )
• Metric condition for Lp norm

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Filtering in QBIC

• Histogram column vectors x, y of length n
• S xi 1, S yi 1
• Difference z (x-y)
• S zi 0
• Contribution of each color bin to a smaller set
  of colors
• VT (c1, c2,.., cn), each ci is a column vector
  of length 3
• xavg VT x, yavg Vty, column vectors of length
  3

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Filtering in QBIC

• Distances
• davg2 (xavg - yavg)T(xavg - yavg) (VT
  z)T(VT z) zTVVt z zTW z
• dhist2 zTA z
• dhist2 gt ?1davg2 , where ?1 is the smallest
  eigenvalue of
• Az ?Wz

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Filtering in QBIC

• Rewrite z to remove the extra condition that S zi
  0.
• z becomes a (n-1) dimensional column vector
• zTA z zTA z and zTW z zTW z
• A and W are (n-1)x(n-1) matrices
• Show that zTA z gt ?1zTW z

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Proof of zTA z gt ?1zTW z

• Minimize wrt z, zTA z, subject to the
  constraint zTW z C.
• Same as minimizing wrt z,
• zTA z - ?(zTW z - C)
• Differentiate wrt z and set to 0
• Az ?W z
• ? and z must be eigenvalues and eigenvectors
  resp. of
• Az ?W z

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Proof of zTA z gt ?1zTW z

• zTA z ?zTW z ?C
• To minimize zTA z , we must choose the
  smallest eigenvalue ?1.
• The minimization of zTA z, under z, subject
  to the constraint zTW z C equals ?1C
• If zTW z C gt 0 then
• zTA z gt ?1C
• If zTW z 0 then
• zTA z gt 0, A is positive semi-definite