Multimedia DBs

PAA and APCA

• 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

0

1

2

3 4

5

6

7

Haar 0

Haar 1

Haar 2

Haar 3

Haar 4

Haar 5

Haar 6

Haar 7

0 20 40 60 80 100 120 140

X

X'DFT

Agrawal, Faloutsos, Swami 1993

Chan & Fu 1999

eigenwave 0

eigenwave 1

eigenwave 2

eigenwave 3

eigenwave 4

eigenwave 5

eigenwave 6

eigenwave 7

Korn, Jagadish, Faloutsos 1997

Feature Spaces

X

X'DWT

0 20 40 60 80 100 120 140

X

X'SVD

0 20 40 60 80 100 120 140

Piecewise Aggregate Approximation (PAA)

valueaxis

time axis

Original time series(n-dimensional vector)S={s1, s2, …, sn}

n’-segment PAA representation (n’-d vector)

S = {sv1 , sv2, …, svn’ }sv1

sv2 sv3sv4

sv5

sv6

sv7

sv8

PAA representation satisfies the lower bounding lemma(Keogh, Chakrabarti, Mehrotra and Pazzani, 2000; Yi and Faloutsos 2000)

Can we improve upon PAA?

n’-segment PAA representation

(n’-d vector)

S = {sv1 , sv2, …, svN }

sv1

sv2 sv3sv4

sv5

sv6

sv7

sv8

sv1

sv2

sv3

sv4

sr1 sr2 sr3 sr4

n’/2-segment APCA representation

(n’-d vector)

S= { sv1, sr1, sv2, sr2, …, svM , srM }

(M is the number of segments = n’/2)

Adaptive Piecewise Constant Approximation (APCA)

1.69

3.02

1.21

1.75

3.77

1.03

Reconstruction error PAA Reconstruction error APCA

APCA approximates original signal better than PAA

Improvement factor =

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.)

Q

M

i iiii svqvsrsr1

21 ))((

DLB(Q’,S)

DLB(Q’,S)

Distance Measure

S

Q

D(Q,S)

n

iii sq

1

2

D(Q,S)

Exact (Euclidean) distance D(Q,S) Lower bounding distance DLB(Q,S)

S

S

Q’

Index on 2M-dimensional APCA space

Any feature-based index structure can used (e.g., R-tree, X-tree, Hybrid Tree)

R1

R3

R2R4

2M-dimensional APCA space

S6

S5

S1

S2 S3

S4

S8

S7

S9

R2 R3 R4

R3 R4

R1

S3 S4 S5 S6 S7 S8 S9S2S1

R2

k-nearest neighbor AlgorithmR1

S7

R3

R2

R4

S1

S2S3

S5

S4

S6

S8

S9

MINDIST(Q,R2)

MINDIST(Q,R4)

MINDIST(Q,R3)

Q

For any node U of the index structure with MBR R, MINDIST(Q,R) D(Q,S) for any data item S under U

Index Modification for MINDIST Computation

APCA point S= { sv1, sr1, sv2, sr2, …, svM, srM }

S1

S2S3

S5

S4 S6

S8S9

R1

R3

R2R4

APCA rectangle S= (L,H) where

L= { smin1, sr1, smin2, sr2, …, sminM, srM } and

H = { smax1, sr1, smax2, sr2, …, smaxM, srM }

sv1

sv2

sv3

sv4

sr1 sr2 sr3 sr4

smax3

smin3

smax1

smin1

smax2

smin2

smax4

smin4

S7

REGION 3

REGION 2

REGION 1

MBR Representation in time-value space

valueaxis

time axis L= { l1, l2, l3, l4 , l5, l6 }

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 }

l1

l2

l3

l4 l6

l5

H= { h1, h2, h3, h4 , h5, h6 }

h1

h2

h3

h4

h5

h6

Regions

M regions associated with each MBR; boundaries of ith region:

REGION i

l(2i-1)

h(2i-1)

h2il(2i-2)+1

h3

h1

h5

h2 h4 h6

valueaxis

time axis

l3

l1

l2 l4

l6

l5

REGION 1

REGION 3

REGION 2

Regions

h3

h1

h5

h2 h4 h6

valueaxis

time axis

l3

l1

l2 l4

l6

l5

REGION 2 t1 t2

REGION 3

REGION 1

• 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)

MINDIST Computation

For time instant t, MINDIST(Q, R, t) =

minregion G active at t MINDIST(Q,G,t)

h3

h1

h5

h2 h4 h6

l3

l1

l2 l4

l6

l5

t1

REGION 3

REGION 2

REGION 1

MINDIST(Q,R,t1)=min(MINDIST(Q, Region1, t1), MINDIST(Q, Region2, t1))=min((qt1 - h1)2 , (qt1 - h3)2 )=(qt1 - h1)2

MINDIST(Q,R) =

n

ttRQMINDIST

1),,(

Lemma3: MINDIST(Q,R) D(Q,C) for any time series C under node U

Approximate Search

• A simpler definition of the distance in the feature space is the following:

• But there is one problem… what?

M

i crki

crcr

k i

ii qcv1

2

1)(

1

1DLB(Q’,S)

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!

Images - color

what is an image?A: 2-d array

Images - color

Color histograms,and distance function

Images - color

Mathematically, the distance function is:

Images - color

Problem: ‘cross-talk’:

• Features are not orthogonal ->

• SAMs will not work properly

• Q: what to do?

• A: feature-extraction question

Images - color

possible answers: • avg red, avg green, avg blue

it turns out that this lower-bounds the histogram distance ->

• no cross-talk• SAMs are applicable

Images - color

performance:

time

selectivity

w/ avg RGB

seq scan

Images - shapes

• distance function: Euclidean, on the area, perimeter, and 20 ‘moments’

• (Q: how to normalize them?

Images - shapes

• distance function: Euclidean, on the area, perimeter, and 20 ‘moments’

• (Q: how to normalize them?

• A: divide by standard deviation)

Images - shapes

• distance function: Euclidean, on the area, perimeter, and 20 ‘moments’

• (Q: other ‘features’ / distance functions?

Images - shapes

• distance function: Euclidean, on the area, perimeter, and 20 ‘moments’

• (Q: other ‘features’ / distance functions?

• A1: turning angle

• A2: dilations/erosions

• A3: ... )

Images - shapes

• distance function: Euclidean, on the area, perimeter, and 20 ‘moments’

• Q: how to do dim. reduction?

Images - shapes

• distance function: Euclidean, on the area, perimeter, and 20 ‘moments’

• Q: how to do dim. reduction?

• A: Karhunen-Loeve (= centered PCA/SVD)

Images - shapes

• Performance: ~10x faster

# of features kept

log(# of I/Os)

all kept