1. Multimedia DBs

2. 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

3. Feature Spaces X X' DFT X X' DWT X X' SVD20 40 60 80
100
120
140 X X'
DFT X X'
DWT 20 40 60 80
100
120
140 X X'
SVD 20 40 60 80
100
120
140
eigenwave 0
eigenwave 1
eigenwave 2
eigenwave 3
eigenwave 4
eigenwave 5
eigenwave 6
eigenwave 7
Haar 0
Haar 1
Haar 2
Haar 3
Haar 4
Haar 5
Haar 6
Haar 7 1 2 3 4 5 6 7
Agrawal, Faloutsos, Swami 1993
Chan & Fu 1999
Korn, Jagadish, Faloutsos 1997

4. Piecewise Aggregate Approximation (PAA)
value
axis
time axis
Original time series
(n-dimensional vector)
S={s1, s2, …, sn}
sv1
sv2
sv3
sv4
sv5
sv6
sv7
sv8
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)

5. Can we improve upon PAA? n’-segment PAA representation (n’-d vector)
sv1
sv2
sv3
sv4
sv5
sv6
sv7
sv8
n’-segment PAA representation
(n’-d vector)
S = {sv1 , sv2, …, svN }
Adaptive Piecewise Constant
Approximation (APCA)
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)

6. APCA approximates original signal better than PAA
Reconstruction error PAA Reconstruction error APCA
Improvement factor =
1.69
3.77
1.21
1.03
3.02
1.75

7. 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.)

8. Distance Measure Lower bounding distance DLB(Q,S)
D(Q,S)
Exact (Euclidean) distance D(Q,S) S Q’ Q
DLB(Q’,S)
DLB(Q’,S)

9. Index on 2M-dimensional APCA spaceR1 R3 R2 R4 R2 R3 R4 R1 S3 S4 S5 S6 S7 S8 S9 S2 S1
2M-dimensional APCA space S6 S5 S1 S2 S3 S4 S8 S7 S9
The k-nearest neighbor search traverse the nodes of the multidimensional index structure in the order of the distance from the query.
We define the node distance as MINDIST which is the minimum distance from the query to any point in the node boundary.
In case of Query Point Movement, MinDist is computed as
the distance between the centroid of relevant points and
the point P which is defined as in this equation.
Any feature-based index structure can used (e.g., R-tree, X-tree, Hybrid Tree)

10. k-nearest neighbor Algorithm
MINDIST(Q,R2)
MINDIST(Q,R3) R1 S7 R3 R2 R4 S1 S2 S3 S5 S4 S6 S8 S9 Q
MINDIST(Q,R4)
The k-nearest neighbor search traverse the nodes of the multidimensional index structure in the order of the distance from the query.
We define the node distance as MINDIST which is the minimum distance from the query to any point in the node boundary.
In case of Query Point Movement, MinDist is computed as
the distance between the centroid of relevant points and
the point P which is defined as in this equation.
For any node U of the index structure with MBR R, MINDIST(Q,R) £ D(Q,S) for any data item S under U

11. Index Modification for MINDIST Computation
APCA point S= { sv1, sr1, sv2, sr2, …, svM, srM }
smax3
smin3 R1
sv3 S2 S5 R3 S3
smax1
smin1
smax2
smin2 S1 S6 S4
sv1 R2
smax4
smin4 R4 S8
sv2 S9
sv4 S7
sr1
sr2
sr3
sr4
APCA rectangle S= (L,H) where
L= { smin1, sr1, smin2, sr2, …, sminM, srM } and
H = { smax1, sr1, smax2, sr2, …, smaxM, srM }

12. 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 }
REGION 2
H= { h1, h2, h3, h4 , h5, h6 } h1 h2 h3 h4 h5 h6
value
axis
time axis l3 l4 l6 l5
REGION 1 l1 l2
REGION 3
L= { l1, l2, l3, l4 , l5, l6 }

13. Regions l(2i-1) h(2i-1) h2i l(2i-2)+1 h3 h5 h2 h4 h6 l3 l1 l2 l4 l6 l5
REGION i
l(2i-1)
h(2i-1)
h2i
l(2i-2)+1
M regions associated with each MBR; boundaries of ith region: h3 h1 h5 h2 h4 h6
value
axis
time axis l3 l1 l2 l4 l6 l5
REGION 1
REGION 3
REGION 2

14. 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 t1 t2 h3
value
axis h1 l3
REGION 3 h5 l1 l5
REGION 1 l2 l4 h2 h4 h6 l6
time axis

15. MINDIST Computation t1 minregion G active at t MINDIST(Q,G,t)
For time instant t, MINDIST(Q, R, t) =
minregion G active at t MINDIST(Q,G,t) t1
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 h2 h4 h6
MINDIST(Q,R) = l6
Lemma3: MINDIST(Q,R) £ D(Q,C) for any time series C under node U

16. But there is one problem… what?
Approximate Search
A simpler definition of the distance in the feature space is the following:
But there is one problem… what?
DLB(Q’,S)

17. 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!

18. Images - color
what is an image?
A: 2-d array

19. Images - color
Color histograms,
and distance function

20. Mathematically, the distance function is:
Images - color
Mathematically, the distance function is:

21. Images - color Problem: ‘cross-talk’:
Features are not orthogonal ->
SAMs will not work properly
Q: what to do?
A: feature-extraction question

22. 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

23. Images - color
time
performance:
seq scan
w/ avg RGB
selectivity

24. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
(Q: how to normalize them?

25. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
(Q: how to normalize them?
A: divide by standard deviation)

26. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
(Q: other ‘features’ / distance functions?

27. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
(Q: other ‘features’ / distance functions?
A1: turning angle
A2: dilations/erosions
A3: ... )

28. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
Q: how to do dim. reduction?

29. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
Q: how to do dim. reduction?
A: Karhunen-Loeve (= centered PCA/SVD)

30. Images - shapes Performance: ~10x faster log(# of I/Os) all kept
# of features kept