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Advanced topics in databases

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Advanced topics in databases

V. Megalooikonomou

Generic Multimedia Indexing

(slides are based on notes by C. Faloutsos)

- Multimedia Indexing
- Spatial Access Methods (SAMs)
- k-d trees
- Point Quadtrees
- MX-Quadtree
- z-ordering
- R-trees

- Generic Multimedia Indexing

- Spatial Access Methods (SAMs)

- Generic Multimedia Indexing
- problem dfn
- Distance function
- Similarity queries – Types
- Requirements (ideal method)
- Basic idea, Lower-bounding
- Gemini approach
- Applications
- 1-D Time sequences
- 2-D Color images

- Given a database of multimedia objects
- Design fast search algorithms that locate objects that match a query object, exactly or approximately
- Objects:
- 1-d time sequences
- Digitized voice or music
- 2-d color images
- 2-d or 3-d gray scale medical images
- Video clips

- Objects:
- E.g.: “Find companies whose stock prices move similarly”

- Generic Multimedia Indexing
- problem dfn
- Distance function
- Similarity queries – Types
- Requirements (ideal method)
- Basic idea, Lower-bounding
- Gemini approach
- Applications
- 1-D Time sequences
- 2-D Color images

- 1st step: provide a measure for the distance between two objects
- Distance function D():
- Given two objects OA, OB the distance (=dis-similarity) of the two objects is denoted by
D(OA, OB)

E.g., Euclidean distance (sum of squared differences) of two equal-length time series

- Given two objects OA, OB the distance (=dis-similarity) of the two objects is denoted by

- Distance function D():

- Generic Multimedia Indexing
- problem dfn
- Distance function
- Similarity queries
- Requirements (ideal method)
- Basic idea, Lower-bounding
- Gemini approach
- Applications
- 1-D Time sequences
- 2-D Color images

std

S1

F(S1)

1

365

day

F(Sn)

Sn

avg

day

1

365

- Similarity queries are classified into:
- Whole match queries:
- Given a collection of N objects O1,…, ON and a query object Q find data objects that are within distance from Q

- Sub-pattern Match:
- Given a collection of N objects O1,…, ON and a query (sub-) object Q and a tolerance identify the parts of the data objects that match the query Q

- Whole match queries:

std

S1

F(S1)

1

365

day

F(Sn)

- Similarity queries are classified into:
- Whole match queries:
- Given a collection of N objects O1,…, ON and a query object Q find data objects that are within distance from Q

- Sub-pattern Match:
- Given a collection of N objects O1,…, ON and a query (sub-) object Q and a tolerance identify the parts of the data objects that match the query Q

- Whole match queries:

Sn

avg

day

1

365

std

S1

F(S1)

1

365

day

F(Sn)

- Similarity queries are classified into:
- Whole match queries:
- Given a collection of N objects O1,…, ON and a query object Q find data objects that are within distance from Q

- Sub-pattern Match:
- Given a collection of N objects O1,…, ON and a query (sub-) object Q and a tolerance identify the parts of the data objects that match the query Q

- Whole match queries:

Sn

avg

day

1

365

- Similarity queries are classified into:
- Whole match queries:
- Sub-pattern Match:

- Whole match queries:

std

S1

F(S1)

1

365

day

F(Sn)

Sn

avg

day

1

365

- Additional types of queries:
- K- Nearest Neighbor queries:
- Given a collection of N objects O1,…, ON and a query object Q find the K most similar data objects to Q

- All pairs queries (or ‘spatial joins’):
- Given a collection of N objects O1,…, ON find all objects that are within distance from each other

- K- Nearest Neighbor queries:

std

S1

F(S1)

1

365

day

F(Sn)

Sn

avg

day

1

365

- Additional types of queries:
- K- Nearest Neighbor queries:
- Given a collection of N objects O1,…, ON and a query object Q find the K most similar data objects to Q

- All pairs queries (or ‘spatial joins’):
- Given a collection of N objects O1,…, ON find all objects that are within distance from each other

- K- Nearest Neighbor queries:

- Generic Multimedia Indexing
- problem dfn
- Distance function
- Similarity queries – Types
- Requirements (ideal method)
- Basic idea, Lower-bounding
- Gemini approach
- Applications
- 1-D Time sequences
- 2-D Color images

- Fast: sequential scanning and distance calculation with each and every object too slow for large databases
- “Correct”: No false dismissals. False alarms are acceptable. Why?
- Small space overhead
- Dynamic: easy to insert, delete, and update objects

- Use k feature extraction functions to map objects into k-dimensional space (applying a mapping F () )
- Use highly fine-tuned database SAMs (Spatial Access Methods) like R-trees to accelerate the search (by pruning out large portions of the database that are not promising)…

- Generic Multimedia Indexing
- problem dfn
- Distance function
- Similarity queries – Types
- Requirements (ideal method)
- Basic idea, Lower-bounding
- Gemini approach
- Applications
- 1-D Time sequences
- 2-D Color images

- Focus on ‘whole match’ queries
- Given a collection of N objects O1,…, ON, a distance/dis-similarity function D(Oi, Oj), and a query object Q find data objects that are within distance from Q

- Focus on ‘whole match’ queries
- Given a collection of N objects O1,…, ON, a distance/dis-similarity function D(Oi, Oj), and a query object Q find data objects that are within distance from Q

May be too slow.. Why?

- Focus on ‘whole match’ queries
- Given a collection of N objects O1,…, ON, a distance/dis-similarity function D(Oi, Oj), and a query object Q find data objects that are within distance from Q

May be too slow.. for the following reasons:

- Distance computation is expensive (e.g., editing distance in DNA strings)
- The Database size N may be huge

- Faster alternative:
- Step 1:a ‘quick and dirty’ test to discard quickly the vast majority of non-qualifying objects
- Step 2: use of SAMs to achieve faster than sequential searching

- Example:
- Database of yearly stock price movements
- Euclidean distance function
- Characterize with a single number (‘feature’)
- Or use two or more features

Feature2

S1

F(S1)

1

365

day

F(Sn)

Sn

Feature1

1

365

day

- A query with tolerance becomes a sphere with radius

- The mapping F() from objects to k-d points should not distort the distances
- D(): distance of two objects
- Df(): distance of their corresponding feature vectors
- Ideally, perfect preservation of distances
- In practice, a guarantee of no false dismissals
- How?

- The mapping F() from objects to k-d points should not distort the distances
- D(): distance of two objects
- Df(): distance of the corresponding feature vectors
- Ideally, perfect preservation of distances
- In practice, a guarantee of no false dismissals
- How? If the distance in f-space matches or underestimates the distance between two objects in the original space

- Let O1, O2 be two objects with distance function D() and F(O1), F(O2), be their feature vectors with distance function Df(), then:
To guarantee no false dismissals for whole match queries, the feature extraction function F() should satisfy:

Df(F(O1), F(O2)) D(O1, O2)

for every pair of objects O1, O2

- Let Q be the query object and O be the qualifying object and be the tolerance.
- Prove: If object O qualifies it will be retrieved by a range query in the f-space
- Or, D(Q, O) Df(F(Q), F(O))
- However, Df(F(Q), F(O)) D(Q, O)
- What about ‘all-pairs’?
- What about ‘nearest-neighbor’ queries?

- Let Q be the query object and O be the qualifying object and be the tolerance.
- Prove: If object O qualifies it will be retrieved by a range query in the f-space
- Or, D(Q, O) Df(F(Q), F(O))
- However, Df(F(Q), F(O)) D(Q, O)
- What about ‘all-pairs’? (‘spatial join’ on f-space)
- What about ‘nearest-neighbor’ queries?

- Let Q be the query object and O be the qualifying object and be the tolerance.
- Prove: If object O qualifies it will be retrieved by a range query in the f-space
- Or, D(Q, O) Df(F(Q), F(O))
- However, Df(F(Q), F(O)) D(Q, O)
- What about ‘all-pairs’? (‘spatial join’ on f-space)
- What about ‘nearest-neighbor’ queries? ??

- Generic Multimedia Indexing
- problem dfn
- Distance function
- Similarity queries – Types
- Requirements (ideal method)
- Basic idea, Lower-bounding
- Gemini approach
- Applications
- 1-D Time sequences
- 2-D Color images

- GEMINI approach:
- Determine distance function D()
- Find one or more numerical feature-extraction functions (to provide a ‘quick and dirty’ test)
- Prove that Df() lower-bounds D() to guarantee no false dismissals
- Use a SAM (e.g., R-tree) to store and retrieve k-d feature vectors

- !!! The methodology focuses on the speed of search only; not on the quality of the results which relies on the distance function

- Applications:
- 1-d time sequences
- 2-d color images

- Problems to solve:
- How to apply the lower-bounding lemma
- ‘Curse of Dimensionality’ (time sequences)
- ‘Cross-talk’ of features (color images)

- Generic Multimedia Indexing
- problem dfn
- Distance function
- Similarity queries – Types
- Requirements (ideal method)
- Basic idea, Lower-bounding
- Gemini approach
- Applications
- 1-D Time sequences
- 2-D Color images

- Distance function: Euclidean distance
- Find features that:
- Preserve/lower-bound the distance
- Carry as much information as possible(reduce false alarms)

- If we are allowed to use only one feature what would this be?

- Distance function: Euclidean distance
- Find features that:
- Preserve/lower-bound the distance
- Carry as much information as possible(reduce false alarms)

- If we are allowed to use only one feature what would this be? The average.
- … extending it…

- Distance function: Euclidean distance
- Find features that:
- Preserve/lower-bound the distance
- Carry as much information as possible(reduce false alarms)

- If we are allowed to use only one feature what would this be? The average.
- … extending it…
- The average of 1st half, of the 2nd half, of the 1st quarter, etc.
- Coefficients of the Fourier transform (DFT), wavelet transform, etc.

- Show that the distance in feature space lower-bounds the actual distance
- What about DFT?

- Show that the distance in feature space lower-bounds the actual distance
- What about DFT?
Parseval’s Theorem: DFT preserves the energy of the signal as well as the distances between two signals.

D(x,y) = D(X,Y)

where X and Y are the Fourier transforms of x and y

- If we keep the first k n coefficients of DFT we lower-bound the actual distance

- Response time improves as the transform concentrates more the energy of the signal
- DFT concentrates the energy for a large class of signals, the colored noises
- Colored noises: skewed energy spectrum that drops as O(f -b)
- Energy spectrum or power spectrum of a signal is the square of the amplitude |Xf| as a function of the frequency f
- b = 2: random walks or brown noise (very predictable)
- b 2: black noises
- b = 1: pink noise
- b = 0: white noise (completely unpredictable)
- Colored noises even in images (photographs)

- Generic Multimedia Indexing
- problem dfn
- Distance function
- Similarity queries – Types
- Requirements (ideal method)
- Basic idea, Lower-bounding
- Gemini approach
- Applications
- 1-D Time sequences
- 2-D Color images

- Image features for Content Based Image Retrieval (CBIR):
- Low Level:
- Color – color histograms
- Texture – directionality, granularity, contrast
- Shape – turning angle, moments of inertia, pattern spectrum
- Position – 2D strings method
- …etc

- Object Level:
- Regions

- Low Level:

- Each color image – a 2-d array of pixels
- Each pixel – 3 color components (R,G,B)
- h colors – each color denoting a point in 3-d color space (as high as 224 colors)
- For each image compute the h-element color histogram – each component is the percentage of pixels that are most similar to that color
- The histogram of image I is defined as:
For a color Ci , Hci(I) represents the number of pixels of color Ci in image I

OR:

For any pixel in image I, Hci(I) represents the possibility of that pixel having color Ci.

- Usually cluster similar colors together and choose one representative color for each ‘color bin’
- Most commercial CBIR systems include color histogram as one of the features (e.g., QBIC of IBM)
- No space information

- One method to measure the distance between two histograms x and y is:
where the color-to-color similarity matrix A has entries aij that describe the similarity between color i and color j

- Two obstacles for using color-histograms as feature vectors in GEMINI:
- ‘Dimensionality curse’ (h is large 64, 128)
- Distance function is quadratic
- It involves all cross terms (‘cross-talk’ among features)
- expensive to compute

- precludes the use of SAMs

- It involves all cross terms (‘cross-talk’ among features)

bright red

pink

orange

x

q

e.g.,64 colors

- 1st step: define the distance function between two color images D()=dh()
- 2nd step: find numerical features (one or more) whose Euclidean distance lower-bounds dh()
- If we allowed to use one numerical feature to describe the color image what should it be?
- Avg. amount for each color component (R,G,B)
- Where … , similarly for G and B
Where P is the number of pixels in the image, R(p) is the red component (intensity) of the p-th pixel

- Given the average color vectors and of two images we define davg() as the Euclidean distance between the 3-d average color vectors
- 3rd step: to prove that the feature distance davg() lower-bounds the actual distance dh()
- Main idea of approach:
- First a filtering using the average (R,G,B) color,
- then a more accurate matching using the full h-element histogram

- pick any pixel p1 of color Ciin the image I
- at distance k away from p1 pick another pixel p2
- what is the probability that p2 is also of color Ci ?

Red ?

k

P2

P1

Image: I

- The auto-correlogram of image I for color Ci , distance k:
- Integrate both color information and space information.

- Pixel Distance Measures
- Use D8 distance (also called chessboard distance):
- Choose distance k=1,3,5,7
- Computation complexity:
- Histogram:
- Correlogram:

- Features Distance Measures:
- D( f(I1) - f(I2) ) is small I1 and I2 are similar.
- Example: f(a)=1000, f(a’)=1050; f(b)=100, f(b’)=150
- For histogram:
- For correlogram:

- If there is no difference between the query and the target images, both methods have good performance.

Correlogram method

Query Image

(512 colors)

1st

2nd

3rd

4th

5th

Histogram method

1st

2nd

3rd

4th

5th

- The correlogram method is more stable to color change than the histogram method.

Query

Correlogram method: 1st

Histogram method: 48th

Target

- The correlogram method is more stable to large appearance change than the histogram method

Query

Correlogram method: 1st

Histogram method: 31th

Target

- The correlogram method is more stable to contrast & brightness change than the histogram method.

Query 3

Query 1

Query 2

Query 4

C: 178th

H: 230th

C: 1st

H: 1st

C: 1st

H: 3rd

C: 5th

H: 18th

Target

- The color correlogram describes the global distribution of local spatial correlations of colors.
- It’s easy to compute
- It’s more stable than the color histogram method

- GEMINI is a popular method
- Whole matching problem
- Should pay attention to:
- Distance functions
- Feature Extraction functions
- Lower Bounding
- Particular application

- Sub-pattern matching?