fast similarity search in image databases cse 6367 – computer vision vassilis athitsos university...
TRANSCRIPT
![Page 1: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/1.jpg)
Fast Similarity Search in Image Databases
CSE 6367 – Computer VisionVassilis Athitsos
University of Texas at Arlington
![Page 2: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/2.jpg)
2
4128 images are generated for each hand shape.
Total: 107,328 images.
A Database of Hand Images
![Page 3: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/3.jpg)
3
Efficiency of the Chamfer Distance
• Computing chamfer distances is slow.– For images with d edge pixels, O(d log d) time.– Comparing input to entire database takes over 4 minutes.
• Must measure 107,328 distances.
input model
![Page 4: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/4.jpg)
4
The Nearest Neighbor Problem
database
![Page 5: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/5.jpg)
5
The Nearest Neighbor Problem
query
• Goal: – find the k nearest
neighbors of query q.database
![Page 6: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/6.jpg)
6
The Nearest Neighbor Problem
• Goal: – find the k nearest
neighbors of query q.
• Brute force time is linear to:– n (size of database).– time it takes to measure a
single distance.
query
database
![Page 7: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/7.jpg)
7
• Goal: – find the k nearest
neighbors of query q.
• Brute force time is linear to:– n (size of database).– time it takes to measure a
single distance.
The Nearest Neighbor Problem
query
database
![Page 8: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/8.jpg)
8
Examples of Expensive Measures
DNA and protein sequences: Smith-Waterman.
Dynamic gestures and time series: Dynamic Time Warping.
Edge images: Chamfer distance, shape context distance.
These measures are non-Euclidean, sometimes non-metric.
![Page 9: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/9.jpg)
9
Embeddings
database
x1
x2
x3
xn
![Page 10: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/10.jpg)
10
database
x1
x2
x3
xn
embedding F
x1x2
x3
x4
xn
Rd
Embeddings
![Page 11: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/11.jpg)
11
database
x1
x2
x3
xn
embedding F
x1x2
x3
x4
xn
q
query
Rd
Embeddings
![Page 12: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/12.jpg)
12
x1x2
x3
x4
xn
q
database
x1
x2
x3
xn
embedding F
q
query
Rd
Embeddings
![Page 13: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/13.jpg)
13
x1x2
x3
x4
xn
Rd
q
Measure distances between vectors (typically much faster).
Caveat: the embedding must preserve similarity structure.
Embeddingsdatabase
x1
x2
x3
xn
embedding F
q
query
![Page 14: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/14.jpg)
14
Reference Object Embeddings
original space X
![Page 15: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/15.jpg)
15
Reference Object Embeddings
original space X
r
r: reference object
![Page 16: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/16.jpg)
16
Reference Object Embeddings
original space X
r: reference object Embedding: F(x) = D(x,r)
D: distance measure in X.
r
![Page 17: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/17.jpg)
17
Reference Object Embeddings
original space X Real lineF
r: reference object Embedding: F(x) = D(x,r)
D: distance measure in X.
r
![Page 18: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/18.jpg)
18
original space X Real lineF
r: reference object Embedding: F(x) = D(x,r)
D: distance measure in X.
F(r) = D(r,r) = 0
r
Reference Object Embeddings
![Page 19: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/19.jpg)
19
original space X Real lineF
r: reference object Embedding: F(x) = D(x,r)
D: distance measure in X.
a
F(r) = D(r,r) = 0 If a and b are similar,
their distances to r are also similar (usually).
br
Reference Object Embeddings
![Page 20: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/20.jpg)
20
original space X Real lineF
r: reference object Embedding: F(x) = D(x,r)
D: distance measure in X.
a
F(r) = D(r,r) = 0 If a and b are similar,
their distances to r are also similar (usually).
br
Reference Object Embeddings
![Page 21: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/21.jpg)
21
F(x) = D(x, Lincoln)
F(Sacramento)....= 1543F(Las Vegas).....= 1232F(Oklahoma City).= 437F(Washington DC).= 1207F(Jacksonville)..= 1344
![Page 22: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/22.jpg)
22
F(x) = (D(x, LA), D(x, Lincoln), D(x, Orlando))
F(Sacramento)....= ( 386, 1543, 2920)F(Las Vegas).....= ( 262, 1232, 2405)F(Oklahoma City).= (1345, 437, 1291)F(Washington DC).= (2657, 1207, 853)F(Jacksonville)..= (2422, 1344, 141)
![Page 23: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/23.jpg)
23
F(x) = (D(x, LA), D(x, Lincoln), D(x, Orlando))
F(Sacramento)....= ( 386, 1543, 2920)F(Las Vegas).....= ( 262, 1232, 2405)F(Oklahoma City).= (1345, 437, 1291)F(Washington DC).= (2657, 1207, 853)F(Jacksonville)..= (2422, 1344, 141)
![Page 24: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/24.jpg)
24
Embedding Hand Images
F(x) = (C(x, R1), C(A, R2), C(A, R3))
R1
R2
R3
image x
x: hand image. C: chamfer distance.
![Page 25: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/25.jpg)
25
Basic Questions
R1
R2
R3
x: hand image. C: chamfer distance.
How many prototypes? Which prototypes? What distance should we
use to compare vectors?
image x
F(x) = (C(x, R1), C(A, R2), C(A, R3))
![Page 26: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/26.jpg)
26
Some Easy Answers.
R1
R2
R3
x: hand image. C: chamfer distance.
How many prototypes? Pick number manually.
Which prototypes? Randomly chosen.
What distance should we use to compare vectors? L1, or Euclidean.
image x
F(x) = (C(x, R1), C(A, R2), C(A, R3))
![Page 27: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/27.jpg)
27
Filter-and-refine Retrieval
Embedding step: Compute distances from query to reference
objects F(q). Filter step:
Find top p matches of F(q) in vector space. Refine step:
Measure exact distance from q to top p matches.
![Page 28: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/28.jpg)
28
Evaluating Embedding Quality
Embedding step: Compute distances from query to reference
objects F(q). Filter step:
Find top p matches of F(q) in vector space. Refine step:
Measure exact distance from q to top p matches.
How often do we find the true nearest neighbor?
![Page 29: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/29.jpg)
29
Evaluating Embedding Quality
Embedding step: Compute distances from query to reference
objects F(q). Filter step:
Find top p matches of F(q) in vector space. Refine step:
Measure exact distance from q to top p matches.
How often do we find the true nearest neighbor?
![Page 30: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/30.jpg)
30
Evaluating Embedding Quality
Embedding step: Compute distances from query to reference
objects F(q). Filter step:
Find top p matches of F(q) in vector space. Refine step:
Measure exact distance from q to top p matches.
How often do we find the true nearest neighbor?
How many exact distance computations do we need?
![Page 31: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/31.jpg)
31
Evaluating Embedding Quality
Embedding step: Compute distances from query to reference
objects F(q). Filter step:
Find top p matches of F(q) in vector space. Refine step:
Measure exact distance from q to top p matches.
How often do we find the true nearest neighbor?
How many exact distance computations do we need?
![Page 32: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/32.jpg)
32
Evaluating Embedding Quality
Embedding step: Compute distances from query to reference
objects F(q). Filter step:
Find top p matches of F(q) in vector space. Refine step:
Measure exact distance from q to top p matches.
How often do we find the true nearest neighbor?
How many exact distance computations do we need?
![Page 33: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/33.jpg)
33
query
Database (107,328 images)
nearestneighborBrute force retrieval time: 260 seconds.
Results: Chamfer Distance on Hand Images
![Page 34: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/34.jpg)
34
Brute Force
Embeddings Embeddings
Accuracy 100% 95% 100%
# of distances 80640 1866 24650
Sec. per query 112 2.6 34
Speed-up factor 1 43 3.27
Query set: 710 real images of hands.
Database: 80,640 synthetic images of hands.
Results: Chamfer Distance on Hand Images
![Page 35: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/35.jpg)
35
Ideal Embedding Behavior
original space X F Rd
Notation: NN(q) is the nearest neighbor of q.
For any q: if a = NN(q), we want F(a) = NN(F(q)).
aq
![Page 36: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/36.jpg)
36
A Quantitative Measure
qa
original space X F Rd
b
If b is not the nearest neighbor of q,F(q) should be closer to F(NN(q)) than to F(b).
For how many triples (q, NN(q), b) does F fail?
![Page 37: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/37.jpg)
37
A Quantitative Measure
original space X F Rd
qa
F fails on five triples.
![Page 38: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/38.jpg)
38
Embeddings Seen As Classifiers
qa
b
Classification task: is qcloser to a or to b?
![Page 39: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/39.jpg)
39
Any embedding F defines a classifier F’(q, a, b). F’ checks if F(q) is closer to F(a) or to F(b).
qa
b
Embeddings Seen As Classifiers
Classification task: is qcloser to a or to b?
![Page 40: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/40.jpg)
40
Given embedding F: X Rd: F’(q, a, b) = ||F(q) – F(b)|| - ||F(q) – F(a)||.
F’(q, a, b) > 0 means “q is closer to a.” F’(q, a, b) < 0 means “q is closer to b.”
qa
b
Classifier Definition
Classification task: is qcloser to a or to b?
![Page 41: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/41.jpg)
41
Given embedding F: X Rd: F’(q, a, b) = ||F(q) – F(b)|| - ||F(q) – F(a)||.
F’(q, a, b) > 0 means “q is closer to a.” F’(q, a, b) < 0 means “q is closer to b.”
Classifier Definition
Goal: build an F such that F’ has low
error rate on triples of type (q, NN(q), b).
![Page 42: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/42.jpg)
42
1D Embeddings as Weak Classifiers
1D embeddings define weak classifiers. Better than a random classifier (50% error rate).
![Page 43: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/43.jpg)
43
1D Embeddings as Weak Classifiers
1D embeddings define weak classifiers. Better than a random classifier (50% error rate).
We can define lots of different classifiers. Every object in the database can be a reference object.
Question: how do we combine many such
classifiers into a single strong classifier?
![Page 44: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/44.jpg)
44
1D Embeddings as Weak Classifiers
1D embeddings define weak classifiers. Better than a random classifier (50% error rate).
We can define lots of different classifiers. Every object in the database can be a reference object.
Question: how do we combine many such
classifiers into a single strong classifier?
Answer: use AdaBoost. AdaBoost is a machine learning method designed for
exactly this problem.
![Page 45: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/45.jpg)
45
Using AdaBoostoriginal space X
Fn
F2
F1
Real line
Output: H = w1F’1 + w2F’2 + … + wdF’d . AdaBoost chooses 1D embeddings and weighs them. Goal: achieve low classification error. AdaBoost trains on triples chosen from the database.
![Page 46: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/46.jpg)
46
From Classifier to Embedding
AdaBoost output H = w1F’1 + w2F’2 + … + wdF’d
What embedding should we use?What distance measure should we use?
![Page 47: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/47.jpg)
47
From Classifier to Embedding
AdaBoost output
BoostMap embedding
H = w1F’1 + w2F’2 + … + wdF’d
F(x) = (F1(x), …, Fd(x)).
![Page 48: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/48.jpg)
48
From Classifier to Embedding
AdaBoost output
BoostMap embedding
Distance measure
D((u1, …, ud), (v1, …, vd)) = i=1 wi|ui – vi|
d
F(x) = (F1(x), …, Fd(x)).
H = w1F’1 + w2F’2 + … + wdF’d
![Page 49: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/49.jpg)
49
From Classifier to Embedding
AdaBoost output
D((u1, …, ud), (v1, …, vd)) = i=1 wi|ui – vi|
d
F(x) = (F1(x), …, Fd(x)).BoostMap embedding
Distance measure
Claim: Let q be closer to a than to b. H misclassifiestriple (q, a, b) if and only if, under distance measure D, F maps q closer to b than to a.
H = w1F’1 + w2F’2 + … + wdF’d
![Page 50: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/50.jpg)
50
Proof
H(q, a, b) =
= wiF’i(q, a, b)
= wi(|Fi(q) - Fi(b)| - |Fi(q) - Fi(a)|)
= (wi|Fi(q) - Fi(b)| - wi|Fi(q) - Fi(a)|)
= D(F(q), F(b)) – D(F(q), F(a)) = F’(q, a, b)
i=1
d
i=1
d
i=1
d
![Page 51: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/51.jpg)
51
Proof
H(q, a, b) =
= wiF’i(q, a, b)
= wi(|Fi(q) - Fi(b)| - |Fi(q) - Fi(a)|)
= (wi|Fi(q) - Fi(b)| - wi|Fi(q) - Fi(a)|)
= D(F(q), F(b)) – D(F(q), F(a)) = F’(q, a, b)
i=1
d
i=1
d
i=1
d
![Page 52: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/52.jpg)
52
Proof
H(q, a, b) =
= wiF’i(q, a, b)
= wi(|Fi(q) - Fi(b)| - |Fi(q) - Fi(a)|)
= (wi|Fi(q) - Fi(b)| - wi|Fi(q) - Fi(a)|)
= D(F(q), F(b)) – D(F(q), F(a)) = F’(q, a, b)
i=1
d
i=1
d
i=1
d
![Page 53: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/53.jpg)
53
Proof
H(q, a, b) =
= wiF’i(q, a, b)
= wi(|Fi(q) - Fi(b)| - |Fi(q) - Fi(a)|)
= (wi|Fi(q) - Fi(b)| - wi|Fi(q) - Fi(a)|)
= D(F(q), F(b)) – D(F(q), F(a)) = F’(q, a, b)
i=1
d
i=1
d
i=1
d
![Page 54: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/54.jpg)
54
Proof
H(q, a, b) =
= wiF’i(q, a, b)
= wi(|Fi(q) - Fi(b)| - |Fi(q) - Fi(a)|)
= (wi|Fi(q) - Fi(b)| - wi|Fi(q) - Fi(a)|)
= D(F(q), F(b)) – D(F(q), F(a)) = F’(q, a, b)
i=1
d
i=1
d
i=1
d
![Page 55: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/55.jpg)
55
Proof
H(q, a, b) =
= wiF’i(q, a, b)
= wi(|Fi(q) - Fi(b)| - |Fi(q) - Fi(a)|)
= (wi|Fi(q) - Fi(b)| - wi|Fi(q) - Fi(a)|)
= D(F(q), F(b)) – D(F(q), F(a)) = F’(q, a, b)
i=1
d
i=1
d
i=1
d
![Page 56: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/56.jpg)
56
Significance of Proof• AdaBoost optimizes a direct measure of
embedding quality.
• We have converted a database indexing problem into a machine learning problem.
![Page 57: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/57.jpg)
57
query
Database (80,640 images)
nearestneighborBrute force retrieval time: 112 seconds.
Results: Chamfer Distance on Hand Images
![Page 58: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/58.jpg)
58
Query set: 710 real images of hands.
Database: 80,640 synthetic images of hands.
Results: Chamfer Distance on Hand Images
Brute Force
Random Reference
ObjectsBoostMap
Accuracy 100% 95% 95%
# of distances 80640 1866 450
Sec. per query 112 2.6 0.63
Speed-up factor 1 43 179
![Page 59: Fast Similarity Search in Image Databases CSE 6367 – Computer Vision Vassilis Athitsos University of Texas at Arlington](https://reader036.vdocuments.us/reader036/viewer/2022062515/56649f565503460f94c7b2bd/html5/thumbnails/59.jpg)
59
Query set: 710 real images of hands.
Database: 80,640 synthetic images of hands.
Results: Chamfer Distance on Hand Images
Brute Force
Random Reference
ObjectsBoostMap
Accuracy 100% 100% 100%
# of distances 80640 24950 5995
Sec. per query 112 34 13.5
Speed-up factor 1 3.23 8.3