lec-08 feature aggregation ii: fisher vector, akula and super vector

Image Analysis & Retrieval

CS/EE 5590 Special Topics (Class Ids: 44873, 44874)

Fall 2016, M/W 4-5:15pm@Bloch 0012

Lec 08

Feature Aggregation II:

Fisher Vector, Super Vector and AKULA

Zhu Li

Dept of CSEE, UMKC

Office: FH560E, Email: [email protected], Ph: x 2346.

http://l.web.umkc.edu/lizhu

p.1Z. Li, Image Analysis & Retrv. 2016

mailto:[email protected]

http://l.web.umkc.edu/lizhu

Outline

ReCap of Lecture 07

Image Retrieval System

BoW

VLAD

Dense SIFT

Fisher Vector Aggregation

AKULA

Summary

Z. Li, Image Analysis & Retrv. 2016 p.2

Precision, Recall, F-measure

Precision, TPR = TP/(TP + FP),

Recall = TP/(TP + FN),

FPR=FP/(TP+FP)

F-measure

= 2*(precision*recall)/(precision + recall)

Precision:is the probability that a

retrieved document is relevant.

Recall:is the probability that a

relevant documentis retrieved in a search .


Why Aggregation ?

Curse of Dimensionality

Decision Boundary / Indexing


+

…..

Bag-of-Words: Histogram Coding

Codebook:

Feature space: Rd, k-means to get k centroids, {𝜇1, 𝜇2, … , 𝜇𝑘}

BoW Hard Encoding:

For n feature points,{x1, x2, …,xn} assignment matrix: kxn, with column only 1-non zero entry

Aggregated dimension: k


k

n

Kernel Code Book Soft Encoding

Kernel Code Book Soft Encoding

Kernel Affinity: 𝐾 𝑥𝑗 , 𝜇𝑘 = 𝑒−𝑘|𝑥𝑗−𝜇𝑘|

2

Assignment Matrix: 𝐴𝑗,𝑘 = 𝐾(𝑥𝑗 , 𝜇𝑘)/ 𝑘𝐾(𝑥𝑗 , 𝜇𝑘)

Encoding: k-dimensional: X(k)= 1

𝑛 𝑗𝐴𝑗,𝑘


VLAD- Vector of Locally Aggregated Descriptors

Aggregate feature difference from the codebook

Hard assignment by finding the NN of feature {xk} to {𝜇𝑘}

Compute aggregated differences

L2 normalize

Final feature: k x d


3

x

v1 v2v3 v4

v5

1

4

2

5

① assign descriptors

② compute x- i

③ vi=sum x- i for cell i

𝑣𝑘 =

∀𝑗,𝑠.𝑡.𝑁𝑁 𝑥𝑗 =𝜇𝑘

𝑥𝑗 − 𝜇𝑘

𝑣𝑘 = 𝑣𝑘/ 𝑣𝑘 2

VLAD on SIFT

Example of aggregating SIFT with VLAD

K=16 codebook entries

Each cell is a SIFT visualized as centroids in blue, and VLAD difference in red

Top row: left image, bottom row: right image, red: code book, blue: encoded VLAD


Outline

ReCap of Lecture 07

Image Retrieval System

BoW

VLAD

Dense SIFT

Fisher Vector Aggregation

AKULA

Summary


One more trick

Recall that SIFT is a powerful descriptor

VL_FEAT: vl_dsift

A dense description of image by computing SIFT descriptor (no spatial-scale space extrema detection) at predetermined grid

Supplement HoG as an alternative texture descriptor


VL_FEAT: vl_dsift

Compute dense SIFT as a texture descriptor for the image

[f, dsift]=vl_dsift(single(rgb2gray(im)), ‘step’, 2);

There’s also a FAST option

[f, dsift]=vl_dsift(single(rgb2gray(im)), ‘fast’, ‘step’, 2);

Huge amount of SIFT data will be generated


Fisher Vector

Fisher Vector and variations:

Winning in image classification:

Winning in the MPEG object re-identification:o SCFV(Scalable Coded Fisher Vec) in CDVS


Codebook: Gaussian Mixture Model (GMM)

GMM is a generative model to express data

Assuming data is generated from with parameters {𝑤𝑘 , 𝜇𝑘 , 𝜎𝑘}


𝑥𝑘 ~

𝑘=1

𝐾

𝑤𝑘𝑁(𝜇𝑘,𝜎𝑘)

𝑁 𝜇𝑘 ,𝜎𝑘 =1

2𝜋𝑑2 Σ𝑘

1/2𝑒−12𝑥−𝜇𝑘

′Σ𝑘−1(𝑥−𝜇𝑘)

A bit of Theory: Fisher Kernel

Encode the derivation from the generative model

Observed feature set, {x1, x2, …,xn} in Rd, e.g, d=128 for SIFT.

How’s these observations derivate from the given GMM model with a set of parameter, 𝜆 = 𝑤𝑘 , 𝜇𝑘 , 𝜎𝑘 ?o i.e, how the parameter, e.g, mean will move to best fit the observation

?


𝜇4𝜇3

𝜇2

𝜇1

X1 +

A bit of Theory: Fisher Kernel

Score function w.r.t. the likelihood function 𝜇𝜆(𝑋) 𝐺𝜆𝑋 = 𝛻𝜆 log𝑢𝜆(𝑋): derivative on the log likelihood

The dimension of score function is m, where m is the number of generative model parameters, m=3 for GMM

Given the observed data X, score function indicate how likelihood function parameter (e.g, mean) should move to better fit the data.

Distance/Derivation of two observation X, Y w.r.t the generative model

Fisher Info Matrix (roughly the covariance in the Mahanolibisdistance)

𝐹𝜆 = 𝐸𝑋 𝐺𝜆𝑋𝐺𝜆𝑋′

Fisher Kernel Distance: normalized by the Fisher Info Matrix:


𝐾𝐹𝐾 𝑋, 𝑌 = 𝐺𝜆𝑋′𝐹𝜆−1𝐺𝜆𝑋

Fisher Vector

KFK(X, Y) is a measure of similarity, w.r.t. the generative model

Similar to the Mahanolibis distance case, we can decompose this kernel as,

That give us a kernel feature mapping of X to Fisher Vector

For observed images features {xt}, can be computed as,


𝐾𝐹𝐾 𝑋,𝑌 = 𝐺𝜆𝑋′𝐹𝜆−1𝐺𝜆𝑋 = 𝐺𝜆

𝑋′𝐿𝜆′𝐿𝜆𝐺𝜆𝑋

GMM Fisher Vector

Encode the derivation from the generative model Observed feature set, {x1, x2, …,xn} in Rd, e.g, d=128 (!) for SIFT.

How’s these observations derivate from the given GMM model with a set of parameter, 𝜃 = 𝑎𝑘, 𝜇𝑘, 𝜎𝑘 ?

GMM Log Likelihood Gradient

Let 𝑤𝑘 =𝑒𝑎𝑘

𝑗 𝑒𝑎𝑗

, Then we have


weight

mean

variance

GMM Fisher Vector VL_FEAT implementation

GMM codebook

For a K-component GMM, we only allow 3K parameters, 𝜋𝑘 , 𝜇𝑘 ,𝜎𝑘 𝑘 = 1. . 𝐾}, i.e, iid Gaussian component

Posterior prob of feature point xi to GMM component k


Σ𝑘 =

𝜎𝑘 0 0 00 𝜎𝑘 0 0….𝜎𝑘

GMM Fisher Vector VL_FEAT implementation

FV encoding

Gradient on the mean, for GMM component k, j=1..D

In the end, we have 2K x D aggregation on the derivation w.r.t. the means and variances


𝐹𝑉 = [𝑢1 , 𝑢2 ,… ,𝑢𝐾 , 𝑣1 , 𝑣2 , … , 𝑣𝐾]

VL_FEAT GMM/FV API

Compute GMM model with VL_FEAT

Prepare data:numPoints = 1000 ; dimension = 2 ;

data = rand(dimension,N) ;

Call vl_gmm:numClusters = 30 ;

[means, covariances, priors] = vl_gmm(data, numClusters) ;

Visualize:figure ;

hold on ;

plot(data(1,:),data(2,:),'r.') ;

for i=1:numClusters

vl_plotframe([means(:,i)' sigmas(1,i) 0 sigmas(2,i)]);

end


VL_FEAT API

FV encoding

encoding = vl_fisher(datatoBeEncoded, means, covariances, priors);

Bonus points:

Encode HoG features with Fisher Vector ?

randomly collect 2~3 images from each class

Stack all HoG features together into an n x 36 data matrix

Compute its GMM

Use this GMM to encode all image HoG features (other than average)


Super Vector Aggregation – Speaker ID

Fisher Vector: Aggregates Features against a GMM

Super Vector: Aggregates GMM against GMM

Ref:

o William M. Campbell, Douglas E. Sturim, Douglas A. Reynolds: Support vector machines using GMM supervectors for speaker verification. IEEE Signal Process. Lett. 13(5): 308-311 (2006)


“Yes, We Can !”

?

Super Vector from MFCC

Motivated from Speaker ID work Speech is a continuous evolution of the vocal tract

Need to extract a sequence of spectra or sequence of spectral coefficients

Use a sliding window - 25 ms window, 10 ms shift


DCTLog|X(ω)|

MFCC

GMM Model from MFCC

GMM on MFCC feature


M

j

s

j

s

j

s

j

s pp1

)()()()( ),|()|( xx

• The acoustic vectors (MFCC) of speaker s is modeled by a

prob. density function parameterized by M

j

s

j

s

j

s

j

s

1

)()()()( },,{

• Gaussian mixture model (GMM) for speaker s:

M

j

s

j

s

j

s

j

s

1

)()()()( },,{

Universal Background Model

UBM GMM Model:


M

j

jjj pp1

)ubm()ubm()ubm()ubm( ),|()|( xx

• The acoustic vectors of a general population is modeled by

another GMM called the universal background model

(UBM):

• Parameters of the UBM

M

jjjj 1

)ubm()ubm()ubm()ubm( },,{

MAP Adaption

Given the UBM GMM, how is the new observation derivate ?

The adapted mean is given by:


Supervector Distance

Assuming we have UBM GMM model

𝜆𝑈𝐵𝑀 = {𝑃𝑘 , 𝜇𝑘, Σ𝑘},

with identical prior and covariance

Then for two utterance samples a and b, with GMM models 𝜆𝑎 = {𝑃𝑘 , 𝜇𝑘

𝑎, Σ𝑘},

𝜆𝑏 = {𝑃𝑘, 𝜇𝑘𝑏, Σ𝑘},

The SV distance is,

It means the means of two models need to be normalized by the UBM covariance induced Mahanolibis distance metric

This is also a linear kernel function scaled by the UBM covariances


𝐾 𝜆𝑎 , 𝜆𝑏 =

𝑘

𝑃𝑘Σ𝑘−(12)𝜇𝑘𝑎

𝑇

( 𝑃𝑘Σ𝑘−(12)𝜇𝑘𝑏)

Supervector Performance in NIST Speaker ID

System 5: Gaussian SV

DCF (Detection Cost Function)


m31491

AKULA – Adaptive KLUster Aggregation

2013/10/25

Abhishek Nagar, Zhu Li, Gaurav Srivastava and Kyungmo Park


Outline

Motivation

Adaptive Aggregation

Results with TM7

Summary


Motivation

Better Aggregation

Fisher Vector and VLAD type aggregation depending on a global model

AKULA removes this dependence, and directly coding the cluster centroids and sift count

SCFV/RVD all having situations where clusters are turned off due to no assignment, this can be avoided in AKULA

SIFT detection & selection K-means AKULA description


Motivation

Better Subspace Choice

Both SCFV and RVD do fixed normalization and PCA projection based on heuristic.

What is the best possible subspace to do the aggregation ?

Using a boosting scheme to keep adding subspaces and aggregations in an iterative fashion, and tune TPR-FPR to the desired operating points on FPR.


CE2: AKULA – Adaptive KLUster Aggregation

AKULA Descriptor: cluster centroids + SIFT count

A2={yc21, yc2

2, …, yc2k ; pc2

1, pc22, …, pc2

k }

Distance metric:

Min centroids distance, weighted by SIFT count

d A1 , A2 =1

𝑘 𝑗=0

𝑘d𝑚𝑖𝑛1 𝑗 𝑤𝑚𝑖𝑛

1 (𝑗) +1

𝑘 𝑖=0

𝑘d𝑚𝑖𝑛2 𝑖 𝑤𝑚𝑖𝑛

2 (𝑖)

A1={yc11, yc1

2, …, yc1k ; pc1

1, pc12, …, pc1

k },

d𝑚𝑖𝑛1 𝑗 = min

𝑖𝑑𝑗,𝑖

d𝑚𝑖𝑛2 𝑖 = min

𝑗𝑑𝑗,𝑖

w𝑚𝑖𝑛1 𝑗 = 𝑤𝑗,𝑖∗ , 𝑖∗ = 𝑎𝑟𝑔min

𝑖𝑑𝑗,𝑖

w𝑚𝑖𝑛2 𝑖 = 𝑤𝑗∗,𝑖 , 𝑗∗ = 𝑎𝑟𝑔min

𝑗𝑑𝑗,𝑖


AKULA implementation in TM7

Inner loop aggregation

Dimension is fixed at 8

Numb of clusters, or nc=8, 16, 32, to hit 64, 128, and 256 bytes

Quantization: scale by ½ and quantized to int8, sift count is 8 bits, total (nc+1)*dim bytes per aggregation



Outer loop subspace optimization by boosting

Initial set of subspace models {Ak} computed from MIR FLICKR data set SIFT extractions by k-means the space to 4096 clusters

Iterative search on subspaces to generate AKULA aggregation that can improve performance in precision-recall

Notice that aggregation is de-coupled in subspace iteration, to allow more DoF in aggregation, to find subspaces that provides complimentary info.

The algorithm is still being debugged, hence only having 1st iteration results in TM7



Outer loop subspace optimization by boosting Initial set of subspace models {Ak} computed from MIR

FLICKR data set SIFT extractions by k-means the space to 4096 clusters

Iterative search on subspaces to generate AKULA aggregation that can improve performance in precision-recall

Notice that aggregation is de-coupled in subspace iteration, to allow more DoF in aggregation, to find subspaces that provides complimentary info.

The algorithm is still being debugged, hence only having 1st iteration results in TM7

Indexing/Hashing is required for AKULA, it involves nc x dim multiplications and additions at this time. A binarization scheme will be considered once its performance is optimized in non-binary form.


GD Only TPR-FPR: AKULA vs SCFV

Data set 1:

AKULA (128bytes, dim=8, nc=16) distance is just 1-way dmin1.*wt

Forcing a weighted sum on SCFV (512 bytes) hamming distances without 2D decision fitting, i.e, count hamming distance between common active clusters, and sum up their distances


GD Only TPR-FPR: AKULA vs SCFV

Data set 2, 3:

AKULA distance is just 1-way dmin1.*wt

AKULA=128bytes, SCFV = 512 bytes.


3D object set: 4 , 5

Data set4, 5:


AKULA in PM

FPR performance:

AKULA rates:

pm rates m akula rates

512 8 64

1K 16 128

2K 16 128

1K_4K 16 128

2K_4K 16 128

4K 16 128

8K 32 256

16K 32 256


TPR@1% FPR

0

20

40

60

80

100

120

1a 1b 1c 2 3 4 5

TPR

(%

)

bitrate: 512

TM7

AKULA

0

20

40

60

80

100

120

1a 1b 1c 2 3 4 5T

PR

(%

)

bitrate: 1k

TM7

AKULA


TPR@1%FPR:

0

20

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100

120

1a 1b 1c 2 3 4 5

TPR

(%

)

bitrate: 2k

TM7

AKULA

0

20

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100

120

1a 1b 1c 2 3 4 5

TPR

(%

)

bitrate: 1k-4k

TM7

AKULA


TPR@1%FPR:

0

20

40

60

80

100

120

1a 1b 1c 2 3 4 5

TPR

(%

)

bitrate: 2k-4k

TM7

AKULA

0

20

40

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100

120

1a 1b 1c 2 3 4 5

TPR

(%

)

bitrate: 4k

TM7

AKULA


TPR@1%FPR:

75

80

85

90

95

100

105

1a 1b 1c 2 3 4 5

TPR

(%

)

bitrate: 8k

TM7

AKULA

80

85

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95

100

105

1a 1b 1c 2 3 4 5

TPR

(%

)

bitrate: 16k

TM7

AKULA


AKULA Localization

Quite some improvements: 2.7%


AKULA Summary

Benefits: Allow more DoF in aggregation optimization,

o by an outer loop boosting scheme for subspace projection optimization

o And an inner loop adaptive clustering without the constraint of the global GMM model

Simple weighted distance sum metric, with no need to tune a multi-dimensional decision boundary

The overall pair wise matching matched up with TM7 SCFV with 2-dimensional decision boundary

In GD only matching outperforms the TM7 GD

Good improvements to the localization accuracy

Light in extraction, but still heavy in pair wise matching, and need binarization scheme and/or indexing scheme to work for retrieval

Future Improvements: Supervector AKULA ?


Lec 08 Summary

Fisher Vector

Aggregate features {Xk} in RD

against GMM

Super Vector

Aggregate GMM against a global GMM (UBM)

AKULA

Direct Aggregation


++ + +

lec-08 feature aggregation ii: fisher vector, akula and super vector

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