Image Retrieval with Fisher Vectors of Binary Features
Yusuke Uchida and Shigeyuki Sakazawa
KDDI R&D Laboratories, Inc.
Saitama, Japan{ys-uchida, sakazawa}@kddilabs.jp

Abstract—Recently, the Fisher vector representation of local
features has attracted much attention because of its effectiveness in both image classification and image retrieval. Another
trend in the area of image retrieval is the use of binary featuresuch as ORB, FREAK, and BRISK. Considering the significant
performance improvement in terms of accuracy in both image
classification and retrieval by the Fisher vector of continuousfeature descriptors, if the Fisher vector were also to be applied
to binary features, we would receive the same benefits in binaryfeature based image retrieval and classification. In this paper,
we derive the closed-form approximation of the Fisher vectorof binary features which are modeled by the Bernoulli mixture
model. In experiments, it is shown that the Fisher vector
representation improves the accuracy of image retrieval by
25% compared with a bag of binary words approach.
Keywords-Fisher vector, Bernoulli mixture model, binaryfeature, image retrieval

I. I NTRODUCTION
With the advancement of both stable interest region detectors [1] and robust and distinctive descriptors [2], local
feature-based image or object retrieval has attracted a great
deal of attention. In local feature based image retrieval orrecognition, each image is first represented by a set of local
feature vectors X = {x1 , · · · , xt , · · · , xT }. A set of feature
vectors X is then encoded into a fixed length vector inorder to calculate (dis)similarity between sets of feature
vectors. The most frequently used method is a bag-of-visual
words (BoVW) representation [3], where feature vectors arequantized into visual words (VWs) using a visual codebook,resulting in a histogram representation of VWs.
Recently, the Fisher vector representation [4] has attracted
much attention because of its effectiveness. The Fisher vector is defined by the gradient of log-likelihood function normalized with the Fisher information matrix. In [4], features
vectors are modeled by the Gaussian mixture model (GMM)
and a closed form approximation is first proposed for the
Fisher information matrix of a GMM. Later, the performanceof the Fisher vector is improved in [5] by using power andℓ2 normalization. Because the Fisher vector can represent
higher order information than the BoVW representation, it
is shown that it can outperform the BoVW representation inboth image classification [5] and image retrieval tasks [6]–[8].
Another trend in the area of image retrieval is the useof binary features such as ORB [9], FREAK [10], andTable IP OSITION OF THIS PAPER .
Feature typeContinuous
Binary

BoVW
[3]
[15]

Fisher Vector[4]This paper

BRISK [11]. Binary features are one or two orders ofmagnitude faster than SIFT or SURF in detection anddescription, while providing comparable performance. Thesebinary features are especially suitable for mobile visual
search or augmented reality on mobile devices [12]. While
the Fisher vector is widely applied to continuous featuredescriptors (e.g., SIFT) which can be modeled by the GMM,
to the best of our knowledge, there has been no attempt toapply the Fisher vector to the recent binary features referred
to above for the purpose of image retrieval. Considering thesignificant performance improvement in terms of accuracy
in both image classification and retrieval by the Fisher vector
of continuous feature descriptors, if the Fisher vector werealso to be applied to binary features, we would receive
the same benefits in binary feature based image retrieval
and classification. In this paper, we derive the closed-form
approximation of the Fisher vector of binary features which
are modeled by the Bernoulli mixture model. Table I shows
the position of this paper. In experiments, we evaluate theeffectiveness of both the Fisher vector of binary features
and their associated normalization approaches. The proposed
Fisher vector representation of binary features is general andnot restricted to image features; it is also expected to be
applied to other modalities such as audio signals [13], [14].
The rest of this paper is organized as follows. In Section
2, recent binary features are briefly introduced. In Section
3, the BoVW and Fisher vector image representations aredescribed. In Section 4, we derive the Fisher vector representation of binary features. In Section 5, the effectivenessof the Fisher vector of binary features is confirmed. Our
conclusions are presented in Section 6.
II. L OCAL BINARY FEATURES
Recently, binary features such as ORB [9], FREAK [10],and BRISK [11] have attracted much attention [16]. Binaryfeatures are one or two orders of magnitude faster than SIFT
or SURF features in extraction, while providing comparable
performance to SIFT and SURF. Resulting binary featurevectors are more compact than SIFT or SURF feature
vectors. In this section, recent binary features are briefly
introduced.
A. Detection
Most of the local binary features employ fast featuredetectors. The ORB feature utilizes the FAST [17] detector, which detects pixels that are brighter or darker thanneighboring pixels based on the accelerated segment test.
The test is optimized to reject candidate pixels very quickly,
realizing extremely fast feature detection. In order to ensure
approximate scale invariance, feature points are detectedfrom an image pyramid. The FREAK and BRISK features
adopt the multi-scale version of the AGAST [18] detector.
Although the AGAST detector is based on the same criteriaas FAST, the detection is accelerated by using an optimal
decision tree in deciding whether each pixel satisfies thecriteria or not.
B. Description
Local binary features extract binary strings from patchesof interest regions instead of extracting gradient-based highdimensional feature vectors like SIFT. Many methods utilize
binary tests in extracting binary strings. The BRIEF descriptor [19], a pioneering work in the area of binary descriptors,
is a bit string description of an image patch constructed froma set of binary intensity tests. Consider the t-th smoothed
image patch pt , a binary test τ for d-th bit is defined by:{
1 if pt (ad ) ≥ pt (bd )xtd = τ (pt ; ad , bd ) =, (1)0 elsewhere ad and bd denote relative positions in the patch
pt , and pt (·) denotes the intensity at the point. Using D
independent tests, we obtain D-bit binary string xt =(xt1 , · · · , xtd , · · · , xtD ) for the patch pt . The ORB feature employs a learning method for de-correlating BRIEF
features under rotational invariance. Although the BRISK
and FREAK features use different sampling patterns fromBRIEF, they are also based on a set of binary intensity tests.
These binary features are designed so that each bit has thesame probability of being 1 or 0, and bits are uncorrelated.
In addition to these methods which extract binary features
directly, there are many methods which encode continuous feature descriptors (e.g., SIFT) into compact binary
codes [20]–[24]. By using these methods, an image can also
be represented as a set of binary features.
III. I MAGE REPRESENTATIONS
In local feature based image retrieval or recognition,
each image is first represented by a set of local features
X = {x1 , · · · , xt , · · · , xT }. A set of features X is thenencoded into a fixed length vector in order to calculate(dis)similarity between sets of features. In this section, two
encoding methods are introduced.

A. Bag-of-Visual Words
The BoVW framework is the de-facto standard way to
encode local features into a fixed length vector. In the BoVW
framework, feature vectors are quantized into VWs using
a visual codebook, resulting in a histogram representationof VWs. Image (dis)similarity is measured by L1 or L2
distance between the normalized histograms. Although it
was first proposed for an image retrieval task [3], it is now
widely used for both image retrieval [25]–[28] and image
classification [29], [30]. In [15], the bag-of-visual words
approach is also applied to binary features.
B. Fisher kernel and Fisher vector
Fisher kernel is a powerful tool for combining the benefits of generative and discriminative approaches [31]. Let
X = {x1 , · · · , xt , · · · , xT } denote the set of T local
feature vectors extracted from an image. We assume that thegeneration process of X can be modeled by a probability
density function p(X|λ) whose parameters are denoted by
λ. In [31], it is proposed to describe X by the gradient GX
λof the log-likelihood function, which is also referred to as
the Fisher score:1∇λ L(X|λ),
(2)
GX
λ =
T
where L(X|λ) denotes the log-likelihood function:
L(X|λ) = log p(X|λ).

(3)

The gradient vector describes the direction in which parameters should be modified to best fit the data [4]. A natural
kernel on these gradients is the Fisher kernel [31], which isbased on the idea of natural gradient [32]:−1 Y
K(X, Y ) = GXλ Fλ Gλ .

(4)

Fλ is the Fisher information matrix of p(X|λ) defined as
Fλ = EX [∇λ L(X|λ) ∇λ L(X|λ)T ].

(5)

Because Fλ−1 is positive semidefinite and symmetric, it
has a Cholesky decomposition Fλ−1 = LTλ Lλ . Therefore
the Fisher kernel is rewritten as a dot-product between
normalized gradient vectors GλX with:
GλX = Lλ GX
λ .

(6)

The normalized gradient vector GλX is referred to as the
Fisher vector of X [5].
In [4], the generation process of feature vectors (SIFT)
are modeled by the GMM, and the diagonal closed-form
approximation of the Fisher vector is derived. Then, theperformance of the Fisher vector is significantly improved in[5] by using power normalization and ℓ2 normalization. The
Fisher vector framework has achieved promising results andis becoming the new standard in both image classification [5]and image retrieval tasks [6]–[8].

While the Fisher vector is widely applied to continuousfeature descriptors (e.g., SIFT) which can be modeled by
the GMM, to the best of our knowledge, there has been noattempt to apply the Fisher vector to recent binary features
such as ORB [9] for the purpose of image retrieval. In
this paper, we derive the closed-form approximation of theFisher vector of binary features which are modeled by the
Bernoulli mixture model, and evaluate the effectiveness ofboth the Fisher vector of binary features and their associatednormalization approaches.
IV. F ISHER VECTOR FOR BINARY FEATURESIn this section, we model binary features with the
Bernoulli distribution, and derive the Fisher vector representation of binary features.
A. Bernoulli mixture modelLet xt ∈ {0, 1}D denote a D-dimensional binary feature out of T binary features X = {x1 , · · · , xt , · · · , xT }extracted from an image. In modeling binary features, it
is straightforward to adopt a single multivariate Bernoulli
distribution. However, although many binary descriptors
are designed so that bits of resulting binary features areuncorrelated [9], there are still strong dependencies amongbits. Therefore, a single multivariate Bernoulli component
will be inadequate to cope with the kind of complex bit
dependencies that often underlie binary features. This drawback is overcome when several Bernoulli components areadequately mixed. In this paper, we propose to model binaryfeatures with a multivariate Bernoulli mixture (BMM). The
use of the BMM instead of a single multivariate Bernoulli
distribution will be justified in the experimental section.
Let λ = {wi , µid , i = 1..N, d = 1..D} denote a set ofparameters for a multivariate Bernoulli mixture model with
N components, and xtd represents the d-th bit of xt . Given
the parameter set λ, the probability density function of T
binary features X is described as:p(X|λ)p(xt |λ)pi (xt |λ)

=

=

=

T
∏
t=1
N
∑
i=1
D
∏

p(xt |λ),wi pi (xt |λ),µxidtd (1 − µid )1−xtd .

(7)

d=1

In order to estimate the values of the parameter set λ,given a set of training binary features x1 , · · · , xs , · · · , xS ,the expectation-maximization (EM) algorithm is applied [33]. In the expectation step, the occupancy probability γs (i) (or posterior probability p(i|xs , λ)) of xs being
generated by the i-th component of BMM is calculated as
wi pi (xs |λ)γs (i) = p(i|xs , λ) = ∑N.
j=1 wj pj (xs |λ)(8)

In the maximization step, the parameters are updated as
Si =

S
∑

1 ∑
γs (i)xsd . (9)Si s=1S

γs (i), wi = Si /S, µid =s=1

In our implementation, parameter wi is initialized with 1/N ,and µid is with uniform distribution U (0.25, 0.75).
B. Deriving the Fisher vector of the BMM
In this section, we derive the Fisher vector of the BMM. In
order to calculate the Fisher vector GλX in Eq. (6), the Fisherscore GX
λ in Eq. (2) and the Fisher information matrix Fλ in
Eq. (5) should be calculated. In this paper, we consider only
the Fisher vector w.r.t. the parameter µid , because the Fisher
vector w.r.t. the weight parameter wi does not contribute to
the performance [4]. The derivation of the Fisher vector w.r.t.
wi is the same as that of GMM [4].
Letting GXµid denote the Fisher score w.r.t. the parameterµid ∈ λ, GXµid is calculated as:
GX
µid

=

T
1 ∑ ∂L(xt |λ)1 ∂L(X|λ)=T ∂µid
T t=1 ∂µid=

1∑1∂pi (xt |λ).
T t=1 pi (xt |λ) ∂µid
T

(10)

Considering that xtd in Eq. (7) can only be 0 or 1, we get:∂pi (xt |λ)= (−1)1−xtd
∂µid

D
∏

µxiete (1 − µie )1−xte . (11)e=1,e̸=dFinally we obtain:1∑(−1)1−xtdγt (i) xtd,
T t=1µid (1 − µid )1−xtd
T

GX
µid =

(12)

where γt (i) is the occupancy probability defined in Eq. (8).
Then, we derive the approximate Fisher information matrix of the BMM under the following three assumptions [4]:(1) the Fisher information matrix Fλ is diagonal, (2) the
number of binary features xt extracted from an image isconstant and equal to T , and (3) the occupancy probability
γs (i) is peaky; there is one index i such that γs (i) ≈ 1 andthat ∀j ̸= i, γs (j) ≈ 0.
As we assume the Fisher information matrix is diagonal,
Eq. (5) is approximated as Fλ ≈ diag(Fµ11 , · · · , FµN D ),where Fµid denotes the Fisher information w.r.t. µid :[()2 ]∂L(X|λ)Fµid = E
.
(13)
∂µid
Then, with the (2) and (3) assumptions, we approximately
obtain:)( ∑N∑N
j=1 wj (1 − µjd )j=1 wj µjd+.
Fµid = T wi
µ2id
(1 − µid )2
(14)

(a)

(b)

(c)

(d)

(e)

Figure 1. (a) all point pairs of 256 binary tests used in the ORB descriptor. (b)-(e) five tests corresponding to the bits with the top five probabilities µidof being 1 (red) and 0 (blue). Four components out of N = 32 components are shown.

Please refer to Appendix for the derivation. Finally, theFisher vector GλX is obtained with the concatenation of−1/2
normalized Fisher scores Fµid GXµid (i = 1..N, d = 1..D).
The Fisher vector is further normalized with power normalization and ℓ2 normalization [5]. Given a Fisher vectorz = GλX , the power-normalized vector f (z) is calculated as
f (z) = sign(z)|z|α .

(15)

In experiments, we set α = 0.5 as recommended in [5]. Afterthe power normalization, ℓ2 normalization is performed tof (z), resulting in the final Fisher vector representation of
the set of binary features.
V. E XPERIMENTIn the experiments, the Stanford mobile visual searchdataset1 is used. It contains camera-phone images of products, CDs, books, outdoor landmarks, business cards, text
documents, museum paintings and video clips. While it
includes eight classes of images, we use general CD classimages in this paper. These images consist of 100 reference
images and 400 query images. In the experiments, becausesome query images are too large (10M pixels), all images
are resized so that the long sides of images are less than 640
pixels, keeping the original aspect ratio.
As an indicator of retrieval performance, mean average
precision (MAP) [25], [27] is used. For each query, aprecision-recall curve is obtained based on the retrieval
results. Average precision is calculated as the area under the
precision-recall curve. Finally, the MAP score is calculated
as the mean of average precisions over all queries.
We adopt the ORB [9] descriptor as a binary feature because of its efficiency. On average, 900 features are extracted
from 4 scales. The parameter set λ is estimated with the EM
algorithm using one million ORB binary features extracted
from the MIR Flickr collection2 .
First, we investigate a clustering results performed in theestimation of the parameter set λ of the BMM with N = 32
components. Figure 1 (a) represents all point pairs of 256
binary tests used in the ORB descriptor. Figure 1 (b)-(e)1 http://www.stanford.edu/∼dmchen/mvs.html
2 http://press.liacs.nl/mirflickr/

represent five tests corresponding to the bits with the top
five probabilities µid of being 1 (red) and 0 (blue) in four
components out of N = 32 components. It implies that somebits of the ORB descriptor are highly correlated and thatthe BMM successfully captures this correlation. The result
justifies the use of the BMM instead of single multivariate
Bernoulli distribution to model binary features.
Then, the performance of the Fisher vector of binaryfeatures is evaluated in terms of image retrieval accuracy.
Dissimilarity between two images is defined by the Euclidean distance between the BoVW or the Fisher vector representations of the images. The following five methods arecompared: (1) bag of binary words approach (BoBW) [15],(2) Fisher vector without normalization (FV), (3) Fisher
vector with ℓ2 normalization (L2 Norm), (4) Fisher vector
with power normalization (P Norm), and (5) Fisher vector
with both power and ℓ2 normalization (P+L2 Norm). For
BoBW, a visual codebook with 1024 centroids is used.
Figure 2 shows a comparison of the Fisher vector and
BoVW representations applied to binary features, where theFisher information matrix is assumed to be the identity
matrix diag(1, · · · , 1) in Figure 2 (a), while the approximate
Fisher information matrix derived in this paper is used in
Figure 2 (b). The accuracy of the Fisher vector without
any normalization (FV) is disappointing compared with the
BoBW framework. A little surprisingly, even if the Fisher
information matrix is approximated to the identity matrix,
the accuracy is improved from 0.623 (BoBW) to 0.712(P+L2 Norm N = 256) in Figure 2 (a). If the proposed
Fisher information matrix is adopted, the accuracy is further
improved from 0.712 to 0.781 (P+L2 Norm N = 512) asshown in Figure 2 (b). This is because the Euclidean metricis not an appropriate metric in the parameter space. We canalso see that the accuracy improves as the number N ofcomponents increases, which is consistent with the case of
SIFT+GMM [8].
VI. C ONCLUSIONSIn this paper, we derived the closed-form approximationof the Fisher vector of binary features which are modeledby the Bernoulli mixture model. The effectiveness of the0.8
P+L2 Norm
L2 Norm
P Norm
FV
BoBW

0.6

Mean average precision

Mean average precision

0.8

0.4

0.6

0.4

0.2

0.2
1

P+L2 Norm
L2 Norm
P Norm
FV
BoBW

2

4

8

16

32

64

128 256 512

1

2

4

8

16

32

64

128 256 512

NN(a) Fλ ≈ diag(1, · · · , 1)(b) Fλ ≈ diag(Fµ11 , · · · , FµN D )Figure 2.

Comparison of the Fisher vector and BoVW representations applied to binary features.

Fisher vector of binary features was confirmed. There weresome interesting observations such as that the performanceof the Fisher vector without power and ℓ2 normalizationis very poor, while the Fisher vector with power and ℓ2
normalization outperforms the BoBW framework even if theFisher information matrix is approximated by the identity
matrix. In future, we will apply the Fisher vector of binary features to image classification problems. We are alsointerested in exploring the scalability of the Fisher vectorof binary features for large-scale image retrieval. We also
expect that the proposed Fisher vector representation canalso be successfully applied to other modalities such as audio
signals.
A PPENDIX
We derive the Fisher information matrix under the following three assumptions: (1) the Fisher information matrix Fλis diagonal, (2) the number of binary features xt extracted
from an image is constant and equal to T , and (3) the
occupancy probability γs (i) is peaky. From Eq. (13), we
get:
(
[(
)2 
)2 ]
T
∑
∂L(X|λ)∂L(xt |λ) Fµid = E
= E
∂µid
∂µid
t=1
[
]
(
)2
T
∑
∂L(xt |λ)
=
E
∂µid
t=1
] [
]
[
∑
∂L(xs |λ)∂L(xt |λ)E. (16)
+2
E
∂µid
∂µid
1≤t<s≤T

If the parameter set λ is estimated with maximum-likelihood
estimation, we have:][∂L(xt |λ)
= 0.
(17)
E
∂µid

Using the value of the Fisher score in Eq. (12), we get:
[(
)2 ]
∫
∂L(xt |λ)γt2 (i)E=p(xt |λ) x
2 dxt
∂µid
(µidtd (1 − µid )1−xtd )xt∫
γ 2 (i)=p(xt |λ) t 2 dxtµid
xtd =1∫γt2 (i)+p(xt |λ)dxt . (18)
(1 − µid )2
xtd =0Using the assumption that the occupancy probability γt (i)is peaky, we approximate γt2 (i) as γt (i) in Eq. (18). Finally,using the following equations,∫N∑p(xt |λ)γt (i)dxt = wi
wj µjd ,xtd =1j=1∫
p(xt |λ)γt (i)dxt= wi

xtd =0

we obtain:
Fµid = T wiN∑wj (1 − µjd ), (19)

j=1

( ∑N

j=1 wj µjd
+
µ2id

∑N

j=1 wj (1 − µjd )(1 − µid )2

)
.
(20)

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