Low Rank Global Geometric Consistency for Partial

Low Rank Global Geometric Consistency for
Partial-Duplicate Image Search
Li Yang
Yang Lin
Zhouchen Lin()
Hongbin Zha
Key Laboratory of Machine Perception (MOE)
School of EECS, Peking University
Beijing, P.R. China 100871
{yangli,linyang}@cis.pku.edu.cn, [email protected], [email protected]
Abstract—All existing feature point based partial-duplicate
image retrieval systems are confronted with the false feature
point matching problem. To resolve this issue, geometric contexts
are widely used to verify the geometric consistency in order to
remove false matches. However, most of the existing methods
focus on local geometric contexts rather than global. Seeking
global contexts has attracted a lot of attention in recent years.
This paper introduces a novel global geometric consistency,
based on the low rankness of squared distance matrices of
feature points, to detect false matches. We cast the problem of
detecting false matches as a problem of decomposing a squared
distance matrix into a low rank matrix, which models the global
geometric consistency, and a sparse matrix, which models the
mismatched feature points. So we arrive at a model of Robust
Principal Component Analysis. Our Low Rank Global Geometric
Consistency (LRGGC) is simple yet effective and theoretically
sound. Extensive experimental results show that our LRGGC is
much more accurate than state of the art geometric verification
methods in detecting false matches and is robust to all kinds of
similarity transformation (scaling, rotation, and translation) and
even slight change in 3D views. Its speed is also highly competitive
even compared with local geometric consistency based methods.
Partial-duplicate image search is to retrieve from a database
images which may contain the same contents as the query
image. Figure 1 shows some example images. The retrieved
images may be altered versions of the query image in color,
contrast, scale, rotation, partial occlusion, etc. They may also
be scenes taken from slightly different viewpoints. With the
rapid development of the Internet and the explosive increase of
digital image/video resources, partial-duplicate image search
has wide applications, such as image/video copyright violation
detection [1], crime prevention, and automatic image annotation [2][3].
Among existing partial-duplicate image search approaches, most of them first utilize the well known
bag-of-visual-words (a.k.a. Bag-of-Features, BoF) method
[3][4][5][6][7][8][9][10][11][12] to match similar local features between each pair of images. BoF can help avoid
expensive evaluation on pairwise similarity between feature
points in two images, thus improves search efficiency. BoF
based matching is achieved by quantizing local features to
visual words and assuming that feature points indexed by the
same visual word are matched. However, as only local features
are considered the ambiguity of visual words and the feature
quantization error result in many false matches among images,
leading to low retrieval precision and recall.
Fig. 1.
Examples of partial-duplicate web images.
An effective way to improve the matching accuracy is
to consider the spatial contexts among the feature points
[6]. There are two classes of methods for removing false
matches by incorporating spatial contexts. The first class tries
to build global geometric consistency. The second depends on
local geometric consistency. The conventional method, random
sample consensus (RANSAC) [6][13][14], is a representative
of the first class, which captures the geometric relationship
among all feature points in entire images. However, since
RANSAC randomly select a set of matched feature points
between two images many times in order to estimate an
optimal transformation (homography), it is very slow. So
RANSAC is usually adopted for re-ranking the top candidate
images. Due to the heavy cost of computing global geometric
consistency, the second class aims at reducing the computation
cost. These methods only verify spatial consistency among
feature points within local areas instead of entire images [11].
In [7], a simpler similarity transformation shown in Eq.
(1) replaces the homography transformation used in RANSAC.
Such a replacement may lead to only slightly worse results but
much faster speed.
cos θ − sin θ
. (1)
sin θ cos θ
In Eq. (1), (x1i , y1i ) and (x2i , y2i ) represent the coordinates
of the ith corresponding feature points between the first and
second images. s, θ, and tx and ty are the scaling factor,
rotation angle, and translations in horizontal and vertical
directions, respectively. Based on this simplified model, in
recent years many approaches, such as [7][8][12][9][10][11],
Matrix D
constructed by
D1 and D2
Original Matches
Correct Global
Corruptions by
False Matches
Detected False Matches
(Red Dashed Line)
Squared Distance
Matrix D1, D2
Fig. 2. The procedure of our LRGGC for detecting false matches. Given initially matched feature points (e.g., by BoF), we first compute the matrices that
record the squared distances between all feature points in each image. Then we stack the two distance matrices into one and apply RPCA to decompose it into
a low rank matrix and a sparse matrix. The low rank one reveals to correctly matched points, while the sparse one helps identify falsely matched points.
have been proposed to detect falsely matched pairs to improve
retrieval performance.
Many of the previous methods, such as [7][8][12], are local
geometric verification approaches. They heavily rely on the
local feature properties, like SIFT [15] features, especially 1D
dominant orientation θb and 1D characteristic scale sb. Then
the parameters θ and s in Eq. (1) could be estimated by the
following equations [7]: s = sb2 /b
s1 , θ = θb2 − θb1 , where
sb2 , sb1 , θ2 , and θ1 come from two corresponding feature points.
In Weak Geometric Consistency (WGC) [7], two histograms
are formed from a collection of s and θ, respectively, for
all corresponding pairs. The peaks of histograms are used
to filter false locally matched features. Obviously, its validity
is based on an assumption that correct local matches should
have similar local scale and rotation changes. Under the
same assumption, Enhanced WGC (E-WGC) [8] utilizes the
histogram of `2 norms of translation vectors instead of scales
and orientations. In [12], motivated by WGC and E-WGC
Wang et al. proposed Strong Geometric Consistency (SGC).
The difference is that SGC divides the matched features into
several groups based on orientation changes and uses the
histogram of 2D translation vectors.
As mentioned in [11], since these approaches only verify
spatial consistency among features within some local areas
instead of entire images, they may fail if there exists global
geometric inconsistency among local areas. In other words,
the local consistency assumption mentioned above may not be
satisfied when feature points distributed in different local areas
of images. Therefore, global geometric verification methods
are needed.
As mentioned before, RANSAC is a representative method
for global geometric verification. However, its expensive computation cost prevents its utility in large scale image search.
So in practice, it is usually applied for re-ranking top-ranked
images. To overcome the computation issue, Spatial Coding
(SC) [9] is proposed to check spatial consistency globally by
using spatial maps to record the spatial relationship among all
matched feature points. It makes the global verification much
more efficient. However, spatial coding is sensitive to rotation.
In order to address this limitation of SC, Geometric Coding
(GC) [11] is proposed. GC utilizes the dominant orientation.
Besides, it makes full use of SIFT features to build the geometric coding maps to encode the spatial relationship between
local feature points and iteratively remove false matches that
cause the largest inconsistency. In this way, GC performs both
more efficiently and effectively than SC.
In this paper, we propose a new global geometric verification approach called Low Rank Global Geometric Consistency
(LRGGC). Our approach is motivated by Robust Principal
Component Analysis (RPCA) [16], which has been proven by
theory and various applications to be effective in detecting
large and sparse noises. We first prove that the matrix that
records the squared distances between all feature points in an
image is low rank. Suppose the matches are all correct, if we
stack the two squared distance matrices into one, it will also be
low rank. If there are mismatches, the stacked matrix will differ
from a low rank one by a sparse matrix. In this way, we cast
the global geometric verification as an RPCA problem, where
the low rank matrix models the globally consistent geometric
relationship among feature points in images, and the sparse
matrix models false matches. We show the procedure of our
LRGGC in Figure 2. Our method has three advantages.
Firstly, LRGGC is a global geometric verification
method. So it will be more robust in removing false
matches than local geometric verification methods.
Second, LRGGC is very simple yet theoretically
sound. It only utilizes the coordinates of the initially
matched feature points. By its design, LRGGC can
naturally handle all kinds of similarity transformation,
including scaling, rotation, and translation. Unlike the
existing global geometric verification methods, such
as SC and GC, LRGGC does not require additional
features, like SIFT, and the corresponding histograms,
in order to reduce their sensitivity to some transformation, such as rotation. By the theory of RPCA, which
LRGGC depends on, LRGGC could detect all false
matches correctly with an overwhelming probability if
there are sufficient number of initially matched points.
That rank(D2 ) ≤ 4 can be proved in the same way.
Third, LRGGC is computationally fast. Its speed
is superior to existing global geometric verification
methods, such as RANSAC and GC, and is even
highly competitive when compared with existing local geometric verification methods, such as WGC,
EWGC, and SGC.
where λ > 0 is the scaling factor. Note that the effects of
rotation and translation are all removed after computing the
squared distances. So if we construct the follow matrix D by
stacking D1 and D2 :
In short, LRGGC combines the robustness of global verification methods and the efficiency of local verification methods,
making it extremely attractive in large scale duplicate-image
In the following, we first introduce our proposed LRGGC
in Section II. We give experimental results in Section III to
compare the effectiveness and speed of representative global
and local verification approaches, using several benchmark
datasets. Finally, we conclude the paper in Section IV.
In this section, we formulate the problem of detecting false
matches as an RPCA problem. We first introduce the basic
model of global geometric consistency, and then model the
violation of global consistency caused by false matches.
A. Modeling Global Geometric Consistency with a Low Rank
Suppose we have correct matchings between two images.
Let a1i = (x1i , y1i )T be the feature points in the first image.
Their matched points in the second image are denoted as a2i =
(x2i , y2i )T , where i = 1, · · · , n. For each image, we compute
a squared distance matrix D1 , D2 ∈ Rn×n :
D1 (i, j) = ka1i − a1j k2 ,
D2 (i, j) = ka2i − a2j k2 .
We have the following theorem.
Theorem 1. The ranks of the two matrices must satisfy:
rank(D1 ) ≤ 4 and rank(D2 ) ≤ 4.
Proof: For D1 , we first write it as
2AT1 A1
eαT1 ,
D1 = α1 e −
where α1 = ka1i k2 i=1 , e is an all-one vector, and A1 =
[a11 a12 · · · a1n ] ∈ R2×n . By this trick, we can reduce the
computing complexity from O(n2 ) to O(n). Next, we note
rank(α1 eT ) = rank(eαT1 ) = 1,
rank(2AT1 A1 ) = rank(A1 ) ≤ 2.
So we conclude that
rank(D1 ) ≤ rank(α1 eT ) + rank(2AT1 A1 ) + rank(eαT1 )
≤ 4.
Another key observation is that if {a2i } are the corresponding points of {a1i } under a similarity transformation (see Eq.
(1)), then we have
D1 = λD2 ,
then D ∈ R2n×n and
rank(D) = rank(D1 ) ≤ 4.
We see that although D may be a big-sized matrix, its rank
is actually very low. If rank(D) ≤ 4, we will have nearly
perfect global geometric consistency among all matched pairs
in images.
B. Modeling False Matches with a Sparse Matrix
When there are mismatches between feature points, (2)
no longer holds. So we cannot expect that (3) is still true.
However, suppose the percentage of false matches is not too
high, rank(D) should not deviate from a rank four matrix
too much. The discrepancy should only reside in the distances
from mismatched points to correctly matched points or the
distances among mismatched points, which should account for
a relatively small percentage in all squared distances. So we
can decompose D into two matrices:
D = A + E,
where A corresponds to the correct global geometric consistency, whose rank does not exceed four, and E accommodates
the corruptions caused by mismatched feature points, which is
a sparse matrix.
Thus, our problem can be formulated as decomposing D
into the sum of a low rank matrix A and a sparse error matrix
E, leading to the following optimization problem:
min rank(A) + γkEk0 , s.t. D = A + E,
where k·k0 represents the `0 -norm (number of nonzero entries
in a matrix) and γ > 0 is a parameter that trades off between
the rank of A and the sparsity of E.
C. Robust Principal Component Analysis
Eq. (5) is exactly the RPCA problem [16]. It is an NP-hard
problem [16]. So Cand`es et al. proposed to solve its convex
surrogate instead:
min kAk∗ + γkEk1 , s.t. D = A + E,
where k · k∗ and k · k1 represent the nuclear norm (sum of
the singular values of a matrix) and `1 -norm (sum of the
absolute values of all entries in a matrix), which are known
to be the convex envelopes of rank and k · k0 , respectively
[16]. This convex program is called Principal Component
Pursuit (PCP). Cand`es et al. proved that under quite general
conditions, e.g., the rank of A is not too high and the number
of non-zero entries in E is not too large, the ground truth
A and E can be recovered with an overwhelming probability
(the probability of failure drops to zero exponentially when the
matrix size increases) [16][17]. Note that it is remarkable that
the magnitudes in E is not assumed. Therefore, our proposed
LRGGC method is theoretically guaranteed to detect the false
matches no matter how wildly the false matches are, if the
number of initially matched points is sufficiently large and the
percentage of false matches is not too high.
PCP can be very efficiently solved by the alternating direction method (ADM) [18][19]. In particular, a few iterations are
sufficient to obtain highly accurate estimates of false matches.
In Figure 2, after false match detection by LRGGC, we
simply count the number of correct matches as the similarity
between the query and the retrieved images to rank all the
images in a database.
2) Local features: We use SIFT [15] as the standard
feature. SIFT detects Difference-of-Gaussian (DoG) key points
and extracts a 128-dimensional SIFT descriptor for each key
point. It also provides the scale and dominant orientation information of the key points. After extracting SIFT descriptors, we
adopt a codebook of 200K visual words to generate initially
matched local feature pairs by quantizing each SIFT descriptor
to a visual words index. The scale, orientation, and coordinates
of each SIFT feature are preserved for further processing.
3) Evaluation Metrics: We adopt the widely used mean
Average Precision (mAP) [6] and average time cost to evaluate
the accuracy and speed of various approaches. The mAP for a
set of queries is the mean of the average precision scores for
each query. It can be computed as follows:
mAP =
AP (q)/Q.
Here Q is the number of queries and the measure AP [6] is
the area under a precision-recall curve of the image retrieval
method. It is computed as follows:
AP =
(P (i) × r(i))/M,
In this section, we compare our LRGGC approach with
existing state of the art geometric verification methods on two
benchmark datasets, in order to validate the effectiveness and
efficiency of our method. All the experiments are performed
on a desktop computer with a 3.8 GHz CPU and 8 GB RAM.
where N is the number of relevant images, M is the number
of all images, and r(i) is an indicator function. r(i) = 1 if
the the i-th
Pi best match is a relevant image; r(i) = 0 if not.
P (i) = j=1 r(j)/i is the precision at the cut-off ranking i
in the relevant image list.
B. Compared Methods
A. Experiment Setup
1) Datasets: Two popular datasets, Holiday dataset [7] and
DupImage dataset [10], are used for evaluation. We also adopt
the MIRflickr1M [20] dataset as distracters.
Holiday dataset. Holiday contains 1491 high resolution near-duplicated images from original personal
photographs taken on holidays. The dataset has totally
500 groups of images, which covers a very large
variety of scene types (natural, man-made, etc.). The
first image of each group is used as the query image,
and the rest of the images in the same group should
be ranked at the top of the search results.
DupImage dataset. DupImage contains 1104 partialduplicate images collected from the Internet. Most of
the images are composite. They are manually divided
into 33 groups. In each groups, images are partial
duplicates of each other. In the same way, we choose
the first image in each groups as the query, and expect
that all the top-ranked images in the same group.
MIRFlickr1M dataset. MIRFlickr1M has one million
un-classified images downloaded from Flickr. The
image retrieval community often use MIRFlickr to test
the robustness and scalability of different approaches
by randomly adding images from this dataset to other
datasets. This would make image retrieval more challenging and realistic.
Several state of the art geometric verification methods are
chosen for comparison. They include the standard BoF method
[5], RANSAC [6], GC [10], WGC [7], E-WGC [8], and SGC
[12]. We use BoF as a basline. RANSAC and GC are global
verification methods and WGC, E-WGC, and SGC are local
verification methods.
C. Performance and Discussion
Figure 3 shows the mAPs of all the methods. Our proposed
LRGGC method significantly outperforms all the state of the
art methods. BoF, as a baseline method, performs the worst.
The reason is that it does not use any geometric consistency
verification. So its results have a lot of false matches. The
local verification methods, WGC, E-WGC, and SGC, perform
not as well as the global verification methods, RANSAC and
GC. This testifies to the validity of using global geometric
consistency to detect false matches. Especially, since RANSAC
is based on more powerful affine transformations, which is
time-consuming, it could only be applied on the top results.
So its performance is highly determined by the recall of top
Figure 4 summarizes the average time costs for one image
query on the two datasets by the compared methods and our
LRGGC approach. The time costs of SIFT feature extraction
and visual word index generation are not included because
they are shared by all methods, which are for detecting false
matches only. As one can see, our method is faster than
DupImage dataset
Holiday dataset
database size
Fig. 3.
The mAPs of the compared methods and our LRGGC approach on two datasets with different numbers of distracters.
Holiday dataset
Global Methods
DupImage dataset
Local Methods
average time cost per query (sec.)
average time cost per query (sec.)
Local Methods
Global Methods
Fig. 4.
database size
The average time costs of the compared methods and our LRGGC approach on the two datasets.
other two global verification methods, RANSAC and GC.
In particular, the advantage of LRGGC over RANSAC is
significant and on the Holiday data set GC’s cost is three
times as long as our LRGGC. Our method is also highly
competitive even compared with local verification methods,
WGC, E-WGC, and SGC.
Figure 3 has shown that global methods all perform better
than the local ones. On the other hand, Figure 4 shows that
RANSAC is too slow for practical applications. Therefore, we
only compare with the remaining global method, GC, on image
retrieval performance. As shown in Figure 5, two images are
selected as queries to demonstrate the performance of image
retrieval. Compared with GC, our approach has much higher
mAPs (Figure 5(a) and (b)) and more relevant images are
ranked to the top (Figure 5(c) and (d)).
By the above experiments, we confirm that our LRGGC
is both more robust and fast, making it highly attractive in
partial-duplicate image search.
We have proposed a new global geometric verification
method called LRGGC for partial-duplicate image search.
LRGGC only uses the coordinates of feature points to efficiently and robustly filter false matches. Our method is simple
yet effective and fast. The underlying RPCA model further
provides LRGGC a solid theoretical ground. The proposed
LRGGC significantly outperforms state of the art methods in
our experiments on two benchmark datasets. In the future,
we will investigate whether LRGGC can help correct false
matches. We will also target on further speeding up LRGGC.
Z. Lin is supported by NSF China (Grant Nos. 61272341,
61231002, 61121002) and MSRA. H. Zha is supported by
the National Basic Research Program of China (973 Program)
2011CB302202. The authors would like to thank Wengang
Zhou for providing their DupImage datasets.
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