Geometric Verification using semi-2D Constraints
for 3D Object Recognition
Kohei Matsuzakil∗ , Yusuke Uchida∗† , Shigeyuki Sakazawa∗ and Shin’ichi Sato‡∗ KDDI R&D Laboratories, Inc., † University of Tokyo, ‡ National Institute of Informatics∗ Saitama, Japan, † Tokyo, Japan, ‡ Tokyo, Japan∗ {ko-matsuzaki, ys-uchida, sakazawa}@kddilabs.jp, ‡ satoh@nii.ac.jp

Abstract—Geometric verification with epipolar geometry often
results in high score for an incorrect image pair due to an
ambiguity of its geometric constraints. The ambiguity is causedby high degree of freedom of epipolar geometry and weak
constraint from the fitting between a point and a line. In orderto mitigate the ambiguity, we propose to filter geometricallyinconsistent components, namely correspondences, a sample, amodel, and inliers in RANSAC-based geometric verification. For
the filtering, we introduce novel semi-2D constraints whose geometric constraint is weaker than full-2D constraint, but stronger
than pure-epipolar constraint. Additionally, an advantage ofthe proposed approach is that it requires only an image pair,neither additional information nor prior learning. Experimentson the public dataset containing 3D object images show that the
proposed approach improves the true positive rate when the false
positive rate is low, and greatly reduces computational time of
the geometric verification for both a correct image pair and anincorrect image pair.

I. I NTRODUCTION
Identifying correct correspondences between two or more
images is an important task in many computer vision problems
such as large scale image retrieval [1], object recognition [2],visual SLAM [3], and structure-from-motion [4]. The 2D-2D
correspondence problem is solved using a global geometricmodel. For example, correspondences on two planes are described by a homography, and two-view of a rigid 3D object
are described by an epipolar geometry (EG).
A typical solution to estimate the geometric model fromcorrespondences containing outliers is RANSAC (RANdom
SAmple Consensus) [5]. RANSAC computes a geometricmodel from a sample drawn randomly from correspondences.
Then it classifies correspondences into inliers and outliers
according to fitting to the model. Eventually, it chooses ageometric model which obtains the greatest number of inliers.
Many of researches about two-view geometry aim to maximize the number of inliers for an correct image pair in order toderive a high quality EG [6]–[8]. In contrast, we aim to reduce
the number of inliers for an incorrect image pair since we areinterested in discrimination of correct / incorrect image pairs
containing a rigid 3D object by thresholding their number of
inliers. It is necessary to increase the number of inliers for acorrect image pair as well as reduce the number of inliers forthe incorrect one in such discrimination. However, it has notbeen sufficiently researched so far for the latter.
Our motivation is to improve a geometric verification (GV)
in a pipeline of large scale image retrieval / recognition. Given

Fig. 1. Illustration of a result of geometric verification with an epipolargeometry on an incorrect image pair. Red circles and black lines represent
inliers and epipolar lines respectively.

a query image containing a rigid object, large scale image
retrieval [1] method retrieves a similar image containing theobject from uncalibrated database images. Many of image
retrieval methods typically represent an image as a set of local
features such as SIFT [9], encode them with such as bag-ofwords indexing [2], and rank their similarity scores between
the query image and the database images. Then, they perform
RANSAC-based GV to image pairs with top-N similarity
scores, and re-rank them according to their GV scores. In thecase of image recognition, an additional GV score thresholding
is performed to determine whether the same object is in the
database or not.
In such a pipeline, input to the GV is often an incorrect
image pair. When we assume an affine matrix or a homographymatrix as the geometric model, the number of inliers on
an incorrect image pair always becomes small. However, as
shown in Figure 1, it has been observed that GV with an EG
often returns a high number of inliers although the image pairis entirely incorrect. This is due to both the high degree of
freedom (DoF) of EG (i.e., seven DoF of fundamental matrix
or five DoF of essential matrix) and ambiguous constraint of
EG based on the fitting between a point and a line. Namely,
this is because there is a case where many correspondences
happen to fit to a certain (mostly geometrically inconsistent)
EG computed from an incorrect sample.
Such accidental fits may happen frequently since epipolarconstraint is satisfied when a point is locates somewhere ona corresponding epipolar line. On the other hand, it rarely
happen such a case with affine constraint or homography
constraint since they are based on the fitting between a point
and a point. Furthermore, if we verify the query image on
the large scale database, it is highly probable that an incorrect
image pair with high number of accidental inliers is found.
For an application containing the recognition stage, it is idealthat false positive (i.e., recognition of incorrect image) does
not exist. However, setting a large threshold simply results indecrease of true positive rate.
In order to address this problem, we propose a novel
approach that reduces accidental inliers in the RANSAC. Thisapproach performs filtering to four components of RANSAC,
namely correspondences, a sample, a model, and inliers individually. It does not only directly reduce the accidental
inliers but also indirectly reduce them by filtering the othercomponents in early stages. Thereby, it significantly reduces
inliers resulting from the incorrect image pair.
The purpose of the proposed approach is to improve theaccuracy of image recognition based on GV. Therefore, we
aim to reduce GV score for the incorrect image pair while
maintaining GV score for the correct one as possible. We
show experimentally that the proposed approach improve
the accuracy of image recognition conclusively. Furthermore,
we show that our filtering of a geometrically inconsistentcomponents achieves the speed-up of the RANSAC.
II. R ELATED WORK
We summarize the RANSAC to estimate the EG [10], [11],and also briefly describe recent extensions of the RANSAC.
A. RANSACRANSAC estimates a model T on data Q based on iteration.
It also classifies the data Q as set of inliers I or set ofoutliers O simultaneously. In each iteration, it randomly selects
a sample, namely s sets of correspondences from Q, thenit computes an EG T from the sample. In the case thatthe model T is assumed as a fundamental matrix, the 7point algorithm [12] (s = 7) or 8-point algorithm [13], [14](s = 8) is typically employed. In the case that the model T
is assumed as an essential matrix, the 5-point algorithm [15](s = 5) is typically employed. For a correspondence m, m′
in homogeneous coordinate system on data Q, it computes asquare of the distance between a point and an epipolar lined = tm′ T m. In the case that d is lower than a specified
threshold value, the correspondence is classified as an inlier;otherwise an outlier. When a set of inliers with maximum
size so far is obtained, the number of inliers |I| and T of thattime are stored. In the standard approach, such processes arerepeated until the number of iterations reaches following:M = log(1 − p)/log[1 − (1 − k)s ](1)

where p is a confidence, k = |I|/|Q| is an inlier ratio. In orderto guarantee the termination, it also terminates in the case thatthe number of iterations reaches a specified maximum number.

B. RANSAC Extensions
DEGENSAC [7] achieves more robust fundamental matrix
estimation against degeneration by detecting a homographydegenerate sample. QDEGSAC [8] achieves more robustmodel estimation on (quasi-)degenerate data without explicit
knowledge about degeneracies. However, these approaches
improve the estimation quality on an correct image pair but
do not reduce the number of inliers on incorrect one.
SCRAMSAC [16] forms a reduced set of correspondenceswith more reliability by checking a spatial consistency ofeach correspondence in a circular region around the correspondence. Then, it achieves a fundamental matrix estimationthat is fast and more robust against degeneration by operating
RANSAC on the reduced set of correspondences. However,
the risk of explosion of the accidental inliers is still remaining
since this approach does not include the explicit countermeasure for them.
Johns et al. [17] achieve a more accurate fundamental
matrix estimation than SCRAMSAC in terms of both image
retrieval and place recognition by learning generative placemodels from a significant number of training images per place.
However, this approach requires many training images withvarious changes of view point, light environment, and so forth
per target. Therefore, it is not easy to collect the training
images in practical use of the large scale image retrieval /recognition applications.
The closest approach to our motivation is SCRAMSAC [16].
However, it is limited to improve the correspondences, thus the
problem of the accidental inliers on an incorrect image pair isstill remaining unsolved. In contrast, we propose to explicitly
filter geometrically inconsistent components and accidental
inliers without any additional information and prior learning.
III. P ROPOSED A PPROACH
The proposed approach introduces novel constraints for
RANSAC-based GV using the components of local feature(i.e., orientation, scale, and coordinates). Figure 2 shows theoverview of the proposed approach. In Figure 2, blocks of thesingle line and blocks of the double line represent the standard
RANSAC process and our original process respectively. Notethat the four processes of the proposed approach (A, B, C, andD in Figure 2) are applicable independently.
The proposed approach firstly filters an initial set of correspondences Q based on the weak geometric consistency(WGC) [18] (A in Figure 2). This filtering improves the inlier
ratio of the set of correspondences on a correct image pair,thus increases the probability of finding the correct solution.
Furthermore, it leads speed-up for the RANSAC by reducing
the size of the resulting set of correspondences Q′ . In all of
the subsequent process, the proposed approach operates on
the Q′ . In the inside of the RANSAC iteration, the proposed
approach detects a geometrically inconsistent sample, thenterminates the iteration early if necessary (B in Figure 2).
The proposed approach detects an EG that has not spatialconsistency with the sample, then terminates the iteration early
if necessary (C in Figure 2). After the model fitting to Q′ , theܳA. Correspondence selection based on WGCܳԢRANSAC Iteration
Draw random sample from ܳԢ

B. Sample relative configuration check
true
false
Compute model ܶ

C. Epipolar geometry check
false
true
Fit the model ܶ to ܳԢ‫ܫ‬D. Inlier relative configuration check‫ܫ‬Ԣ

Meet a termination criterion
true
false
‫ܫ‬Ԣǡ ܶFig. 2. Overview of the proposed approach

proposed approach check the spatial consistency of resulting
set of inliers I against the sample and EG (D in Figure 2).
The proposed approach filters the I through this check, thendetermines the final set of inliers I ′ .
The filters of B, C, and D in Figure 2 are based on semi2D constraints that always can be satisfied if the target is 2D
object. That is to say, these constraints result in a geometricconstraint that is weaker than a full-2D constraint, but stronger
than an pure-epipolar constraint. Although these constraints
involve the risk of rejecting correct components (i.e., a sample,
an EG, and inliers) ascribable to the parallax, most of thecases may find the sufficient solution through the RANSAC
iterations. In the following, we describe the four elements ofthe proposed approach.

We apply this assumption to the correspondence selection.
That is to say, we vote the Q to the two-dimensional bins
consisting of orientation difference and scale ratio. Then we
filter the correspondences excepting correspondences belonging to a bin with a maximum voting score. For softer voting,
we establish the bins with overlapping half of resolution withregard to orientation difference and scale ratio. When we set
the resolution of orientation difference to 30 degrees, the 24
bins with overlapping every 15 degrees are established. We
name this method the correspondence selection based on WGC(CSW).
By using CSW, it is expected that voting score of each bin
becomes random, and the size of Q′ is significantly reduced
consequently on an incorrect image pair. Reducing the size of
Q′ result in reducing the possibility of the accidental inliers.
On the other hands, potential inliers concentrate on a specificbin on a correct image pair, maintaining the final number of
inliers. As a result of improved inlier ratio, expected numberof iteration until finding the correct solution gets fewer.
B. Sample Ralative Configuration Check
The filter named the sample relative configuration check(SRCC) is based on the assumption that the relative configuration of correct sample (i.e., s sets of correspondences) has
a consistency between a correct image pair. Geometricallyinconsistent sample is rejected by using this filter, thus thepotential inliers resulting from the sample are rejected.
As with shown in Figure 3, s C2 corresponding triangles
can be formed from the sample. For each corresponding
triangle, SRCC detects the inside-out. Specifically, it examines
the direction of rotation (i.e., clockwise or anti-clockwise)from the three vertexes of the triangle. It determines thatthe corresponding triangles do not have the inside-out if theirdirection is match.
For the 3D object, corresponding triangles from correct sample can generate the inside-out ascribable to parallax. However,
it is expected that the frequency is significantly less thanthe one from the incorrect sample. Therefore, SRCC counts
the number of inside-out from among the s C2 corresponding
triangles, then it returns true if the count is lower than thethreshold value T Htri ; otherwise it returns false.

݉ଶA. Correspondence Selection based on WGC
WGC [18] is originally proposed to improve a scoring algorithm in the pipeline of large scale image retrieval. Specifically,
it votes the matching descriptors to the bins of orientationdifference and the bins of scale ratio. Then it improves theaccuracy with regard to image retrieval by filtering the bins
excepting a bin with a maximum voting score. WGC isbased on the assumption that correct correspondences havea consistency with regard to orientation difference and scaleratio.

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݉Ԣଶ

݉ସ
݉ଷ

݉ହ

݉Ԣଵ

݉Ԣଷ

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݉Ԣହ

Fig. 3. An example of sample configuration in the case of s = 5. Colored
circles represent the 5 sets of correspondences mi , m′i (i = 1, 2, ..., 5). The
5 C2 = 10 corresponding triangles are formed from them. The dotted line of
the cubes represent the target 3D object

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Fig. 4. Illustration of the relative configuration of components. White quadrangles, colored circles, black straight lines, and dotted lines represent epipoles
projected onto the image, sample correspondences, epipolar lines, and equidistant curves from the epipole respectively. Red stars and blue stars represent a
correct inlier correspondence and an incorrect inlier correspondence respectively.

This filter has a high possibility to return a wrong result if
the sample points are dense in extremely narrow range or the
sample points are almost on a same straight line such as on
the character string image. As an exception process to avoid
such cases, SRCC rejects the too less scattered sample in thex-y space. Specifically, it returns false if the area of the convex
hull of the sample is extremely small on each image.
C. Epipolar Geometry Check
The filter named the epipolar geometry check（EGC） isbased on the assumption that the relative configuration of
epipoles from correct EG and the sample has a consistency
between a correct image pair. Geometrically inconsistent EG isrejected by using this filter, thus the potential inliers resulting
from the EG are rejected.
As with shown in Figure 4, s sets of epipolar lines arecomputed from an EG and s sets of sample correspondenceson each image. It is known that the epipole e on the image
can be calculated from the fundamental matrix [12]. Let mi
denote sample points (i = 1, 2, ..., s). Let oi and di denoteorientations and distances from e to mi respectively.
In Figure 4, the corresponding orientations oi and o′iare arranged in the same order from end to end (i.e.,o1 , o2 , o4 , o3 , o5 ) in the image pair. The corresponding distances di and d′i are also the same (i.e., d3 , d2 , d1 , d5 , d4 ). In
this way, EGC generates permutations of the sample correspondence ID with respect to orientation and distance in theimage pair. If the permutations with regard of both orientationand distance are match, EGC returns true.
In Figure 4, both epipoles are located in the same side of
the object, but they can be located in the opposite sides of theobject with each other. Taking account into such a case, EGC
returns true if the permutations with regard of both orientationand distance are match in reverse order.

If the epipole is located in the inside of the convex hullof the sample correspondences, it means that the camera is
not moving only back and forth, then epipolar lines become
radially. In that case, EGC returns true if the circular permutations are match with respect to orientation since it does notfind both ends.
D. Inlier Relative Configuration Check
The filter named the inlier relative configuration check(IRCC) is based on the assumption that the relative configuration of correct inlier, the epipoles, and the sample has
a consistency between a correct image pair. Geometricallyinconsistent inliers are rejected by using this filter.
As with shown in Figure 4, an image can be disjointed tounequal regions by using both epipolar lines and equidistantcurves from the epipole. We propose to accept the inlier only
if both inlier points are located in the corresponding region ofeach image.
Specifically, IRCC firstly computes the orientation anddistance from the epipole to the inlier points on each image.
Then, it finds the sample correspondence IDs of both sides ofa point of inlier with regard of both orientation and distanceon each image. If these sample correspondence IDs are match
between the image pair, the inlier is accepted; otherwise theinlier is re-classified to outlier.
For example, an inlier represented by red stars in Figure 4
is accepted since they are located in the corresponding divided
region of each image. On the contrary, an inlier represented
by blue stars is re-classified to outlier since they are locatedin the incorresponding divided region of each image.
IV. E XPERIMENTS
We experimentally evaluate the proposed approach (Prop),
then compare it with SCRAMSAC [16].

A. Dataset and Experimental Setup
In our experiments, we use the University of Kentucky
Benchmark1 (UKB) which is a standard object retrieval /recognition benchmark. UKB consists of 10,200 images taking
2,550 objects from four different viewpoints. We generate
4 C2 = 6 correct image pairs per object from the four images
taking the same object. For the incorrect image pairs, we
generate 2, 550 × 6 sets of random image pairs from UKB
as there is no correct image pair. Therefore, we conduct theGV on 2, 550 × 6 × 2 sets of image pair.
We regard that the GV score exceeds a certain thresholdvalue as a ”positive”. We define the ”recognition rate” as true
positive rate when the false positive rate = 0, and ”T Hrec ” asthe threshold value of that time.
For local features, we adopt the ORB [19] which is efficientand suitable for mobile devices. On average, 900 features
are extracted from 8 scales. In order to give the initial
set of correspondences to each image pair, we perform thenearest neighbor matching with cross-check method. These
correspondences are firstly matched to the nearest neighbor tothe other side from one side, then they are filtered excepting
the nearest neighbor to one side from other side.
We assume the fundamental matrix as the model of
RANSAC. In order to compute it, we employ the 7-point
algorithm [12]. The threshold value with respect to the distance
between a point and an epipolar line is set to three. In thecase that we assume the homography matrix as the modelof RANSAC, the threshold value with respect to the reprojection error is set to three. The maximum number of
iterations of RANSAC is set to 10,000. In order to draw acorrect sample with fewer iterations, we employ the PROSAC[20] strategy with respect to the sample drawing in theProp. For parameters specific to SCRAMSAC, we employ
the same values of experiments in the literature [16] (i.e.,smin = 0.5, smax = 2, θ = 0.55, r = 7 in [16]).
All experiments are performed on a 3.6 GHz Intel Corei7 with 4 GB of RAM. Computation time includes only theprocess shown in Figure 2, and does not include time for the
local feature extraction and initial correspondences generation.
For our CSW (A in Figure 2), the resolution with regardof orientation difference and scale ratio is set to 60 degrees
and four-fold respectively. They are fairly coarse resolutions
to avoid filtering correct correspondences. For our SRCC (Bin Figure 2), the threshold value T Htri with regard of thenumber of inside-out is set to three. These parameters provided
excellent results in our preliminary experiments.

TABLE II MPACT OF E ACH E LEMENT OF THE P ROPOSED A PPROACH
Computation time [ms]Recognitionrate

T HrecBase(EG)

0.695

Base(EG)+CSW

0.705

Base(EG)+SRCC

0.766

39.1

13.6

13.1

Base(EG)+EGC

0.745

23.2

245.9

400.2

Base(EG)+IRCC

0.797

18.2

404.8

644.2

correct pairincorrect pair

68.6

170.5

294.6

57.2

37.3

137.8

element to the Base(EG) respectively. We summarize therecognition rate, T Hrec , and computation time on each of thecorrect / incorrect image pair obtained from these methods inTable I.
We can see that all elements result in higher recognition rate and smaller T Hrec than those of the Base(EG).
Base(EG)+CSW reduces the probability of accidental inliers by reducing the size of the Q. Base(EG)+SRCC andBase(EG)+EGC indirectly reduce accidental inliers by rejecting geometrically inconsistent sample and EG respectively.
Base(EG)+IRCC directly reduces accidental inliers.
In regard to the computation time, it is shown that
Base(EG)+CSW and Base(EG)+SRCC achieve the speedup compared with the Base(EG), while Base(EG)+EGC andBase(EG)+IRCC are computationally expensive. However,thet later two have negligible impact on computation time of
the Prop since the most iterations are terminated in an earlystage by combining with the other elements. This fact is shownin the next subsection.
In general, the process on the correct pair is faster thanthe one on the incorrect pair since the former terminates in an
early stage when it obtains a high inlier rate k in Equation 1. In
contrast, note that only Base(EG)+SRCC achieves comparable
results on the incorrect pair. This is because the frequency withwhich the iteration is early terminated by SRCC is higher on
the incorrect pair.
C. Compare the Proposed Method with SCRAMSAC

We compare the proposed approach (Prop) with SCRAMSAC [16]. Given the same initial set of correspondences
extracted from the same image pair, we measure the GV performances of them. As a reference of the full-2D constraint, we
also measure the PROSAC [20] with a homography constraint.
Let ”Base(H)” denote this method.
As an indicator of GV performance, we plot the ROC
B. Impact of Each Element of the Proposed Approach(Receiver Operating Characteristic) curve. Its vertical line
We evaluate each element of the Prop, CSW (A in Figure shows the true positive rate and the horizontal one shows2), SRCC (B in Figure 2), EGC (C in Figure 2), and IRCC false positive rate. For reference, we also plot the results of(D in Figure 2). We employ the PROSAC [20] with an EG as Base(EG) in the previous subsection. All results are showna baseline method. Let ”Base(EG)” denote this method. Let in Figure 5. Plot shows the resulting average value. We”Base(EG)+CSW”，”Base(EG)+SRCC”，”Base(EG)+EGC”， summarize the recognition rate, T Hrec , and computation timeand ”Base(EG)+IRCC” denote the method added to each on each of the correct / incorrect image pair obtained fromthese methods in Table II.
1 http://vis.uky.edu/ stewe/ukbenchTABLE IIC OMPARE THE P ROPOSED M ETHOD WITH SCRAMSAC

1

True Positive Rate

0.95
0.9
0.85

Recognitionrate

T HrecBase(H)

0.768

SCRAMSAC [16]

0.735

Prop

0.840

Computation time [ms]correct pairincorrect pair

17.3

183.5

343.1

47.0

15.5

20.2

13.2

11.6

9.8

Base(H)

0.8

Base(EG)

V. C ONCLUSIONSCRAMSAC

0.75

Prop
0.7
0

0.2

0.4

0.6

0.8

1

False Positive Rate
Fig. 5. ROC curves

In this paper, we propose a novel geometric verification
using the semi-2D constraints intended for 3D object. The proposed approach reduces the accidental inliers on the incorrect
image pair without additional information and prior learning.
Experimental results show that the proposed approach issuperior to recent geometric verification approaches in termsof image retrieval / recognition.
R EFERENCES

As can be seen from Figure 5, Prop has the highest true
positive rate when false positive rate is low. This is becausethat Prop achieves to explicitly reduce the accidental inliers
on an incorrect image pair. On the other hand, SCRAMSAC
has the highest true positive rate when false positive rate ishigh. Such characteristic of Prop provides the more reliable
results with respect to the practical use of the large scale image
retrieval / recognition applications.
As can be seen from Table II, SCRASAC has lower true
positive rate in the case of false positive rate = 0 (i.e.,
recognition rate) than Base(H). In contrast, Prop always has
higher true positive rate than Base(H). This is because that
Prop imposes a geometric constraint that is weaker than full2D constraint, but stronger than pure-epipolar constraint forthe database containing 3D objects. That is to say, the full-2D
constraint often does not fit to the 3D object on the correct
image pair, and the pure-epipolar constraint often explode theaccidental inliers on the incorrect one. In contrast, Prop is able
to find the EG to fit the 3D object on the correct image pairwhile reducing the accidental inliers on the incorrect one.
In regard to the computation time, as can be seen from Table
II, Prop is faster than SCRAMSAC on both correct / incorrect
image pair. The main reason for this is the early rejectionof sample with our SRCC (B in Figure 2). On the incorrect
image pair, Prop rejects almost all samples, then it does not
perform the subsequent process. Therefore, Prop on incorrect
image pair is faster than the one on correct image pair. On
the other hands, Prop often perform the subsequent process
which is computationally expensive on the correct image pair,but it often terminates early by obtaining a result with highinlier rate k in Equation 1. As a result, Prop is fast even on
the correct image pair.
Therefore, Prop is superior as the GV method in the context
of large scale image retrieval / recognition, in terms of bothaccuracy and efficiency.

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