Fast Directional Chamfer Matching∗

Ming-Yu Liu† Oncel Tuzel‡ Ashok Veeraraghavan‡ Rama Chellappa††University of Maryland College Park ‡Mitsubishi Electric Research Labs

{mingyliu,rama}@umiacs.umd.edu {oncel,veerarag}@merl.com

Abstract

We study the object localization problem in images givena single hand-drawn example or a gallery of shapes as theobject model. Although many shape matching algorithmshave been proposed for the problem over the decades,chamfer matching remains to be the preferred method whenspeed and robustness are considered. In this paper, we sig-nificantly improve the accuracy of chamfer matching whilereducing the computational time from linear to sublinear(shown empirically). Specifically, we incorporate edge ori-entation information in the matching algorithm such thatthe resulting cost function is piecewise smooth and the costvariation is tightly bounded. Moreover, we present a sublin-ear time algorithm for exact computation of the directionalchamfer matching score using techniques from 3D distancetransforms and directional integral images. In addition, thesmooth cost function allows to bound the cost distributionof large neighborhoods and skip the bad hypotheses within.Experiments show that the proposed approach improves thespeed of the original chamfer matching upto an order of45x, and it is much faster than many state of art techniqueswhile the accuracy is comparable.

1. Introduction

Humans utilize extensive prior information in the formof multiple visual cues including color, texture and shapeto recognize objects. Machine vision algorithms attempt toimitate human perception system by learning similar pri-ors based on exemplars. However, collecting the requiredtraining data is a tedious task and significantly limits the ef-fectiveness of these algorithms in many cases. For instance,it is much more appealing to search an image collection us-ing a single exemplar supplied by a user than learning everypossible object class beforehand. Yet, recognition of objectsin images using only a few exemplars remains to be a verychallenging problem.

The common approach to tackle this problem is to uti-lize features that exhibit the least variability within ob-

∗Partially supported by MURI from the Office of Naval Researchunderthe GrantN00014 − 08−I−0638.

ject classes and across imaging conditions together with asimilarity measure that models the maximum invariances.Among the aforementioned visual cues, shape informationlargely satisfies invariances; also in many cases only a sin-gle hand-drawn contour is discriminative enough to recog-nize and localize an object in a cluttered image. Fast and ac-curate shape matching has numerous applications includingobject recognition and localization, image retrieval, pose es-timation, and tracking.

1.1. Related Work

An extensive literature exists for shape matching andhere we overview only a few. Several researchers proposedshape representations and similarity measures invariant todeformation or articulation of objects [3, 12]. They success-fully handle intra-class variations and achieved good perfor-mance in object recognition. However, these methods typ-ically require clean segmentation of shapes which rendersthem unsuitable while dealing with cluttered images.

More recent studies focus on recognition and localiza-tion of object shapes in cluttered images. In [4], the shapematching problem is posed as finding the optimal corre-spondences between feature points which then leads to aninteger quadratic programming problem. In [10], a con-tour segment network framework is described where shapematching is formulated as finding paths on the network sim-ilar to model outlines. In [8], Ferrari et. al. proposed afamily of scale invariant local shape descriptors (pair-of-adjacent-segment feature) which are formed by k-connectednear straight contour fragments. These descriptors are laterutilized in a shape matching framework [9] through a votingscheme on a Hough space. The solution is then iterativelyrefined using a point registration algorithm. Zhu et. al. [20]formulated shape detection as subset selection on a set ofsalient contours. The selection problem is approximatelysolved using a two-stage linear programming. In [7], a hi-erarchical object contour representation was proposed to-gether with a dynamic programming approach for match-ing. In [15], a multi-stage approach is employed wherecoarse detections via matching subsets of contour segmentsare pruned by building the entire contour using dynamic

programming.These algorithms provided impressive results for match-

ing shapes in cluttered images. However, they share a com-mon drawback, high computational complexity, that makesthem unsuitable for time critical applications. Althoughproposed decades ago, chamfer matching [1] remains to bethe preferred method when speed and accuracy are consid-ered as discussed in [18]. There exist several new variantsof chamfer matching mainly to improve the cost functionusing orientation information [11, 16]. We discuss moreabout these approaches in comparison to our formulation inthe following sections.

1.2. Contributions

In this paper, we greatly improve the accuracy of cham-fer matching while significantly reducing its computationalcost. We propose an alternative approach for incorporat-ing edge orientation and solve the matching problem in anorientation augmented space. This formulation plays anactive role in defining edge correspondences which resultin more robust matching in highly cluttered environments.The resulting cost function is smooth and the cost variationis tightly bounded.

The best computational complexity for existing cham-fer matching algorithms is linear in the number of templateedge points, even without the orientation term. We optimizethe directional matching cost in three stages: (1) We presenta linear representation of the template edges. (2) We thendescribe a three dimensional distance transform represen-tation. (3) Finally, we present a directional integral imagerepresentation over distance transforms. Using these inter-mediate structures, exact computation of the matching scorecan be performed in sublinear time in the number of edgepoints. In the presence of many shape templates, the mem-ory requirement also reduces drastically. In addition, thesmooth cost function allows to bound the cost distributionof large neighborhoods and skip the bad hypotheses within.

2. Chamfer Matching

Chamfer matching (CM) [1] is a popular technique tofind the best alignment between two edge maps. LetU ={ui} andV = {vj} be the sets of template and query imageedge maps respectively. The chamfer distance betweenU

andV is given by the average of distances between eachpointui ∈ U and its nearest edge inV

dCM (U, V ) =1

n

∑

ui∈U

minvj∈V

|ui − vj |. (1)

wheren = |U |.LetW be a warping function defined on the image plane

parameterized withs. For instance, if it is a 2D Euclideantransformation,s ∈ SE(2), s = (θ, tx, ty), wheretx and

ty are the translations alongx andy axis respectively andθis the in-plane rotation angle. Its action on image points isgiven via the transformation

W(x; s) =

(

cos(θ) − sin(θ)sin(θ) cos(θ)

)

x+

(

txty

)

. (2)

The best alignment parameters ∈ SE(2) between thetwo edge maps is then given by

s = arg mins∈SE(2)

dCM (W(U ; s), V ) (3)

whereW(U ; s) = {W(ui, s)}.Chamfer matching provides a fairly smooth measure of

fitness, and can tolerate small rotations, misalignments, oc-clusions, and deformations. The matching cost can be com-puted efficiently via a distance transform imageDTV (x) =minvj∈V |x − vj | which specifies the distance from eachpixel to the nearest edge pixel inV . The distance transformcan be computed in two passes over the image [6] and usingwhich the cost function (1) can be evaluated in linear timeO(n) via dCM (U, V ) = 1

n

∑

ui∈U DTV (ui).

3. Directional Chamfer Matching

Chamfer matching becomes less reliable in the presenceof background clutter. To improve robustness, several vari-ants of chamfer matching have been introduced by incorpo-rating edge orientation information into the matching cost.In [5, 11], the template and query image edges are quantizedinto discrete orientation channels and individual matchingscores across channels are summed. Although this methodalleviates the problem of cluttered scenes, the cost functionis very sensitive to the number of orientation channels andbecomes discontinuous in channel boundaries. In [16], thechamfer distance is augmented with an additional cost fororientation mismatch which is given by the average differ-ence in orientations between template edges and their near-est edge points in the query image. The method is calledoriented chamfer matching and throughout the paper we usethe abbreviation OCM.

Instead of an explicit formulation of the orientation mis-match, we generalize the chamfer distance to points inR

3

for matching directional edge pixels. Each edge pointx isaugmented with a direction termφ(x) and the directionalchamfer matching (DCM) score is given by

dDCM (U, V ) =1

n

∑

ui∈U

minvj∈V

|ui−vj|+λ|φ(ui)−φ(vj)|

(4)whereλ is a weighting factor between location and ori-entation terms. Note that the directionsφ(x) are com-puted at moduloπ, and the orientation error gives theminimum circular difference between the two directionsmin{|φ(x1)− φ(x2)|, ||φ(x1)− φ(x2)| − π|}.

(a) (b)Figure 1. Matching costs per edge point for (a) Oriented chamfermatching [16]; (b) Directional chamfer matching. DCM jointlyminimizes location and orientation errors whereas in [16] the lo-cation error is augmented with the orientation error of the nearestedge point.

(a) (b)Figure 2. Linear representation. (a) Edge image. The image con-tains 1242 edge points. (b) Linear representation of the edge im-age. The image contains 60 line segments.

In Figure 1, we present a comparison of the proposedcost function with [16]. It can be easily verified that theproposed matching cost is a piecewise smooth function ofboth the translationtx, ty and the rotationθ of the templateedges. Therefore, matching is more robust against clutter,missing edges and small misalignments.

4. Search Optimization

To the best of our knowledge, the lowest computationalcomplexity for the existing chamfer matching algorithms islinear in the number of template edge points, even withoutthe directional term. In this section, we present a sublin-ear time algorithm for exact computation of the 3D chamfermatching score (4).

4.1. Linear Representation

The edge map of a scene does not follow an unstruc-tured binary pattern. Instead, the object contours complywith certain continuity constraints which can be retained byconcatenating line segments of various lengths, orientationsand translations. Here, we represent an edge image witha collection ofm-line segments. Compared with a set ofpoints which has cardinalityn, its linear representation ismore concise. It requires onlyO(m) memory to store anedge map wherem << n.

We use a variant of RANSAC algorithm to compute thelinear representation of an edge map. The outline of the al-gorithm is as follows. The algorithm initially hypothesizesa variety of lines by selecting a small subset of points andtheir directions. The support of a line is given by the set of

(a) (b) (c) (d) (e)Figure 3. Computation of the integral distance transform tensor.(a) The input edge map. (b) Edges are quantized into discreteori-entation channels. (c) Two dimensional distance transformof eachorientation channel. (d) The three dimensional distance transformDT3 is updated based on the orientation cost. (e)DT3 tensoris integrated along the discrete edge orientations and integral dis-tance transform tensor,IDT3, is computed.

points which satisfy the line equation within a small resid-ual and form a continuous structure. The line segment withthe largest support is retained and the procedure is iteratedwith the reduced set until the support becomes smaller thana few points.

The algorithm only retains points with certain structureand support, therefore the noise is filtered. In addition, thedirections recovered through the line fitting procedure aremore precise compared to those computed using local oper-ators such as image gradients. An example of linear repre-sentation is given in Figure 2 where a set of 1242 points ismodeled with 60 line segments.

4.2. Three-Dimensional Distance Transform

The matching score given in (4) requires finding theminimum cost match over location and orientation termsfor each template edge point. Therefore the computationalcomplexity of the brute-force algorithm is quadratic in thenumber of template and query image edge points. Here wepresent a three-dimensional distance transform representa-tion (DT 3) to compute the matching cost in linear time. Wenote that, a similar structure was also used in [14] for fastevaluation of Hausdorff distances.

This representation is a three dimensional image tensorwhere the first two dimensions are the locations on the im-age plane and the third dimension is the quantized edgeorientations. The orientations are quantized intoq discretechannelsΦ = {φi} evenly in[0 π) range. Each elementof the tensor encodes the minimum distance to an edge pointin the joint location and orientation space:

DT 3V (x, φ(x)) = minvj∈V

|x− vj |+ λ|φ(x)− φ(vj)|. (5)

whereφ(x) is the nearest quantization level in orientationspace toφ(x) in Φ.

We present an algorithm to computeDT 3 tensor inO(q)passes over the image by solving two dynamic programsconsecutively. Equation (5) can be rewritten as

DT 3V (x, φ(x)) = minφi∈Φ

(

DTV {φi}

+ λ|φ(x)− φi|)

(6)

whereDTV {φi}

is the two dimensional distance transform

of the edge points inV having orientationφi.Initially, we computeq two dimensional distance trans-

formsDTV {φi}

which requiresO(q) passes over the imageusing the standard distance transform algorithm [6]. Sub-sequently, theDT 3V tensor (6) is computed by solving asecond dynamic program for each location separately. Thetensor is initialized with the two dimensional distance trans-formsDT 3V (x, φi) = DT

V {φi}(x) and is updated with a

forward recursion

DT 3V (x, φi) = min {DT 3V (x, φi), (7)

DT 3V (x, φi−1) + λ|φi−1 − φi|}

and a backward recursion

DT 3V (x, φi) = min {DT 3V (x, φi), (8)

DT 3V (x, φi+1) + λ|φi+1 − φi|}

for each locationx. Unlike the standard distance transformalgorithm, a special condition is needed for handling thecircular orientation distance. The forward and backward re-cursions do not terminate after a full cycle,i = 1 . . . q ori = q . . . 1 respectively, but the costs are continued to be up-dated in a circular form until the cost for a tensor entry is notchanged. Note that, at most one and a half cycles are neededfor each of the forward and backward recursions, thereforethe worst computational cost isO(q) passes over the image.Using the three dimensional distance transform representa-tion DT 3V the directional chamfer matching score of anytemplateU can be computed in linear time via

dDCM (U, V ) =1

n

∑

ui∈U

DT 3V (ui, φ(ui)). (9)

4.3. Distance Transform Integral

Let LU = {l[sj,ej ]}j=1...m be the linear representationof template edge pointsU wheresj and ej are the startand end locations of thej-th line respectively. For ease innotation, we sometimes refer to a line with only its indexlj = l[sj,ej ]. We assume that the line segments only have

directions among theq discrete channelsΦ, which is en-forced while computing the linear representation. Althoughit might be argued that the discreet line directions introducequantization artifacts, in fact the linear representationgivenin Figure 2b is generated using onlyq = 60 directions and it

is difficult to observe the difference from the original edgeimage (Figure 2a).

All the points on a line segment are associated with thesame orientation which is the direction of the lineφ(lj).Hence the directional chamfer matching score (9) can berearranged as

dDCM (U, V ) =1

n

∑

lj∈LU

∑

ui∈lj

DT 3V (ui, φ(lj)). (10)

In this formulation, thei-th orientation channel of theDT 3V tensor,DT 3V (x, φi), is only evaluated for summingover the points of line segments having directionφi.

Integral images are intermediate image representationsused for fast calculation of region sums [19] and linearsums [2]. Here we present a tensor of integral distancetransform representation (IDT 3V ) to evaluate the summa-tion of costs over any line segment inO(1) operations. Foreach orientation channeli, we compute the one-directionalintegral alongφi (Figure 3).

Letx0 be the intersection of an image boundary with theline passing throughx and having the directionφi. Eachentry ofIDT 3V tensor is given by

IDT 3V (x, φi) =∑

xj∈l[x0,x]

DT 3V (xj , φi). (11)

The IDT 3V tensor can be computed in one pass overtheDT 3V tensor. Using this representation, the directionalchamfer matching score of any templateU can be computedin O(m) operations via

dDCM (U, V ) =1

n

∑

l[sj ,ej ]∈LU

[IDT 3V (ej , φ(l[sj ,ej ]))−

IDT 3V (sj , φ(l[sj,ej ]))]. (12)

The O(m) complexity is only an upper bound on thenumber of computations. For object detection or local-ization we would like to retain only the hypotheses wherethe matching costs are less than the detection threshold orequivalently for localization less than the best hypothesis.We order the template lines with respect to their supportsand start the summation from the line with the largest sup-port. A hypothesis is eliminated during the summation ifthe cost is larger than the detection threshold or current besthypothesis. The supports of the line segments show expo-nential decay, therefore for majority of the hypotheses onlya few arithmetic operations are performed. Weempiricallyshow that the number of evaluated line segments issublin-ear in the number of template pointsn.

4.4. Optimized Region Search

The proposed DCM cost function (4) is smooth. More-over, the variation of the matching cost is bounded and only

changes smoothly in the spatial domain. We utilize this factto significantly reduce the amount of hypotheses evaluatedfrom an image. Letδ ∈ R

2 be the translation of the modelU in the image plane. The DCM cost variation due to trans-lation then becomes

dDCM (U + δ, V )

= 1n

∑

ui∈U minvj∈V |ui + δ − vj |+ λ|φ(ui)− φ(vj)|

≤ 1n

∑

ui∈U minvj∈V |ui − vj |+ |δ|+ λ|φ(ui)− φ(vj)|

= |δ|+ dDCM (U, V ). (13)

Therefore, the variation of the DCM cost is bounded by thespatial translation|dDCM (U+δ, V )−dDCM (U, V )| ≤ |δ|.If the targeting matching cost isǫ and the cost of the currenthypothesis isψ, i.e. ǫ < ψ, there can not be a detectionwithin the|ψ − ǫ| pixel range of the current hypothesis andwe can skip the evaluation of the hypotheses within this re-gion.

5. Experiments

We conducted three sets of experiments on challengingsynthetic and real datasets. Note that, in all our experimentswe emphasize the speed and the improved accuracy of ourapproach compared to chamfer matching. Although the per-formance of proposed approach is comparable with state ofart methods, it can also be utilized to quickly retrieve accu-rate initial hypotheses which can then be refined using moreexpensive point registration algorithms. In all our experi-ments we usedq = 60 orientation channels and 6 degreeserror in orientation corresponds to 1 pixel distance.

5.1. Object Detection and Localization

In the first experiment, we performed object detectionand localization on the ETHZ shape class dataset [10]. Thedataset contains255 images where each image contains oneor more objects from five different classes: apple logos, bot-tles, giraffes, mugs, and swans. The objects in the datasethave large variations in appearance, viewpoint, size, andnon-rigid deformation. We followed the experimental setupproposed in [10, 9] in which a single hand-drawn shape foreach class was used to detect and localize its instances inthe dataset.

Our detection system is based on scanning using a slid-ing window, i.e., we retain all the hypotheses where the costfunction is less than the detection threshold. To illustrate thespeed of the algorithm, we densely sampled the hypothesisspace and searched the images at 8 different scales and 3different aspect ratios. The ratio between two consecutivescales was1.2 and the aspect ratio was1.1. We performednon-maxima suppression by retaining only the lowest costhypothesis among the ones which had significant overlap.

In Figure 4, we report false positive per image vs. detec-tion rate. The curve is generated via altering the detection

Algorithm DCM OCM CMTime (µs) 0.40 51.50 17.59

Table 1. Hypothesis evaluation time comparison. The evaluationtime is averaged over the5 hand-drawing shapes used to detectobject in the ETHZ dataset

threshold for the matching cost. We compared our approachwith the oriented chamfer matching [16] and two recentstudies proposed by Ferrari et. al. [10, 9]. Our approachis significantly superior to the oriented chamfer matchingat all the false positive rates and comparable to [9] whereour results are better for two classes (giraffes and bottles)and slightly worse for the swans class while for two otherclasses, the numbers are almost identical. As shown inthe detection examples (Figure 4), object localization is ex-tremely accurate. We note that in [20] and [15], slightlybetter performances were reported on this dataset. As theseresults were presented in different formats we could not in-clude them in our graphs.

In Table 1, we present the mean evaluation time ofmatching costs per hypothesis. The average number ofpoints in the shape templates were1610, computed over fiveclasses. Similarly, on average our linear representation in-cluded39 lines per class. The number of lines per class isonly an upper bound on the number of computations. Sincethe algorithm retrieves only the hypotheses having a smallercost than the detection threshold, the summation was termi-nated for a hypothesis as the cost reached this value. On av-erage only14 lines were evaluated per hypothesis. The re-sults indicate that the proposed method improves the speedof chamfer matching by43x and of the oriented chamfermatching by 127x. Note that, the speed up is more signif-icant for larger size templates since our cost computationis insensitive to the template size whereas the cost of theoriginal chamfer matching increases linearly.

On average, we evaluated1.05 million hypotheses perimage which took0.42 seconds. Using the bounding tech-nique presented in Section 4.4, we further reduced the aver-age processing time per image to0.39 seconds where ap-proximately91% of the hypotheses were skipped. Notethat, while using the bounding function we needed to com-pute the full cost function (summation over all lines) foreach evaluated hypothesis. Therefore, the speedups is notproportional.

5.2. Human Pose Estimation

In the second experiment, we utilized the derived shapematching framework for human pose estimation which is avery challenging task due to large variations in appearancesof human poses. As proposed in [13], we matched a galleryof human shapes with known poses to the given observa-tion. Due to articulation, the size of the pose gallery neededfor accurate pose estimation is quite large. Hence, it be-comes increasingly important to have an efficient matching

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Figure 4. ROC curve and several localization results on the ETHZ shape dataset. The images are searched using a single hand-drawn shapeshown on the side. The proposed approach achieved performance comparable to [9].

algorithm which can cope with background clutter.

The experiments were also performed on the HumanEvadataset [17] which contains video sequences of multiple hu-man subjects performing various activities captured fromdifferent viewing directions. The ground truth locations ofhuman joints at each image were extracted using the at-tached markers. Shape gallery templates were acquired intwo steps. First, we computed the human silhouettes viaHumanEva background subtraction code. Then, using theCanny edges around the extracted silhouette outlines, weobtained the shape templates. We included all the imagesfrom an action sequence (about 1,000 - 2,000) to the shapegallery which is then used to estimate human poses in differ-ent sequences. As we extracted Canny edges directly fromthe test images, they included significant amount of back-ground clutter. The best shape template together with itsscale and location is then retrieved via the matching frame-

work. We quantitatively evaluated the mean absolute errorbetween the marker locations of the ground truth and the es-timated pose on the image plane. The results are presentedin Table 2 where we observe significantly improved accu-racy compared to chamfer matching and the oriented cham-fer matching. The proposed approach can evaluate morethan 1.1 million hypotheses per second whereas chamfermatching and the oriented chamfer matching can evaluate31000 and 14000 hypotheses per second respectively. Sev-eral pose estimation examples are given in Figure 5.

5.3. Synthetic Experiments

In the last experiment, we estimated the 3D pose of sev-eral industrial parts using synthetic data. The 3D CADmodels of the objects were given in advance. We decom-posed the 3D rotation matrix into in-plane and out-of-planerotations, and generated a gallery of shape templates from

Algorithm Walking Jogging Boxing Avg.DCM 7.3 12.5 9.7 9.8OCM 15.0 15.3 13.6 14.6CM 9.3 13.6 10.6 11.2

Table 2. Pose estimation errors on three action sequences. Errorsare measured as the mean absolute pixel distance from the groundtruth marker locations.

Figure 5. Human pose estimation results. First row: Walkingse-quence. Second row: Jogging and boxing sequences. Estimatedposes and contours are overlayed onto the images.

the uniformly sampled out-of-plane rotations (300 poses foreach object) via rendering the 3D CAD model and detectingthe depth edges (edges due to depth discontinuities). Twosamples among the 300 shape templates for each object areshown in the first row of Figure 6.

We also generated a synthetic test set with a similar pro-cedure. We randomly generated 3D pose parameters foreach object and simultaneously inserted all the objects inthe scene. A few of the generated test images are shownin the second row of Figure 6. As seen in the images, theobjects have large overlaps, therefore a significant amountof the edges are occluded. Moreover, the images were cor-rupted with noise via adding uniformly sampled line seg-ments and a small fraction of the detected edges were alsoremoved. The test set included 500 images.

We retrieved the best gallery pose together with in-planerotation and translation parameters via the proposed shapematching algorithm. Several pose estimation results for fivedifferent objects are shown in the third row of Figure 6. Full3D pose of the objects were then recovered for a knowndepth using the estimated in-plane transformation parame-ters together with the out-of-plane rotation parameters thatgenerated the gallery templates. An estimate was labeled ascorrect if the three estimated angles were within 10 degreesand the position was within 5 mm. of the ground truth pose.As seen in Table 3, on average, our approach reduces esti-mation errors by75% compared to chamfer matching and40% compared to orientated chamfer matching. In this ex-

Algorithm Circuit Ellipse T-Nut Knob Wheel Avg.DCM 0.03 0.05 0.11 0.04 0.08 0.06OCM 0.05 0.14 0.17 0.04 0.17 0.11CM 0.11 0.26 0.34 0.26 0.22 0.24

Table 3. Miss rates for synthetic 3D pose estimation experiment.

Figure 7. Empirical evidence of sublinearity. See text for details.

periment, we were able to estimate the 3D pose of an objectin .71 seconds via the proposed approach whereas the sameprocess took29.1 and65.3 seconds via chamfer matchingand the oriented chamfer matching respectively.

5.4. Empirical Evidence for Sublinear Complexity

When the template shapes are scaled, sublinear com-plexity trivially holds. In this case, as the number of edgepoints,n, increases the cardinality of the linear representa-tion,m, remains constant which implies constant complex-ity for matching. We also provide empirical evidence thatthe matching complexity is a sublinear function of the num-ber of template points in a more general setup. As we dis-cussed before, theO(m) complexity is only an upper boundon the number of evaluations, and on average we need toevaluate only a fraction of those lines to find the minimumcost match. Empirically, we evaluatem lines which fit 20%- 30% of the template points and most of the energy is con-centrated in only a few lines. In Figure 7, we plot the num-ber of template points,n, versus the fraction of evaluatedlines to points,m

n, wherem is selected as the number of

lines that fit30% of template points. The curve is generatedusing 1000 shape images from the MPEG-7 dataset. Weobserve that as the number of template points increases thefraction of evaluated lines decreases, which provides empir-ical evidence that the algorithm is sublinear in the numberof template points (< O(n)). Note that, there is no biasfactor involved sincen vs. m curve passes through zero.

6. Conclusion

We presented a novel approach for improving the accu-racy of chamfer matching while significantly reducing itscomputational cost. We proposed an alternative approachfor incorporating the edge orientation in the cost functionand solved the matching problem in the orientation aug-mented space. The novel cost function is smooth and canbe computed in sublinear time in the size of the shape

Figure 6. 3D pose estimation. First row: Samples from the shape gallery. Second row: Query images. Third row: Pose estimation results.

template. We demonstrated the superior performance ofthe algorithm on three challenging applications where weachieved speedups up to an order of 45x while drasticallyreducing the matching errors.

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