Homography can be^{ }estimated from corresponding points, lines, or textures. But in some^{ }scenarios these features are not always available. As a supplement,^{ }the newly developed contour-based method is presented, which can be^{ }used to estimate the homography between any two planar contours^{ }of an image sequence. The proposed method improves the random^{ }sampling consensus (RANSAC) method into an iterative form. In the^{ }iterations, the random sample process of RANSAC is constrained by^{ }the previous iteration results. It reduces the blind sample process^{ }and ensures fast convergence speed. The experimental results demonstrate that^{ }the proposed method is effective. ©2010 Society of Photo-Optical Instrumentation Engineers^{ }
Homography estimation takes an important^{ }role in many applications such as camera calibration, metric rectification,^{ }3-D reconstruction, visual servo, image registration, and pattern recognition.^{1}^{,}^{2}^{,}^{3}^{,}^{4}^{,}^{5}^{,}^{6}^{,}^{7}^{,}^{8}^{,}^{9}^{,}^{10}^{,}^{11}^{,}^{12} Many^{ }researchers have put their efforts on homography estimation. Homography can^{ }be described by a 3×3 matrix, called a homography matrix,^{ }and the matrix can be formulated by parameterized equations. By^{ }solving the equations with the known corresponding features, the homography^{ }can be obtained. These corresponding features include points, lines, conics,^{ }contours, textures, etc. According to corresponding features, homography estimation methods^{ }can be roughly classified into several categories: frequency-domain-based methods, high^{ }primitives-based methods, and low primitives-based methods.^{ }
The frequency-domain-based methods often formulate^{ }homography in the frequency domain, and then utilize the known^{ }corresponding features in the frequency domain to compute homography. The^{ }Fourier transformation of corresponding textures is often used as features,^{12}^{,}^{13}^{ }and Fourier descriptors (FDs)^{14}^{,}^{15}^{,}^{16} of corresponding contours are also used^{ }as features to compute homography.^{17}^{,}^{18}^{,}^{19}^{ }
Unlike those methods mentioned before, the^{ }high-order primitives-based methods use high-order geometrical primitives such as conics^{ }and polygons, to solve the homography estimation problem.^{20}^{,}^{21}^{,}^{22} These methods^{ }have demonstrated better results in many applications than low-order primitives-based^{ }methods. But in many scenarios these features are not always^{ }available, so it limits the applications of the high-order primitives-based^{ }methods.^{ }
Compared with the high-order primitives-based method, low primitives-based methods are^{ }widely used because low primitives, such as points and lines,^{ }are easily obtained in most cases. Within this category, direct^{ }linear transformation (DLT) has been popularly used. To enhance the^{ }numerical stability of DLT, normalization procedures are often needed.^{23} But^{ }due to noise, mismatch inevitably exists, which will deteriorate the^{ }result of DLT. So robust methods like maximum likelihood estimates^{ }and random sample consensus (RANSAC)^{24} are proposed. But accurate corresponding^{ }features are hard to obtain in real situations, so some^{ }nonlinear methods like Levenberg-Marquardt^{25}^{,}^{26} and Gauss-Newton^{25} are normally applied to^{ }refine the results of these robust methods.^{ }
Contours have many good^{ }properties for homography estimation, e.g., they exist in most scenarios,^{ }and are easy to abstract and match. Therefore, our emphases^{ }are placed on the research of homography estimation from planar^{ }contours. The RANSAC method can be applied to estimate homography^{ }from contours, as is suggested in Ref. 24. However, if^{ }RANSAC is directly used, the computation is huge. The huge^{ }computation often arises from the blind random sample processes of^{ }RANSAC. To reduce the blind sample processes and ensure precision^{ }of the result, we present a contour-based method that combines^{ }RANSAC with the iterative closest point (ICP) algorithm.^{27} The newly^{ }developed method takes an iterative form similar to ICP. However,^{ }unlike ICP, in the iterations of the proposed method, RANSAC^{ }is used to estimate homography. Moreover, the random sample processes^{ }in RANSAC of the current iteration are constrained by the^{ }result of the previous iteration. It reduces the blind sample^{ }process and therefore ensures fast convergence speed. The experimental results^{ }demonstrate that the proposed method is effective.^{ }
This work is organized^{ }as follows. Section I is the introduction; Sec. ^{ }II describes the Iterative RANSAC homography estimation; Sec. III^{ }shows the implementation; and Sec. IV is the conclusions.^{ }
Given^{ }two contours denoted by X and Y, respectively, the two^{ }contours are generated by the same planar object in two^{ }different views. The homography between the two contours can be^{ }formulated as
where H is a 3×3 matrix, called the homography^{ }matrix, the s_{x}'th point of contour X and the s_{y}'th^{ }point of contour Y are assumed to be a corresponding^{ }point pair, and y[s_{y}] and x[s_{y}] are the homogeneous coordinates^{ }of the two points. The RANSAC method can be applied^{ }to estimate the homography of the two contours, as is^{ }suggested in Ref. 24. The direct way to apply RANSAC^{ }to estimate the homography between contour X and Y is^{ }summarized as follows.^{ }
Random sample four points A_{1}, A_{2}, A_{3},^{ }and A_{4} in X (any three of them are not^{ }on the same line), and find the four corresponding points^{ }B_{1}, B_{2}, B_{3}, and B_{4} in Y using Eq. (1)^{ }with the initial homography estimation H_{initial}.^{ }
Define the neighborhood point^{ }set _{s} of point B_{s} (B_{s}Y), P_{s} satisfying PY and^{ }d(P,B_{s})<d_{0}, where d_{0} is a constant. In the neighborhood point^{ }set _{s} of B_{s}, randomly selected one point B (s=1,2,3,4).^{ }Assuming A_{s} and B (s=1,2,3,4) are a corresponding point pair,^{ }compute the new homography H_{new} by solving Eq. (1).^{ }
If^{ }H_{new} satisfies the convergence conditions, then it ends and H_{new}^{ }is the final estimation, otherwise go to step 1.^{ }
If the^{ }initial homography estimation H_{initial} is not known, then B_{1}, B_{2},^{ }B_{3}, and B_{4} should be random sampled from the whole^{ }contour Y, and the neighborhood point set _{s} of point^{ }B_{s} (s=1,2,3,4) should be the whole contour. Therefore, when H_{initial}^{ }is not known, the computation is huge. But if the^{ }two contours are abstracted from the two successive frames of^{ }an image sequence, and then the difference between the two^{ }contours is small, then it is reasonable to assume
where (C_{X1},C_{X2})^{ }and (C_{Y1},C_{Y2}) are the mass centers of contour X and^{ }Y, respectively. Denote the point number of the neighborhood point^{ }set _{s} by N_{s} (s=1,2,3,4). The value of N_{s} determines^{ }the convergence speed. If N_{s} is larger, the speed is^{ }slower because the blind random sample processes and the error^{ }computing in RANSAC waste most of the time. On the^{ }contrary, if N_{s} is smaller, the speed may be faster,^{ }but the correct corresponding point may not be included in^{ }the neighborhood point set, so it will affect the precision^{ }of the estimation. It is a dilemma to choose the^{ }value of N_{s} to balance the precision and computation complexity.^{ }Moreover, even if N_{s} is given, it is hard to^{ }judge whether the correct corresponding points are included in the^{ }neighborhood sets.^{ }
Usually the upper limit of the sample times of^{ }RANSAC can be estimated by the following equation:^{24}
where is^{ }the probability that RANSAC in some iteration selects only inliers,^{ } is the point number needed for estimating a model,^{ }and w is the probability of choosing an inlier. In^{ }our case =4, and in a neighborhood point set, there^{ }is only one inlier point, i.e., the corresponding point, so^{ }with Eq. (2) the upper limit of the sample times^{ }of RANSAC in our case is
where N_{0}=max_{1s4}(N_{s}). Because there is^{ }only one inlier point in one neighborhood set, the probability^{ }of the outlier is high and therefore the upper limit^{ }of the sample times of RANSAC computed from Eq. (3)^{ }is huge.^{ }
The upper limit of sample times in RANSAC determines^{ }the computation complexity and stability. To reduce the blind sample^{ }process and ensure the precision of the result, we present^{ }a method that combines the RANSAC with the iterative closest^{ }point (ICP) algorithm.^{27} The newly developed method takes an iterative^{ }form like ICP, and moreover, in the iteration process, the^{ }random sample processes in RANSAC are constrained by the result^{ }of the previous iteration. It reduces the blind sample and^{ }therefore ensures fast convergence speed.^{ }
To illustrate the proposed method clearly, we first put forward^{ }the simplified method to estimate the homography between contours X^{ }and Y. The simplified method is summarized as follows.^{ }
With^{ }the initial estimation H_{initial}, calculate the initial error E_{initial} using^{ }the method showed in Fig. 1, and random sample four^{ }points A_{1}, A_{2}, A_{3}, and A_{4} from contour X, of^{ }which any three points are not on the same line.^{ }
Figure 1.In contour Y, by solving Eq. (4), find the four^{ }corresponding points B_{1}, B_{2}, B_{3}, and B_{4} of the four^{ }selected points A_{1}, A_{2}, A_{3}, and A_{4}.
where A[s] and B[s]^{ }are the homogeneous coordinates of A_{s} and B_{s} (s=1,2,3,4), respectively.^{ }If the point with coordinate B[s] calculated by Eq. (4)^{ }is not on contour Y, then choose the point of^{ }Y nearest to B[s] as point B_{s}.^{ }
Find the neighborhood^{ }point set _{s} of B_{s}. In _{s}, randomly select one^{ }point B (s=1,2,3,4). Assuming A_{s} and B (s=1,2,3,4) is a^{ }corresponding point pair, compute the new homography H_{new} by solving^{ }Eq. (5) and recomputing the new error E_{new} with the^{ }procedure shown in the flowchart of Fig. 1.
Repeat step 2^{ }until E_{new}<E_{initial}.^{ }
If E_{new} is small enough, then it ends^{ }and H_{new} is the final estimation, otherwise let H_{initial}=H_{new}, E_{initial}=E_{new},^{ }and go to step 1.^{ }
Equations (4),(5) are directly obtained from^{ }Eq. (1). By reorganizing Eq. (4) or (5), Eq. (6)^{ }is obtained.^{ }
where U_{s}=(U U 1)^{T}, V_{s}=(V V 1)^{T} are homogeneous coordinates of the s'th^{ }corresponding point pair and
is a homography matrix and
Then h can^{ }be solved from Eq. (6) by some numerical methods like^{ }the SVD-based method.^{23}^{ }
The computation of the simplified iterative RANSAC homography^{ }estimation method can be estimated. Denote B as the correct^{ }corresponding point of point A_{s} (s=1,2,3,4). Assume there are D_{s}^{ }point intervals between the correct corresponding point B and the^{ }assumed corresponding point B_{s} or B (s=1,2,3,4). Denote D_{max}=max_{1s4}(D_{s}). In^{ }step 2, there are two possible choices to select the^{ }assumed corresponding point B (s=1,2,3,4). One is nearer to the^{ }correct corresponding point, and the other is farther away from^{ }it. Usually three of the four points chosen nearer to^{ }the correct match points will make step 2 finish. We^{ }assume that three of the four points are chosen nearer^{ }to the correct match points, so with Eq. (2) it^{ }needs at most log(1−)/log[1−0.5^{3}] times of samples to make Step^{ }2 finish. Assume D_{max} only decreases one point when step^{ }2 finished. Then the upper limit of the sample times^{ }of the simplified method is
where we assume N_{0}=max_{1s4}(N_{s}) and D^{ }is the value of D_{max} in step 1. The value^{ }of N_{0} in the traditional RANSAC should be selected large^{ }enough to ensure that the corresponding points are included in^{ }the neighborhood point set, but in the proposed method, N_{0}^{ }is smaller and therefore the proposed method needs less computation.^{ }For example, if =0.99 and D=7, for the traditional RANSAC^{ }N_{0} should be at least 2D to ensure that the^{ }corresponding points are included in the neighborhood point set and^{ }the upper limit of the sample times computed with Eq.^{ }(3) is huge. The huge upper limit of sample times^{ }indicates unstable computation complexity. For the proposed method, usually N_{0}=3^{ }is enough, and therefore with Eq. (7), the upper limit^{ }of the sample times of the simplified method is 261.^{ }The smaller upper limit of sample times of the proposed^{ }method indicates more stable computation complexity.^{ }
In every new iteration, the^{ }points B_{1}, B_{2}, B_{3}, and B_{4} in the second contour^{ }are recomputed according to the previous estimation. It ensures the^{ }right searching direction. The random sampling process used to search^{ }the corresponding points is limited in the small point sets,^{ }which are determined by the estimation of the previous iteration.^{ }It reduces the blind sample processes, so the iterative form^{ }of RANSAC converges fast and the computation is stable.^{ }
Although the simplified method can reduce computation^{ }complexity, some practical issues needed to be noticed when applying^{ }it. Above all, how to choose N_{s} (s=1,2,3,4), the point^{ }number of the neighborhood point set, needs to be arranged^{ }because N_{s} is related to the computation complexity. The Monte^{ }Carlo test^{28}^{,}^{29} is used to illustrate the relationship between N_{s}^{ }and the amount of computation. With the two contours generated^{ }by computer without noise, the simplified iterative RANSAC method is^{ }tested by the 100-run Monte Carlo tests, that is, to^{ }carry out the simplified method 100 times respectively and calculate^{ }the average error with the sample times. The results are^{ }shown in Fig. 2. The sample times can be considered^{ }as the index of computation complexity. Figure 2 shows that^{ }larger N_{s} leads to more computation. On the contrary, if^{ }N_{s} is too small, for example N_{s}=3 (s=1,2,3,4), if all^{ }the three points are stained by noise and all the^{ }tried samples cannot satisfy the ending condition, then the simplified^{ }method will be probably trapped in an endless loop in^{ }step 2. So it is important to choose a proper^{ }N_{s} for reducing computation, and some techniques are needed to^{ }deal with the occurrence of endless loop.^{ }
Figure 2.If we randomly select^{ }a point in the neighborhood set _{s} (s=1,2,3,4), there exist^{ }two choices: nearer to the correct match point and farther^{ }away from it. The first choice may make the estimation^{ }more accurate, and this direction is considered to be a^{ }correct direction. The probability of randomly selecting a point in^{ }the set _{s} lying in the correct direction is 1/2,^{ }so if we randomly sample four points from the four^{ }neighborhoods, the probability of all the four sampled points lying^{ }in the correct directions is 1/16. Assume the probability that^{ }one point in contour Y is stained by noise and^{ }becomes an outlier point . An outlier point is the^{ }point that will cause large error if used to compute^{ }homography.^{ }
The probability that the successive N_{s}/2 points of _{s} (s=1,2,3,4)^{ }lying in the proper direction are all outliers is
For less^{ }computation, N_{s} should be smaller; however, N_{s} should be big^{ }enough to make p tend to zero to avoid endless^{ }loop. When is 0.1, 0.2, and 0.3, respectively, the^{ }relationship of p and N_{s} is shown in Fig. 3.^{ }Figure 3 shows that p decreases sharply when N_{s} increases,^{ }so when endless loop occurs it is effective to increase^{ }N_{s} a little to help it jump out of endless^{ }loop. Given p and , a reference to choose N_{s}^{ }is given in Eq. (9).
^{ }
Figure 3.For detecting and dealing with endless^{ }loop, the threshold N_{t} is used. If in an iteration^{ }the sample times exceed N_{t}, then the endless loop is^{ }considered to occur. Increasing N_{s} a little will help to^{ }jump out of endless loop. The maximum sampling choice in^{ }step 2 for an iteration is N, so N_{t} can^{ }be set to N. In fact, N is too large;^{ }in our experiments, N_{t} is set to N/16.^{ }
If at least^{ }one of the four sampled points A_{1}, A_{2}, A_{3}, or^{ }A_{4} in contour X in step 1 is an outlier,^{ }endless loop may occur too. If after adjusting N_{s} the^{ }endless loop occurs again, then it can be considered that^{ }at least one of the four sampled points A_{1}, A_{2},^{ }A_{3}, or A_{4} is an outlier. If so, we need^{ }to resample four points in contour X and restart the^{ }estimation. It will help to skip the outliers in contour^{ }X.^{ }
To improve the precision of the estimation, interpolation is needed.^{ }The B-spline interpolation is suggested. In theory, denser interpolation leads^{ }to more accurate results, but from our experience, when the^{ }density of interpolation reaches a certain level, increasing the density^{ }of interpolation is no good but increases computation. In our^{ }experiments, three interpolation times are used.^{ }
The whole iterative RANSAC homography estimation considering noise, endless^{ }loop, and other problems is summarized as follows.^{ }
Set H_{initial}^{ }and computed E_{initial}; set N_{s} according to Eq. (5); set^{ }H_{current}=H_{initial}, E_{current}=E_{initial}.^{ }
Random select four points A_{1}, A_{2}, A_{3}, and^{ }A_{4} in the first contour, any three of which are^{ }not on the same line; let count=0.^{ }
With H_{current} using^{ }Eq. (4), compute the four corresponding points B_{1}, B_{2}, B_{3},^{ }and B_{4} in the second contour; let N=N_{s}.^{ }
Find the^{ }neighborhood point set _{s} of B_{s} with N points (s=1,2,3,4).^{ }Randomly select one point B in the neighborhood point set^{ }_{s} (s=1,2,3,4), respectively. Assuming A_{s} and B are a corresponding^{ }point pair, obtain the new homography H_{new} by solving Eq.^{ }(5) and compute the new error E_{new}.
Count=count+1^{ }
if E_{new} is^{ }small enough then end and H_{new} is the final estimation^{ }
if^{ }E_{new}<E_{current}, then E_{current}=E_{new}, H_{current}=H_{new}, count=0, go to step 2^{ }
if count>N/16,^{ }then N=N+2, count=0^{ }
if N>N_{s}+4, then go to step 1^{ }
if E_{new}E_{current}^{ }then repeat step 3.^{ }
^{ }Unlike the simplified method, some techniques are^{ }added to deal with endless loop and to dynamically adjust^{ }the point number of the neighborhood set. In fact, step^{ }3 is the process using RANSAC to find the corresponding^{ }points. A new iteration begins if and only if the^{ }current error in step 3 is smaller than the error^{ }of the previous iteration. It ensures that the error is^{ }not increasing with the iterations, so the proposed method will^{ }converge.^{ }
Compared with the normal RANSAC method, we need not care^{ }whether the correct corresponding points are included in the neighborhood^{ }set, because the neighborhood point sets refreshed in the iterations^{ }will move toward the correct corresponding points gradually. Therefore, N_{s}^{ }can be set smaller than in the normal RANSAC method,^{ }and accordingly the computation is smaller.^{ }
Supposed an image sequence of a planar object has^{ }been obtained. Denote the homography between any two successive frames^{ }by H_{m,m+1} (m=1,2,3,…). H_{m,m+1} can be estimated by the proposed^{ }method from the two contours in the m'th frame and^{ }the (m+1)'th frame with the initial estimation
where (C_{X1},C_{X2}) and (C_{y1},C_{y2})^{ }are the mass centers of the contours in the m'th^{ }and the (m+1)'th frame, respectively. The homography between the first^{ }frame and the m'th frame (m>2), denoted by H_{1,m}, can^{ }be estimated by the proposed method from the two contours^{ }in the first and the m'th frame with H_{initial}=H_{1,m−1}, where^{ }H_{1,m−1} can be estimated by
With the transitivity of homography, the^{ }homography between frame s and frame t (t>s) is H_{s,t}=HH_{1,t}.^{ }To obtain precise estimation of H_{s,t}, we estimate H_{s,t} from^{ }the contours in the s'th and the t'th frame using^{ }the proposed method with H_{initial}=HH_{1,t}.^{ }
First we test the proposed method^{ }by estimating the homography from two successive contours, that is,^{ }to estimate H_{m,m+1}. Some example contours are used to generate^{ }the successive contours. The example contours used in the experiment^{ }are selected after carefully considering the complexity, the variety of^{ }curvature, the number of points, etc. Some of the example^{ }contours are shown in Fig. 4. A pair of successive^{ }contours, called a contour pair, is obtained by a virtual^{ }camera. Assume the virtual camera is moving-around the plane of^{ }example contour (which coincides with the Z=0 plane). Obtain one^{ }contour of the contour pair by imaging the example contour^{ }with the virtual camera at the three given rotation angles^{ }around axis X, Y, Z, denoted by , , and^{ }, respectively. After that, add three small angles _{1}, _{2},^{ }and _{3} to , , and , respectively, to obtain^{ }the other contour of the contour pair. Assume the frame^{ }rate of the virtual camera is more than 20 Hz and^{ }the rotation speed of the virtual camera around each axis^{ }is no more than 200 deg per second so _{s} (s=1,2,3)^{ }is limited less than 10 deg. Other contour pairs can be^{ }obtained by changing (,,) and _{s} (s=1,2,3). The advantage with^{ }this scheme of generating contours is that it covers the^{ }most realistic poses at which actual images are generally taken.^{ }For each example contour, at (,,)=(−60 deg+5 degm,−60 deg+5 degn,0 deg), where m=1,2,…24, n=1,2,…24, obtain^{ }contour pairs. It means 576 contour pairs are obtained for^{ }each example contour. We add Gaussian noise with standard deviation^{ }{2,4,6} pixels along each coordinate to % ({2,5,8}) points of^{ }the contour pairs. For each example contour, the 576 contour^{ }pairs are used to estimate the homography by the proposed^{ }method with the initial homography estimation
In the experiment, the proposed^{ }method converges fast and the average sample times at convergence^{ }times at different noise levels are shown in Table I.^{ }
Figure 4.With^{ }some selected contour pairs, we compare the proposed method with^{ }the normal RANSAC method. If N_{s} is not properly set,^{ }the normal RANSAC method will not converge at the given^{ }precision. To make the comparison feasible, some contour pairs are^{ }selected, with which the normal RANSAC method does converge when^{ }N_{s}9. The average sample times of the proposed method and^{ }the normal RANSAC method are obtained by 50-run Monte Carlo^{ }tests, respectively. The results are shown in Table II. Table^{ }II showed that with the same precision, our method outperformed^{ }the normal RANSAC method.^{ }
We also use real image sequences to^{ }test the proposed method. The image sequences used in this^{ }experiment are obtained by translating and rotating a digital camera.^{ }Different combinations of rotation and translation with different speeds are^{ }considered when obtaining these sequences. In this experiment, the homography^{ }between the first frame and the m'th (m>90) frame, i.e.,^{ }H_{1,m}, is estimated. Some results are shown in Fig. 5.^{ }
Figure 5.A contour-based homography estimation method is^{ }presented. The newly developed method takes an iterative form, which^{ }combines RANSAC and ICP. The random sample process in RANSAC^{ }is constrained by the result of the previous iteration. It^{ }reduces the blind sample process and ensures fast convergence speed.^{ }The experimental results show that our technique is effective. It^{ }is found that the distortion caused by camera lens and^{ }the precision of the contour abstraction method will affect the^{ }estimation results. It will be our future work to improve^{ }the technique by considering these factors.^{ }
Ming Zhai is currently a PhD candidate^{ }in the School of Electronic, Information, and Electrical Engineering at^{ }Shanghai Jiao Tong University. He obtained his BS and MS^{ }degrees from An Hui University, Heifei, China, in 2001 and^{ }2005, respectively. His major research interests include image processing, computer^{ }vision, and related system development.
Shan Fu is a professor in^{ }the School of Aeronautics and Astronautics at Shanghai Jiao Tong^{ }University. He obtained his first degree in electronic engineering from^{ }the Northwestern Polytechnic University in 1985, and PhD from Heriot-Watt^{ }University in 1995. His long-time research interest is in the^{ }area of computer vision/image processing and related system development, which^{ }has been closely linked to engineering/industry applications, such as computerized^{ }visual inspection and metrology, experimental mechanics, and structural material engineering.
Zhongliang^{ }Jing received his BS, MS, and PhD degrees, all in^{ }electronics and information technology, from Northwestern Polytechnic University, China, in^{ }1983, 1988, and 1994, respectively. He was elected as a^{ }Cheung Kong Scholar in 1999. He is currently the Associate^{ }Dean of the Institute of Aerospace Science and Technology of^{ }Shanghai Jiao Tong University. His research interests include information fusion,^{ }optimal control theory, target tracking, and aerospace control.
Fig. 1. The flowchart for computing the error. N is^{ }the point number of contour X, D_{0} is the threshold^{ }to judge the outlier, N_{out} is the counter for the^{ }outlier, and n_{0} is the max acceptable proportion of outliers. First citation in article
Fig. 2. The^{ }result of 100-run Monte Carlo tests for the simplified iterative^{ }RANSAC when N_{s} is set to 5, 7, 9, respectively.^{ }Smaller N_{s} converges faster. First citation in article
Fig. 3. The relationship of p and N_{s} when^{ } is given. First citation in article
Fig. 4. Some example contours used in the experiments. The^{ }contours vary in their shape, curvature properties, number of points,^{ }etc. First citation in article
Fig. 5. Some experimental results of real image sequences. Six groups of^{ }results are presented. A_{i} (i=1,2,…,6) is the first frame of^{ }the six image sequences, respectively, and the contours labeled by^{ }green are used to estimate the homography; B_{i} (i=1,2,…,6) is^{ }another frame of the six sequences, respectively; C_{i} (i=1,2,…,6) are^{ }the overlap result of A_{i} with B_{i} wrapped by the^{ }estimated homography, respectively. (Color online only.) First citation in article
Table I. The average sample times in different^{ }noise levels. | |||
2 | 4 | 6 | |
2 | 24.6 | 25.1 | 26.0 |
5 | 39.3 | 43.2 | 45.5 |
8 | 80.5 | 87.1 | 90.3 |
Table II. Some average sample times of the proposed method and^{ }the normal RANSAC method for the selected contour pairs (the^{ }noise level is =4, =5). | ||||||
The example contour used to generate the^{ }contour pair | Fig. 4 (a) | Fig. 4 (b) | Fig. 4 (c) | Fig. 4 (d) | Fig. 4 (e) | Fig. 4 (f) |
The average sample^{ }times of the propose method | 39.5 | 53.3 | 47.5 | 39.0 | 48.5 | 55.2 |
The average sample times of the normal RANSAC | 403.8 | 588.2 | 441.1 | 503.9 | 527.2 | 602.9 |
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