基于轮廓的对齐.pdf

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Automatic object and image alignment using Fourier Descriptors

Introduction

Related work

Image alignment using Fourier Descriptors

Edge detection and linking

Edge correspondence

Fourier Descriptors

Properties and normalization

Correspondence identification method

Correspondence reliability rank

Edge correspondence pseudo-code

Transformation parameters estimation

Refinement of the transformation

Control points and ICP method

Results

Discussion

Conclusions

Acknowledgments

References

Available online at www.sciencedirect.com Image and Vision Computing 26 (2008) 1196–1206 www.elsevier.com/locate/imavis Automatic object and image alignment using Fourier Descriptors Weisheng Duan a, Falko Kuester b,*, Jean-Luc Gaudiot c, Omar Hammami d a Ion Beam Application S.A., Avenue Albert Einstein, 4, B-1348 Louvain-La-Neuve, Belgium b Department of Structural Engineering, University of California San Diego, La Jolla, CA 92093-0436, USA c Department of Electrical Engineering and Computer Science, University of California Irvine, Irvine, CA 92697-2625, USA d E´ cole Nationale Supe´ rieure de Techniques Avance´es, 32 Boulevard Victor, 75739 Paris Cedex 15, France Received 8 September 2004; received in revised form 5 November 2007; accepted 19 January 2008 Abstract This paper presents a new edge-based technique for image alignment, combining Fourier Descriptors (FD) and the Iterative Closest Point (ICP) computation into an accurate and robust processing pipeline. Once edges are identiﬁed in the reference and target images, Fourier Descriptors are used to simultaneously determine edge correspondence and estimate the transformation parameters. Subse- quently, an ICP computation is applied to further improve the alignment results. Using Fourier Descriptors in combination with a reli- able distance matrix, corresponding edge pairs can be reliably detected for all identiﬁed edges. Ó 2008 Published by Elsevier B.V. Keywords: Image alignment; Edge detection; Fourier Descriptors; Correspondence; Transformation; Iterative Closest Point 1. Introduction Fast, reliable and accurate image alignment algorithms are fundamentally important to a broad range of domains, including computer vision, remote sensing, and biomedical imaging. The aim is to ﬁnd transformations between image pairs, that allow us to compensate for spatial and temporal variations within the observed environment. The observed changes may result from movement of the vision-based sensor, the use of multiple sensors, movement of objects or features within the scene, or a combination thereof. In many cases, image alignment is the ﬁrst step before further image comparisons and analysis can take place. For exam- ple, computer vision and remote sensing applications may require identiﬁcation, correlation, and tracking of selected objects to aid in navigation and scene recovery. More com- plex challenges are raised in the ﬁeld of biomedical imag- ing, where the alignment of images acquired with * Corresponding author. Tel.: +1 858 534 9953; fax: +1 858 822 4633. E-mail addresses: wduan@iba.be (W. Duan), fkuester@ucsd.edu (F. Kuester), gaudiot@uci.edu (J.-L. Gaudiot), hammami@ensta.fr (O. Hammami). 0262-8856/$ - see front matter Ó 2008 Published by Elsevier B.V. doi:10.1016/j.imavis.2008.01.009 diﬀerent imaging techniques such as MRI, fMRI, CT, and PET poses a challenging and time consuming problem. New techniques that can aid with the rapid identiﬁcation of relevant features and co-registration of data sets obtained with diﬀerent imaging modalities will greatly aid in the analysis and subsequent diagnosis of diseases. Typically, we have two images to which we shall refer as the reference image and the target image, respectively. The task of the image alignment process is to ﬁnd the spatial transformation between the target image and the reference that puts the target image into the best possible spatial cor- respondence with the reference image. Following the notion of common alignment algorithm [1], image align- ment methods can be broadly divided into two main cate- gories: (i) area-based techniques and (ii) feature-based techniques. In this paper, we propose a simple, accurate, and robust algorithm, which is based on edge features. It may be divided into four steps, (1) identiﬁcation of features or object edges contained in the two images utilizing an appropriate edge detection algorithm, (2) correlation of the detected edges to determine the corresponding pairs that represent the same object, (3) initial estimation of fea-

W. Duan et al. / Image and Vision Computing 26 (2008) 1196–1206 1197 ture transformations based on edge correspondence, and (4) reﬁnement of the estimate using an Iterative Closest Point (ICP) method, which makes the alignment, adding accuracy and robustness. In this paper, we have generalized Fourier Descriptors and exploited them for edge corre- spondence (Step 2) as well as the transformation estimation (Step 3), allowing us to perform two crucial computational steps at the same time. This results in a clean and eﬃcient algorithm for image alignment. 2. Related work Recently, researchers have paid close attention to image alignment and a broad range of methods has been devel- oped in two major categories: (1) area-based methods and (2) feature-based methods. Area-based methods such as mutual information methods [2,3] work directly with the raw image data and remove the need for image fea- tures, making these methods particularly interesting. How- ever, the use of raw data without any reduction makes them calculation intensive, especially when an entire image is used. Therefore, we have focused our work on the fea- ture-based methods. [4] proposed a reﬁnement process, Three types of features, region features, line features and point features [1] are used by most feature-based methods. Generally, region features are detected by image segmentation techniques, which can signiﬁcantly image alignment quality. Goshtasby inﬂuence the ﬁnal et al. in which the image alignment is done iteratively with segmentation to improve the ﬁnal alignment quality. Line features can be extracted by standard edge detectors, such as Canny [5] and Laplacian of Gaussian [6]. Corners are widely used as the point features for image alignment and many corner detectors exist [7]. However, the num- ber of detected corners can be very high and the posi- tions of the detected corners are not always accurate resulting in a slow and less robust alignment process. Compared to regions and corners, edges (line features) are more distinct, stable, easier to detect and are natural features for image alignment. Edge-based image align- ment methods commonly use shape descriptors to deter- mine the correspondence among the detected edges. In [4], Goshtasby et al. proposed a method using shape matrices, while in [8], Li et al. applied a method based on chain-code correlation. A widely cited approach in the literature is based on Invariant Moments based descriptors [9,10]. While all of these methods can provide relatively reliable correspondence results, the use of Fou- rier Descriptors [11] is appealing due to its simplicity. As a shape representation method, Fourier Descriptors are usually used for content-based image retrieval [12]. Because they are simple, insensitive to translation, rotation, and scaling, we apply them to image alignment as the edge similarity measurement. It should be noted that Fourier Descriptors can also be used to estimate the transformation between the target image and the reference image, which greatly simpliﬁes the image alignment process. This estima- tion also solves the initialization problem of the standard Iterated Closest Point (ICP) algorithm [13,14], which is well known for giving accurate and robust alignments for 2D and 3D point sets. The combination of Fourier Descriptors with ICP provides an elegant, accurate, and robust image alignment method. 3. Image alignment using Fourier Descriptors An overview of our method is shown in Fig. 1 and high- lights the use of Fourier Descriptors in combination with ICP computation. As stated earlier, Fourier Descriptors are used to determine the correspondence of detected edges as well as to estimate the transformation between the target image and the reference image, while ICP is applied to reﬁne the transformation. 3.1. Edge detection and linking Edges are considered natural and easily detectable image features, providing compact and rich information in images including object location, shape, size, and texture. The selection of an edge-based image alignment criterion allows us to leverage from mature edge detection techniques such as Canny [5] and Laplacian of Gaussian [6]. The Canny edge detector [5], which was designed to be an optimal edge detector, was used for our ﬁrst processing step. The Canny method is optimal since it ﬁnds edges by looking for global maxima of the image’s gradient. Strong and weak edges are detected, using two diﬀerent thresholds and weak edges appear in the output only if they are con- nected to strong edges. This method is therefore less likely than others to be deceived by noise, and more likely to detect true weak edges. While Canny gives us a reliable edge image, the pixels in the edge image are independent and as a result, the object information in images cannot be directly oﬀered by the Canny detector. This necessitates an edge linking step [15] following edge detection. Using an edge linking method, we then can assemble edge pixels into meaningful object boundaries. To summarize, with the selection of proper parameters, the Canny edge detector followed by an edge linking pro- cess identiﬁes appropriate edges for use in our image align- ment process. Fig. 2 shows an edge detection and linking example. 3.2. Edge correspondence Edge correspondence is the identiﬁcation of the corre- sponding object boundary pairs depicting the same object in the two images. Therefore, the objective is to measure the similarity of two edges obtained from the reference and target image pair. In this case, we use Fourier Descrip- tors as an abstraction of edges which is invariant against translation, rotation, and scaling, while still representing the essential form of edges.

1198 W. Duan et al. / Image and Vision Computing 26 (2008) 1196–1206 Fig. 1. An overview of the image alignment process. a b c 50 100 150 200 250 300 50 100 150 200 250 Fig. 2. Edge detection and linking example. (a) Original image (256 301). (b) Edge detection by Canny method. Parameters used: low threshold 0.3, high threshold 0.6, sigma 0.6. (c) The linked edges. One color presents one lined edge. 3.2.1. Fourier Descriptors This method was ﬁrst introduced by Zahn and Roskies [11]. Among many existing shape representation methods, Fourier Descriptors (FDs) achieve both good representa- tion and good normalization and directly support corre- spondence analysis. where n ¼ N =2; . . . ; N =2 1. Using Discrete Fourier Transformation (DFT), zðnÞ can be transformed into the frequency domain. Without any loss, the result can be transformed back into the spatial domain via the Inverse Discrete Fourier Transformation (IDFT). DFT and IDFT are deﬁned as aðkÞ and zðnÞ, respectively: Deﬁnition. Let C be an edge composed of N points pðnÞ ¼ ðxðnÞ; yðnÞÞ [16]. From these points, we can deﬁne a discrete complex function zðnÞ as zðnÞ ¼ xðnÞ þ jyðnÞ, aðkÞ ¼ 1 N zðnÞej2pkn=N k ¼ N =2; . . . ; N =2 1 ð1Þ X N =21 n¼N =2

X N =21 k¼N =2 zðnÞ ¼ aðkÞej2pkn=N n ¼ N =2; . . . ; N =2 1 ð2Þ W. Duan et al. / Image and Vision Computing 26 (2008) 1196–1206 1199 The coeﬃcients aðkÞ are also called Fourier coeﬃcients. They contain the information about the edges in the Fou- rier domain. 3.2.1.1. Properties and normalization. Certain geometric transformations of the edge function zðnÞ can be related to simple operations of aðkÞ in the Fourier domain. Trans- lation aﬀects only the center coeﬃcient að0Þ, while the other coeﬃcients retain their values. Scaling of the edge with a factor s leads to a scaling of all the coeﬃcients aðkÞ by the same factor. Rotating the edge by an angle b yields a constant phase shift of b in the Fourier domain for all aðkÞ [12]. The Fourier coeﬃcients aðkÞ must be normalized to make them invariant to the translation, rotation, and scal- ing of objects. According to the properties of FDs, omit- ting the Fourier coeﬃcient að0Þ and using the other coeﬃcients aðN =2Þ . . . að1Þ að1Þ . . . aðN =2 1Þ, FDs will be invariant against translation. Taking the magnitude of each Fourier coeﬃcient makes FDs rotation invariant; dividing all Fourier coeﬃcients by the magnitude of að1Þ could make FDs scaling-invariant. The normalized FDs used is and correspondence. determine then to edge similarity 3.2.2. Correspondence identiﬁcation method Given any detected edge pair, one edge from each image, we deﬁne their similarity by the Euclidean dis- tance between their normalized Fourier Descriptors. One immediate problem is that, in most cases, these two edges contain a diﬀerent number of points, and so their Fourier Descriptors are not of the same size, and therefore, the Euclidean distance cannot be calculated directly. To solve this problem the longer edge is ﬁrst sampled such that it has the same number of points as the shorter one before applying Fourier Descriptors to these two edges. Experiments showed that this sampling method works much better than interpolating the shorter edge to align it with the longer one. Finally, the normal- ized distance between these two edges is calculated by dividing their total distance by the point number of the sampled edge. To get the most reliable correspondence, a distance matrix is computed. In this matrix D, the element dij is the normalized distance between edge Ci in the refer- ence image and edge Cj in the target image. When dij is simultaneously minimal in its row (the ith row) and its column (the jth column), i.e., when Ci is the most sim- ilar edge in the reference image for Cj and at the same time Cj is the most similar edge in the target image for Ci, we decide that the correspondence between edge Ci and edge Cj is reliable and obtain one corresponding edge pair. 3.2.3. Correspondence reliability rank After ﬁnding all the reliable corresponding edge pairs, the corresponding edge pairs are then ranked by the nor- malized distance. This allows the ﬁrst edge pair to become the most reliable pair among the corresponding pairs with the last one being the least reliable. 3.2.4. Edge correspondence pseudo-code The pseudo-code for the edge correspondence determi- nation can then be formulated as: for each edge in reference image for each edge in target image sample the longer edge calculate the normalized Fourier Descriptors of these two edges calculate the Euclidean distance between the two Fourier Descriptors normalize the Euclidean distance with the number of edge points end for end for for each edge J in reference image ﬁnd its most similar edge K in target image (smallest normalized distance) for K, in the reference image, ﬁnd its most similar edge J0 if J ¼ J0; J and K are reliable corresponding edges if J 6¼ J0, go to next edge end for rank the corresponding edge pairs by the normalized distance 3.3. Transformation parameters estimation By exploiting the characteristics of Fourier Descriptors, the transformation parameters between the target image and the reference image can be estimated from any corre- sponding edge pair. To determine edge correspondence the Fourier Descriptors are normalized as described in Sec- tion 3.2.1. Without normalization, the FD can also be used to recover the transformation parameters. For example, suppose a1 and a2 are the Fourier Descriptors of two cor- responding edges, then a1ð0Þ and a2ð0Þ are the centroid points of the two edges. The translation between these two edges could be obtained by t ¼ a1ð0Þ a2ð0Þ and the scaling factor s ¼j a1ð1Þ j = j a2ð1Þ j. To get the rotation angle b, the orientation of the basic ellipse /e [12] is used, which is deﬁned by the Fourier coeﬃcients að1Þ ¼ r1ej/1 and að1Þ ¼ r1ej/1 , as zeðnÞ ¼ að1Þej2pn=N þ að1Þej2pn=N ð3Þ e / ¼ ð/1 /1Þ=2, we can rewrite With the abbreviation e e this as /þj2pn=NÞ /þj2pn=NÞ þ r1ejð zeðnÞ ¼ ejð/1þ/1Þ=2½r1ejð ð4Þ

1200 W. Duan et al. / Image and Vision Computing 26 (2008) 1196–1206 which shows that the orientation /e of the basic ellipse is: ð5Þ /e ¼ ð/1 þ /1Þ=2 Now, the rotation angle b between the two corresponding edges can be obtained from b ¼ /1e /2e. This method leads to an ambiguity of p radians, which can be eliminated with an additional test discussed in the next section. As a result we can simultaneously determine edge corre- spondence and estimate transformation parameters (trans- lation, scale) utilizing Fourier Descriptors. Combination of the second and third alignment step sim- pliﬁes the algorithm and improves overall performance. rotation, 3.4. Reﬁnement of the transformation From the estimated transformation parameters, we can derive the transformation matrix and use it to align the tar- get image with the reference. This alignment can be further reﬁned with the Iterative Closest Point (ICP) technique [13], which is widely used for aligning 2D and 3D point sets. The ICP algorithm has many advantages; it is simple, general-purpose, and also highly accurate. However, one must provide a good initial estimate for the transformation before initiating it. In our case a reliable transformation estimate can be easily obtained using the Fourier Descrip- tors, which makes ICP very attractive for the reﬁnement of our image alignment. The simplest ICP algorithm is an iteration process [16]. At each step, it attempts to reﬁne the current transforma- tion between the two point sets and repeats until it con- verges. The base point set is named ‘model’ Q and the other point set is named ‘data’ P. As the ﬁrst step of each iteration, for each point p in P, it ﬁnds the closest point q in Q. As the second step, it estimates the transformation T from the closest point pairs ðp; qÞ and updates P by apply- ing T to it. These two steps are repeated until convergence. The ﬂow chart of the presented ICP algorithm is given in Fig. 3. A good initial estimate of the transformation must be provided to ensure convergence to the global minimum (the correct alignment). The ICP algorithm can be directly applied to reﬁne the image alignment step. In this case, the initial transforma- tion is provided by Fourier Descriptors as discussed in Sec- tion 3.3, where the ‘model’ Q is one of the edge points in the reference image and the ‘data’ P is the corresponding edge points in the target image. The p radians ambiguity for the rotation angle estima- tion that is encountered with the FD-based technique (Sec- tion 3.3) can be eliminated via a simple computation. Before entering the iterations of ICP, assume the possible angles are b and b1 ¼ b p, we thus must apply the trans- formations T and T 1 derived from b and b1, respectively, on ‘data’ P. We then calculate the distance d between TðPÞ and Q, and also the distance d 1 between T 1ðPÞ and Q. Now we compare d and d1; if d < d1, we take b as the rotation angle; while if d > d1, we choose b1. The role Fig. 3. Flow chart of ICP algorithm. of the ICP process is to signiﬁcantly decrease the error of the initial image alignment iteration by iteration and ﬁnally give the most accurate possible result of the image alignment. 3.5. Control points and ICP method Estimating the transformation from a corresponding edge pair, followed by the ICP algorithm to reﬁne the ini- tial alignment, provides an optimal image alignment pro- cess. However, as ICP is applied to the selected edge pair, it does a better job of guaranteeing the local align- ment quality than the global quality. It is also possible to use the ICP method to achieve a global alignment quality improvement. This can be accomplished by using all of the detected edges in the reference image as the ‘model’ Q and the detected edges in the target image as the ‘data’ P. The tradeoﬀ is that this will make the ICP method ineﬃcient since thousands of edge points may be involved in P, and ﬁnding the closest point q in Q for each point in P is computation- ally expensive and as a result time consuming. We addressed this problem by using the sampled edge points in the target image as the ‘data’ P. The sampling guaran- tees that there are at least two points remaining for each detected edge. As a result, when using fewer points to reduce the ICP computational complexity all the detected edges are still taken into account, thereby improving the global alignment quality. The remaining problem that has to be solved is that of estimating the initial transformation. For this we use a point-based method [17], which ﬁrst deﬁnes two corre- sponding control point sets in the two images and then

W. Duan et al. / Image and Vision Computing 26 (2008) 1196–1206 1201 derives the transformation parameters from them. In fact, the transformation parameters can be eﬃciently estimated [18] once the corresponding control points have been deﬁned. We deﬁne the corresponding control points as the cen- troid points of the detected edges, that is, two centroid points of one corresponding edge pair compose one corre- sponding control point pair. This provides a reliable yet simple estimate of the transformation. Alternatively, we also could use the transformation estimation from any pair of corresponding edges based on the Fourier Descriptors method. In summary, the combination of Fourier Descriptors with the ICP method operates as follows: (1) Edge detection and linking (2) Edge correspondence identiﬁcation using Fourier Descriptors (3) Control point selection from the centroid points of corresponding edges (4) Transformation estimation based on control points or one corresponding edge pair (5) Sampling of edge points in the target image and application of ICP method to reﬁne the estimated transformation It should be mentioned that if the detected edges in the images are the boundaries of the objects, then the same objects in the target image and in the reference image are well aligned by our algorithm. 4. Results The proposed algorithm for image alignment was imple- mented in Matlab and a snapshot of the ﬁnal application is shown in Fig. 4. We tested the Fourier Descriptors in terms of edge correspondence and image alignment parameters estimate. Fig. 2(a) shows a real image with 256 301 pixel resolution, which was used as the reference image. A set of 72 target images were generated by rotating Fig. 2(a) from 0° to 360° by 5° increments for each step. For each of the 72 target images the transformation matrix needed to con- vert it back into the reference image was computed. The obtained results are shown in Fig. 5. Fig. 5(a) shows that the estimated rotation angles are very close to the ones that were applied. From Fig. 5(b), we can see that the estimated scaling values recover the real scaling factor of s ¼ 1. Table 1 shows a quantitative analysis of the results and highlights the capability of Fourier Descriptors for the estimation of image alignment parameters. This can be largely attributed to the reliability of edge correspondence algorithm. The combination of Fourier Descriptors and the reliable distance matrix method leads to reliable identiﬁcation of corresponding edge pairs among all detected edges. The most signiﬁcant aspect of the developed Fourier Descriptor based approach is that it can accurately esti- mate the diﬀerent transformations that were applied to multiple objects in the same image. This makes it versatile and applicable to diﬀerent types of image alignment prob- lems. An example is shown in Fig. 6 and Table 2. In Fig. 6, (a) is the same reference image as used in the other tests; (b) Fig. 4. GUI of the developed image alignment software.

1202 W. Duan et al. / Image and Vision Computing 26 (2008) 1196–1206 a l e g n A d e a m t i t s E 400 350 300 250 200 150 100 50 0 0 b 2 l e a c S d e a m t i t s E 100 200 300 400 Rotation Angle 1.5 1 0.5 0 0 100 200 300 400 Rotation Angle Fig. 5. Transformation estimate results computed with Fourier Descriptors. (a) The rotation angle estimation. X-axis: applied rotation angle in Fig. 2(a) from 0° to 360°, Y-axis: the estimated angle. (b) The scaling factor estimation. X-axis: applied rotation angle in Fig. 2(a) from 0° to 360°, Y-axis: the estimated scaling factor. Table 1 Quantity analysis of Fourier Descriptors’ capability in transformation estimate Maximum error of angle estimation 1.2526° Mean error of angle estimation 0.4176° Maximum error of scale estimation 0.0189 Mean error of scale estimation 0.0078 Detailed results are shown in Fig. 5. a c 50 100 150 5 200 250 300 9 8 2 6 1 3 4 7 50 100 150 200 250 b d 50 100 150 200 250 300 1 5 3 6 8 2 4 9 7 50 100 150 200 250 Fig. 6. General capability test of Fourier Descriptors in the alignment parameters estimate. is the target image obtained by applying a diﬀerent trans- formation for each object in the reference image; (c) is the linked reference edge image; (d) is the linked target edge image. The correspondence among the detected edges in (c) and (d) are indicated by the color bar between them and also by the numbers around the edges, i.e., two identical colors or numbered edges indicate one corresponding edge pair. Table 2 gives the numerical results of this test. Focus- ing on just one object pair to explain the results, one can observe that at ﬁrst, the pencil in (a) (reference image) was extracted, and then it was scaled by a factor 1.2, rotated counter-clock-wise by 270°, displaced 104 pixels along the X-axis and 21 pixels along the Y-axis to get the corresponding pencil in (b) (target image). The edges are highlighted in green and labeled 4 in (c) and (d). Subse- quently the Fourier Descriptor based technique was applied for these edges to estimate the transformation between them. The resulting estimates are, scale = 1.2104, rotation = 269.7688° displace- ment = (104.4, 20.59) pixels which are shown in Table 2 by the row labeled ‘4th edge pair’. The other results in Table 2 can be interpreted in the same way as explained for the pencil pair. As can be seen from Table 2, for all the corresponding pairs the estimated parameters are very close to the ones applied. The mean errors of scaling estimation, angle estimation and translation estimation are scale ¼ 0:00276; rot ¼ 0:39818 and trans ¼ ð0:17222; 0:37667Þ pixels. The accuracy of these results illustrate the performance of the Fourier Descriptors for the image alignment parameters estimate. (counter-clock-wise), The complete image alignment pipeline was tested in combination with Fig. 7(a), which served as the reference image. Fig. 7(a) is a rather complex image having poorly deﬁned edges (the small rectangle block), overlapping edges (the pencil and the square block) and also a texture rich object (the disk). The target image, Fig. 7(b), is the 270° rotation of Fig. 7(a) that was subsequently scaled by a fac- tor of 0.85. After the Canny edge detection and edge corre- spondence determination, the corresponding edge pairs are shown in Fig. 8. In the edge correspondence determination process, after ﬁnding all the reliable corresponding edge pairs, the normalized distance between each two corre- sponding edges is calculated by dividing their total distance

W. Duan et al. / Image and Vision Computing 26 (2008) 1196–1206 1203 Table 2 Alignment parameter estimates for Fig. 6, based on Fourier Descriptors Edge pair Scaling Used Estimated 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 1.0 1.0 1.0 1.2 1.0 1.0 0.7 1.0 0.5 1.0 1.0 1.0 1.2053 0.9995 1.0110 0.6953 1.0033 0.5000 Angle (°) Used Estimated 0 0 0 270 0 60 0 0 0 0.0 0.0 0.0 269.8153 0.7973 61.2721 0.7312 0.2109 0.3874 Error 0.0 0.0 0.0 0.0053 0.0005 0.0110 0.0047 0.0033 0.0 Error 0.0 0.0 0.0 0.1847 0.7973 1.2721 0.7312 0.2109 0.3874 Mean error 0.00276 0.39818 Translation (pixel) Used Estimated Error (0, 0) (0, 0) (120,117) (104, 21) (120,117) (7, 57) (0, 0) (0, 0) (30, 110) (0.17222, 0.37667) (0.0, 0.0) (0.0, 0.0) (120.0,117.0) (104.15, 20.63) (120.0,117.0) (7.94, 59.63) (0.42,0.04) (0.0, 0.04) (30.04, 110.31) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) (0.15, 0.37) (0.0, 0.0) (0.94, 2.63) (0.42, 0.04) (0.0, 0.04) (0.04, 0.31) tion from these control points, and then apply the ICP method to reﬁne the alignment. Fig. 9(a) is the direct over- lap of the two edge images after edge correspondence deter- mination; Fig. 9(b) shows the result after applying the estimated transformation on the target edge image and Fig. 9(c) shows the result after 12 iterations of ICP reﬁne- ment process. Table 3 gives the numerical error results of the transformation estimation and ICP reﬁnement process. From Fig. 9(a) and (b), we can see that just after applying the estimated transformation on the reference edge image, the target edge image is already well aligned with the refer- ence one. The mean distance from one edge point in the target image to its closest point in the reference image is 3.51 pixels. After the ICP process, the target image is much better aligned with the reference image in Fig. 9(c) than in Fig. 9(b), showing that the image alignment quality is improved by the ICP method. The mean distance from one edge point in the target image to its closest point in the reference image is well reduced from 3.51 pixels to 0.75 pixels. Because of the good initial transformation esti- mate, the ICP process converges after only 12 iterations. In fact, after only two iterations, the mean distance is already below 1 pixel. From the quick converge and the small alignment error, we can see that the proposed algorithm is very eﬃcient and accurate. Fig. 7. Input images used for the complete image alignment pipeline tests. (a) Rather complex reference image having poorly deﬁned edges, overlapping edges and also texture rich objects. (b) Target image obtained by 270° rotation of (a) and then scaled by the factor 0.85. by the point number of the sampled edge. All the corre- sponding edge pairs are then ranked by the normalized dis- tance. This allows the ﬁrst edge pair to become the most reliable pair among the corresponding pairs with the last one being the least reliable. The correspondence reliability result is shown by the color bar in Fig. 8. From the corre- sponding edge pairs, we determine the centroid points as the corresponding control points, estimate the transforma- a b l ) s e x p ( i Y 50 100 150 200 250 300 2 5 1 3 4 l ) s e x p ( i Y 2 5 3 1 50 100 150 200 4 50 100 150 X (pixels) 200 250 50 100 150 X (pixels) 200 250 Fig. 8. Corresponding edge identiﬁcation. Identical colors and labels in (a) and (b) present corresponding edge pairs. The color bar between (a) and (b) is the correspondence reliability indicator. Smaller numbers indicate a more reliable pair and bigger numbers a less reliable pair. The black crosses inside the edges are the centroid points, which are used to estimate the transformation between reference and target image.

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