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An affine invariant relative attitude relationship descriptor for shape matching based on ratio histograms
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 209 (2012)
Abstract
A novel shape descriptor, named as ratio histograms (Rhistogram), is proposed to represent the relative attitude relationship between two independent shapes. For a pair of two shapes, the shapes are treated as the longitudinal segments parallel to the line connecting centroids of the two shapes, and the Rhistogram is composed of the length ratios of collinear longitudinal segments. Rhistogram is theoretically affine invariant due to collinear distance invariance of the affine transformation. In addition, as the computation of the length ratio weakens the noise contribution, Rhistogram is robust to noise. Based on the Rhistogram, the shapematching algorithm includes two major phases: preprocessing and matching. The first phase, which can be processed offline, is trying to obtain the Rhistograms of all original shape pairs. In the second phase, for each transformed shape pair, its Rhistogram is computed and the candidate matched shape pair with minimal Rhistogram matching error is found. Subsequently, a voting strategy, which further improves the accuracy of shape matching, is adopted for the candidate corresponding shape pairs. Experimental results demonstrate that the proposed Rhistogram is robust and efficient.
Introduction
Shape matching plays an important role in image processing and computer vision applications. Since the images taken from different viewpoints usually suffer from perspective distortions, the matching algorithm should be capable of dealing with them. Numerous methods, such as spectral transform[1, 2], moment invariants[3, 4], isoarea normalization[5], time series[6], Bsplines[7], curvature scale space (CSS)[8, 9], shape contexts[10], shape signature[11], diagonals of orthogonal projection matrices (DOPM)[12], multiscale oriented corner correlation[13], etc., have been proposed for shape matching under affine transformations. However, most of these methods make an assumption that a sparse set of boundary points or interest points have been extracted beforehand. Besides, many traditional matching methods only analyze the properties of a separate shape and the spatial relationships between different shapes are ignored.
The relative position between the shapes often helps in image understanding and shape matching. In some recent related study such as[14], the descriptors extracted from spatial relationship between shapes are used for shape matching. Krishnapuram et al.[15] first proposed a descriptor named histogram of angles. A histogram of all possible angles between point pairs in the regions is used to describe the directional spatial relations between two shapes. The histogram is computationally expensive and can only deal with raster data. To overcome its limitation, Fhistogram[16, 17], generalizing from the histogram of angles, is proposed. Different from the histogram of angles treating the 2D object as a set of points, Fhistogram method handles the shapes as longitudinal segments. Besides the spatial relationship between two shapes, the orientation and the size of the shapes are captured by Fhistogram as well. Furthermore, Fhistogram is capable of processing both raster and vector data. As a result, Fhistogram can effectively be used for shape matching. However, the descriptor is not affine invariant. The Fhistogram of a shape pair varies with the viewing orientation. Thus to compare two Fhistograms, the difference should be normalized by searching for several parameters, which are equivalent to the parameters of the geometric transformation between the images. The parameter searching process leads to an expensive computational cost.
Motivated by Fhistogram, in this article, a novel affine invariant histogram, named as ratio histograms (Rhistogram), is proposed to describe the relative attitude relationship between two shapes. For a pair of two independent shapes, the longitudinal segments, which are the intersections of shapes and the lines parallel to the line connecting the centroids of the two shapes, are treated as primitives. Then, the Rhistogram is composed of the length ratios of collinear longitudinal segments from the two shapes. This descriptor has a clear physical interpretation and can be applied to shape matching without searching for affine transformation parameters. In contrast to Fhistogram which treats longitudinal segments in a number of directions during the range [0, 2π], Rhistogram treats the longitudinal segments of shapes with only one fixed direction; thereby the computational complexity is significantly reduced. Moreover, an efficient shapematching algorithm based on Rhistogram is developed in this article. The algorithm is divided into two phases. The Rhistograms of original shape pairs are first obtained in the offline preprocessing phase. Then, in the matching phase, the algorithm first seeks the candidate correspondences between shape pairs based on Rhistogram matching, and a novel schema is designed to testify the matched shapes by a voting of the candidate corresponding shape pairs. Accordingly, the accuracy of shape matching is obviously improved.
This study is based on the assumption that there are at least two independent shapes in the image, and the topology is reserved while the image is being transformed. The assumption is valid as objects are composed of several shapes, among which the relative positions are reserved while mapped.
The outline of this article is as follows. Rhistogram and its affine invariance are introduced in “Rhistogram and its fundamental properties” section. In “Shape matching” section, the matching algorithm based on Rhistogram is described. Experimental results are presented in “Experiments” section. Finally, “Conclusions” section concludes the article and suggests the future work.
Rhistogram and its fundamental properties
In this section, Rhistogram, which represents the relative attitude relationship between two shapes, is first defined, and then its symmetry and affine invariance are explored.
Definition of Rhistogram
As shown in Figure1, two shapes A and B, which are located in an original positively oriented orthonormal frame$\left(O,\overrightarrow{i},\overrightarrow{j}\right)$, are denoted as${E}^{A}$ and${E}^{B}$ respectively. To measure the attitude of A relative to B, we first define a relative positively oriented orthonormal frame$\left(O,{\overrightarrow{x}}_{\mathit{AB}},{\overrightarrow{y}}_{\mathit{AB}}\right)$ (Figure1a), in which, the${x}_{\mathit{AB}}\text{axis}$ has the same direction with${\overrightarrow{v}}_{{c}_{B}{c}_{A}}$. c_{ A } and c_{ B } are the centroids of A and B, respectively.$l\left(v\right)$ is the line parallel to the${x}_{\mathit{AB}}\text{axis}$ with v the intercept on${y}_{\mathit{AB}}\text{axis}$. For any real number v, the intersection${E}^{A}\cap l\left(v\right)$, denoted by${E}^{A}\left(v\right)$, is a longitudinal segment of _{ A }. In this article, as the term “shape” denotes a 2D plane which may have holes in it or may consist of many connected components,${E}^{A}\left(v\right)$ e.g.,${E}^{A}\left({{v}_{N}}_{1}\right)$ in Figure1a is the union of a finite number of disjoint segments. Similarly,${E}^{B}\left(v\right)\phantom{\rule{0.5em}{0ex}}\left({E}^{B}\cap l\left(v\right)\right)$ stands for a longitudinal segment of shape B.
The Rhistogram of A with respect to B, whose elements are the length ratios of the collinear longitudinal segments, is represented by R_{ AB } as
In Equation (1), L(ξ) denotes the length of the longitudinal segment ξ, N is the number of v that are sampled equidistantly in the range of$\left[{v}_{min},{v}_{max}\right]$,${v}_{min}$ and${v}_{max}$ are the minimum and maximum intercepts of the valid lines l(v) which have nonempty intersections with both A and B, i.e.,
${v}_{min}^{A}$ and${v}_{max}^{A}$ are the minimum and maximum intercepts of the lines l(v) which have nonempty intersections with A. Similarly,${v}_{min}^{B}$ and${v}_{max}^{B}$ are the minimum and maximum intercepts of l(v) which have nonempty intersections with B.
Similarly, the Rhistogram of B relative to A is
Note that, the R_{ BA } is calculated based on the relative positively oriented orthonormal frame$\left(O,{\overrightarrow{x}}_{\mathit{BA}},{\overrightarrow{y}}_{\mathit{BA}}\right)$ (Figure1b), where the${x}_{\mathit{BA}}\text{axis}$ has the same direction with${\overrightarrow{v}}_{{c}_{B}{c}_{A}}$.
Symmetry and affine invariance
Here, desirable properties (i.e., symmetry and affine invariance) of Rhistogram for shape matching are proved.
Symmetry
As shown in Figure1, the direction of${x}_{\mathit{AB}}\text{axis}$ of the frame$\left(O,{\overrightarrow{x}}_{\mathit{AB}},{\overrightarrow{y}}_{\mathit{AB}}\right)$ (Figure1a) is opposite to the direction of${x}_{\mathit{BA}}\text{axis}$ of the frame$\left(O,{\overrightarrow{x}}_{\mathit{BA}},{\overrightarrow{y}}_{\mathit{BA}}\right)$ (Figure1b). Consequently, the lines l(v_{ min }) and l(v_{ max }) in$\left(O,{\overrightarrow{x}}_{\mathit{AB}},{\overrightarrow{y}}_{\mathit{AB}}\right)$ for R_{ AB } are interchanged into the lines l(v_{ max }) and l(v_{ min }) in$\left(O,{\overrightarrow{x}}_{\mathit{BA}},{\overrightarrow{y}}_{\mathit{BA}}\right)$ while R_{ BA } is being computed. Therefore, the symmetry of Rhistogram, which describes the mathematical link between R_{ AB } and R_{ BA }, is deduced as
Affine invariance
The general 2D affine transformation$T=\left\{{A}_{T},b\right\}$ transforms point p in the original image into its corresponding point q in the transformed image by q$q={A}_{T}p+b$, where${b}_{2\times 1}$ is the translation vector and${{A}_{T}}_{2\times 2}$ is the affine transformation matrix between the two images. Rotation, scaling and shearing, considered as the special cases of affine transform, are represented as the following matrices:
If two shapes A and B are transformed into A′ and B′ by an affine transformation, we have
Equation 6 reveals that a shape pair preserves its Rhistogram while the affine mapping.
Obviously, the Rhistogram is invariant to translation transformation, thus, we only explain the invariance of the Rhistogram to affine transformation matrix. As shown in Figure2, in an original positively oriented orthonormal frame$\left(O,\overrightarrow{i},\overrightarrow{j}\right)$, two shapes A and B (Figure2a) are transformed into A′ and B′ (Figure2b) by an affine transformation matrix A_{ T }.$\left(O,{\overrightarrow{x}}_{\mathit{AB}},{\overrightarrow{y}}_{\mathit{AB}}\right)$ and$\left(O,{\overrightarrow{x}}_{A\text{'}B\text{'}},{\overrightarrow{y}}_{A\text{'}B\text{'}}\right)$ are the relative frames for R_{ AB } and R_{ A′B′ }. c_{ A }, c_{ B }, c_{ A′ } and c_{ B′ } are centroids of A, B, A′ and B′, respectively. It can be deduced that, if the projection of a point on y_{ AB }axis is known, the projection of the mapped point on y_{ A’B’ }axis can be obtained by
In Equation (7), det(A_{ T }), which is the determinate of the transformation matrix, is positive.$dis\left(\cdot ,\cdot \right)$ is the distance between two points, v is the projection of a point on y_{ AB }axis, and v′ is the projection of the mapped point on y_{ A’B’ }axis.
Eq. 7 reveals that the projection of the transformed points on y_{ A’B’ }axis is directly proportional to the projection of the original points on y_{ AB }axis. Moreover,${E}^{A}\left(v\right)$, representing the intersection${E}^{A}\cap l\left(v\right)$, can be treated as the union of points of shape A with the projection value of v on y_{ AB }axis. Therefore, we have
where, “⇔” denotes the correspondence relationship. As introduced in “Definition of Rhistogram” section,${v}_{min}$,${v}_{max}$,${{v}^{\prime}}_{min}$ and${{v}^{\prime}}_{max}$ are the minimum and maximum values of intercepts of valid lines for original and transformed shape pairs. So far, as${E}^{A}\left({v}_{n}\right)$ and${E}^{B}\left({v}_{n}\right)$ are collinear, the affine invariance of Rhistogram is proved due to the collinear distance invariance[18] of the affine transformation.
Shape matching
For convenience, we call the shape in an original image as template shape, and the shape in an input image as input shape. The efficient shape matching algorithm has a preprocessing phase and a matching phase. In the preprocessing phase, Rhistograms are obtained from all shape pairs in the original image. As the preprocessing phase can be executed offline, the complexity of our algorithm can remarkably be reduced compared with the straightforward approach. Subsequently, in the matching phase, the candidate corresponding shape pairs are first discovered by Rhistogram matching, and the correspondences between the template and input shapes are obtained by candidate corresponding shape pairs voting.
Preprocessing
In the preprocessing phase, our aim is to obtain the Rhistograms of the template shape pairs in the original image. Assume we are given an original image where m template shapes$\left({M}_{1},{M}_{2},\dots ,{M}_{m}\right)$ have been extracted. For each ordered template shape pair$\left({M}_{i},{M}_{j}\right)$, its Rhistogram is computed through the following steps:

(1)
Extract the centroids c _{ i } and c _{ j } of the two template shapes.

(2)
A relative positively oriented orthonormal frame $\left(O,{\overrightarrow{x}}_{{c}_{i}{c}_{j}},{\overrightarrow{y}}_{{c}_{i}{c}_{j}}\right)$, whose ${x}_{{c}_{i}{c}_{j}}\u2013\text{axis}$ has the same direction with ${\overrightarrow{v}}_{{c}_{i}{c}_{j}}$, is built up. Then, $\left[{v}_{min},{v}_{max}\right]$, the range of intercepts of the valid lines, can be obtained. The valid lines are parallel to the xaxis and have nonempty intersections with both M _{ i } and M _{ j }.

(3)
For each number $\text{n}\in \left[0,\text{N}\right]$, ${R}_{{M}_{i}{M}_{j}}\left(n\right)=L\left({E}^{{M}_{i}}\left({v}_{n}\right)\right)/L\left({E}^{{M}_{j}}\left({v}_{n}\right)\right)\left({v}_{n}={v}_{\mathit{min}}+n\left({v}_{\mathit{max}}{v}_{\mathit{min}}\right)/N\right)$.

(4)
Due to the symmetry of Rhistogram, the Rhistogram of the ordered shape pair $\left({M}_{j},{M}_{i}\right)$ can be obtained as ${R}_{{M}_{j}{M}_{i}}\left(n\right)=1/{R}_{{M}_{i}{M}_{j}}\left(Nn\right)$.
The complexity of the preprocessing step is$O\left({m}^{2}\right)$.
Matching
In the matching phase, we are given a transformed image where t input shapes$\left({T}_{1},{T}_{2},\dots ,{T}_{t}\right)$ are extracted. In this stage, the corresponding shape pairs with the minimal descriptor matching error are first discovered. The matching error between descriptors, which is measured via the normalized L1 distance, is given by
where${R}_{{M}_{i}{M}_{j}}\left(n\right)$ and${R}_{{T}_{k}{T}_{s}}\left(n\right)$ are the Rhistograms of a template shape pair (M_{ i }, M_{ j }) and an input shape pair (T_{ k }, T_{ s }), respectively. The smaller the err, the better the localization of the corresponding shape pairs is.
Theoretically, we have
Unfortunately, the matched shape pair in dependence on a single corresponding descriptor pair is sometimes not reliable, as the presence of noise, and/or because of the inaccuracy in shape extracting. To improve the accuracy of shape matching, for each input shape, its corresponding template shape is established by voting. In the following, we show how to confirm the template shape corresponding to the input shape T_{ k }:

(1).
For each input shape pair$\left({T}_{k},{T}_{s}\right),s\in \left[1,t\right],s\ne k$, its corresponding template shape pair is${\left({M}_{i\text{'}},{M}_{j\text{'}}\right)}_{\mathit{opt}}=\underset{i,j\in \left[1,m\right],i\ne j}{min}\phantom{\rule{0.2em}{0ex}}err\left({R}_{{T}_{k}{T}_{s}},{R}_{{M}_{i}{M}_{j}}\right)$. Then, the template shape M_{ i′ } is casted a vote.

(2).
The template shape that scores the largest number of votes is considered as the corresponding one to T_{ k }.
Experiments
In this section, the performance of Rhistogram is tested and compared with three stateoftheart methods, including zeroorder Fhistogram[16, 17], global DOPM[12] and CSS[8, 9]. The sensitivity of descriptors to affine transformations and noise are first compared, and the performances of descriptors on shape matching and object recognition are further tested.
Sensitivity to affine transformation and noise
The affine invariance of Rhistogram has theoretically been proven in “Symmetry and affine invariance” section, here the examples of this property are provided. The 46 test shapes of 5 groups (see Figure3) are chosen from the MPEG7 CEshape1 database, and the shapes in each group are placed randomly to create 10 different original images. Consequently, the following statistic results are counted based on 50 original and their transformed images. The transformed images are obtained according to different affine transformations and various noise levels. The examples of the transformed shapes are shown in Figure4.
To estimate the sensitivity of descriptors and the performance to each transformation parameter and noise separately, for each group of experiment, only one parameter is changed while others are fixed.
The sensitivity of Rhistogram to affine transformations is evaluated by the matching error between descriptors of the corresponding template and input shape pairs. Simultaneously, the global DOPM and the normalized zero order Fhistogram are taken for comparison.
First, the behavior of different descriptors in relation to rotation is reported. Figure5a depicts the matching error under different values of rotation θ ranging from 10° to 180° with 10° intervals, showing that the matching error floats slightly while the θ changes, and Rhistogram outperforms to global DOPM and Fhistogram.
Then the behavior of descriptors in relation to scaling is evaluated. The original images are transformed by different nonuniform scaling while values of${s}_{x}/{s}_{y}$ changing from 0.25 to 4 in steps of 0.25. Figure5b depicts that the matching errors of the descriptors increase while the degree of the shape distortion increases, whereas Rhistogram is less sensitive to nonuniform scaling than global DOPM.
The sensitivity of descriptors with respect to shearing is shown in Figure5c. The larger the absolute value of k, the higher the matching error is. Rhistogram is more robust to shearing.
Finally, the robustness of the descriptors to noise is observed. To obtain noisy shapes, the coordinates of points on the contours are shifted in the range of [–r, r, and simultaneously with the direction vertical to the tangent of the point. The signaltonoise ratio (SNR) is defined as[5]
where d is the average distance between all the points on the contour and its centroid. The noise with different values of SNR, which changes from 20 to 45 dB in steps of 5 dB, is added onto shapes. The matching errors obtained from the noisy cases are summarized in Figure5d. It points that the errors of all descriptors decreases as SNR increase, and Rhistogram has the best performance against noise.
Rhistogram for shape matching
The original and transformed images in “Sensitivity to affine transformation and noise” section are still taken for experiments. Considering each shape in the original images, the matched shapes in the transformed images should be observed. The correct matching rate (CMR) is calculated as
where OM is the number of correct correspondences that are observed, and AM is the actual number of correspondences between original and input shapes.
The performance of Rhistogram on shape matching is compared with that of global DOPM, zeroorder Fhistogram, and affine length CSS. Moreover, for Rhistogram and Fhistogram algorithm, besides the results of shape matching, the results of descriptor matching are also counted.
First, the performance of different descriptors is compared while the values of rotation change from 0° to 180° with 10° intervals. Figure6a indicates that Rhistogram is more robust to rotation. Furthermore, the CMRs of shape matching using Rhistogram and Fhistogram, which are both 100%, are higher than CMRs of their descriptor matching. It proves the validity of our shape matching algorithm based on voting.
Then, the effect of scaling (Figure6b) and shearing (Figure6c) upon descriptors is observed. The values of scaling change from 0.25 to 4, with 0.25 intervals, and the shearing factor k is associated with different values from –3 to 3, with 1 intervals. Rhistogram is most robust to nonuniform scaling and shearing, and the zeroorder Fhistogram performs worst as it is not invariant to nonuniform scaling and shearing.
Finally, Figure6d shows the performance of descriptors against varying noise levels, denoting that CMR s of all descriptors increase as SNR increases, and Rhistogram is most robust to noise.
In addition, the semilog scale plots of the running times of descriptors with respect to affine transformations and noise are depicted in Figure7. From Figure7, the following observations are obtained: (1) the computational costs of Rhistogram and the Fhistogram are nearly regardless of rotation, shearing, and noise; however, they increase as the scale of shape increases; (2) the computational cost of Rhistogram is much less than that of Fhistogram; (3) the computational cost of global DOPM and CSS, which are decided by the sampled curvature, nearly invariant while the shape is transformed; (4) DOPM has the lowest computational cost while Fhistogram has the highest computational complexity.
As a summary of this section, we found that (1) Rhistogram has the best performance in robustness to affine transformations and noise; (2) for Rhistogram and Fhistogram, the CMR of shape matching is higher than that of descriptor matching because of the voting strategy; (3) the computational complexity of Rhistogram is much lower than that of Fhistogram.
Rhistogram for object recognition
As indicated in “Introduction” section, Rhistogram is based on the premise that there are at least two shapes in an image, and the shapes preserve their relative position while the image is being transformed. However, the Rhistogram can be used for object recognition since objects are usually composed of several shapes with fixed topology. Therefore, in this section, we test the performance of Rhistogram on object recognition. Different from the experiments finding the correspondences between the shapes in “Rhistogram for shape matching” section, here, the task is to discover the matched objects with maxima corresponding shape pairs.
The licenses (see Figure8a) which are searched from Internet are taken as template objects. The input objects in Figure8b are obtained by arbitrary affine transformations on the template objects. The independent shapes of letters and numbers, whose contours are labeled by red color in Figure8, are obtained by simply image segmentation as they are uniform. The results of the license recognition are given in Table1. The notations “Y” and “N” represent the correct and wrong results of object recognition, respectively. In addition, the shapematching results of Rhistogram, Fhistogram, DOPM, and CSS are 100, 10, 57.5, and 60%, respectively. Experiments validate that Rhistogram is superior to Fhistogram, DOPM, and CSS for object recognition under affine transformations, whereas Fhistogram performs worst as it is not affine invariant.
Conclusions
In this article, a novel affine invariant descriptor, Rhistogram, is proposed to describe the relative attitude relationship between the shapes. The shapes are handled as the longitudinal segments parallel to the line connecting centroids of two shapes, and the Rhistogram is constructed by the length ratios of collinear longitudinal segments from two shapes. In the shapematching algorithm, the Rhistograms of original shape pairs are first found in the offline preprocessing phase. Then in the matching phase, to improve the shapematching accuracy, a voting strategy is applied to the candidate corresponding shape pairs, which are discovered by Rhistogram matching. There are four advantages of the proposed algorithm. First, the contours of the shapes do not need to be extracted; second, the new descriptor is robust to affine transformation and noise; third, it is simple with low computational complexity; finally, it guarantees high shape matching accuracy by voting for all candidate correspondences with minimal error of Rhistogram matching. The Rhistogram of a shape pair is insensitive to the distance along the line connecting the centroids of the shapes, which results in that the shape pairs with the same attitudes but different distances generate undistinguishable Rhistograms. One solution to this limitation is to add distance information to Rhistogram, which will be investigated in our future research.
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This work was supported by the National Natural Science Foundation of China under Grant 61201338.
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Wang, W., Xiong, B., Sun, H. et al. An affine invariant relative attitude relationship descriptor for shape matching based on ratio histograms. EURASIP J. Adv. Signal Process. 2012, 209 (2012). https://doi.org/10.1186/168761802012209
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Keywords
 Affine invariant
 Attitude relationship
 Rhistogram
 Shape matching