1 Introduction
Establishing visual correspondences is a core problem in computer vision. A common approach is to detect keypoints in different images and construct keypoints’ local descriptors for matching. The challenge lies in representing keypoints with discriminative descriptors, which are also invariant to photometric and geometric transformations.
Numerous methods have been proposed in the literature to tackle such problems in a certain degree. The scale invariance is often achieved by estimating the characteristic scales of keypoints. The pioneer work is done by Lindeberg
[11], who proposes a systematic methodology for automatic scale selection by detecting the keypoints in multiscale representations. Local extremas over scales of different combinations of normalized derivatives indicate the presence of characteristic local structures. Lowe [13] extends the idea of Lindeberg by selecting scale invariant keypoints in DifferenceofGaussian (DoG) scale space. Other alternatives are SURF [4], BRISK [10], HarrisLaplacian and HessianLaplacian [16]. Since these methods are not designed for affine invariance, their performances drop quickly under significant viewpoint changes. To deal with the distortion induced by viewpoint changes, some researchers propose to detect regions covariant to the affine transformations. Popular methods include HarrisAffine [16], HessianAffine [15], MSER [14], EBR and IBR [21]. They estimate the shapes of elliptical regions and normalize the local neighborhoods into circular regions to achieve affine invariance. Since the estimation of elliptical regions are not accurate, ASIFT [19]proposes to simulate all image views under the full affine space and match the SIFT features extracted in all these simulated views to establish correspondences. It improves the matching performance at the cost of a huge computational complexity.
This paper aims to tackle the affine distortion by developing a novel Affine Subspace Representation (ASR) descriptor, which effectively models the inherent information of a local patch among multiviews. Thus it can be combined with any detector to match images with viewpoint changes, while traditional methods usually rely on an affineinvariant detector, such as HarrisAffine + SIFT. Rather than estimating the local affine transformation, the main innovation of this paper lies in directly building descriptor by exploring the local patch information under multiple views. Firstly, PCA (Principle Component Analysis) is applied to all the warped patches of a keypoint under various viewpoints to obtain a set of patch representations. These representations are referred to as PCApatch vectors in this paper. Secondly, each set of PCApatch vectors is represented by a lowdimensional linear subspace under the assumption that PCApatch vectors computed from various affinewarped patches of the same keypoint span a linear subspace. Finally, the proposed Affine Subspace Representation (ASR) descriptor is obtained by using a subspacetopoint mapping. Such a linear subspace representation could efficiently capture the underlying local information of a keypoint under multiple views, making it capable of dealing with affine distortions. The workflow our method is summarized in Fig. 1, each step of which will be elaborated in Section 3. To speedup the computation, a fast approximate algorithm is proposed by removing most of its computational cost to an offline learning stage (the details will be introduced in Section 3.3). This is the second contribution of this paper. Experimental evaluations on image matching with various transformations have demonstrated that the proposed descriptors can achieve stateoftheart performance. Moreover, when dealing with images with viewpoint changes, ASR performs rather well without a dedicated affine detector, validating the effectiveness of the proposed method.
The rest of this paper is organized as follows: Section 2 gives an overview of related works. The construction of the proposed ASR descriptor as well as its fast computation algorithm are elaborated in Section 3. Some details in implementation is given in Section 4. Experimental evaluations are reported in Section 5 and finally we conclude the paper in Section 6.
2 Related Work
Lindeberg and Garding [12]
presented a methodology for reducing affine shape distortion. The suggested approach is to adapt the shape of smoothing kernel to the local image structure by measuring the second moment matrix. They also developep a method for extracting bloblike affine features with an iterative estimation of local structures. Based on the work of Lindeberg, Baumberg
[3] adapted the local shapes of keypoints at fixed scales and locations , while Mikolajczyk and Schmid [16] iteratively estimated the affine shape as well as the location and scale. Tuytelaars and Van Gool [21] proposed two affine invariant detectors. The geometrybased method detects Harris corners and extracts edges close to such keypoints. Several functions are then chosen to determine a parallelogram spanned by the nearby two edges of the keypoint. The intensitybased method extracts local extremas in intensity as anchor points. An intensity function along rays emanating from these anchor points is used to select points where this function reaches an extremum. All these selected points are linked to enclose an affine covariant region which is further replaced by an ellipse having the same shape moments up to the second moments. Matas et al. [14] developed an efficient affine invariant detector based on the concept of extremal regions. The proposed maximally stable extremal regions are produced by a watershed algorithm and their boundaries are used to fit elliptical regions.Since the accuracy of affine shape estimation is not guaranteed, Morel and Yu [19] presented a new framework for affine invariant image matching named ASIFT. They simulated all possible affine distortion caused by the change of camera optical axis orientation from a frontal position, and extract SIFT features on all these simulated views. The SIFT features on all simulated views are matched to find correspondences. Since ASIFT has to compute SIFT on lots of simulated views and make use of an exhaustive search on all possible views, it suffers a huge computational complexity. Although a similar view simulation method of ASIFT is used in our method, here it is for a totally different purpose: warping local patch of a keypoint under multiple views to extract PCApatch vectors for keypoint description. Therefore, our method does not suffer from the huge computational burden as in ASIFT. Hintersoisser et al. [9]
proposed two learning based methods to deal with full perspective transformation. The first method trains a Fern classifier
[20] with patches seen under different viewing conditions in order to deal with perspective variations, while the second one uses a simple nearest neighbors classifier on a set of “mean patches”that encodes the average of the keypoints appearance over a limited set of poses. However, an important limitation of these two methods is that they can not scale well with the size of keypoints database. Moreover, they both need a frontoparallel view for training and the camera internal parameters for computing the camera pose relative to the keypoint.The most related work to this paper is SLS [8], which describes each pixel as a set of SIFT descriptors extracted at multiple scales. Our work extends SLS to deal with affine variations. Moreover, we propose to use PCApatch vector as a compact intermediate representation of the warped patch instead of SIFT. The main advantages are twofolds: (a) fast, because PCApatch vector is fast to compute while computing SIFT is much slower; (b) since computing PCA vector is a linear operation, it leads to the proposed fast algorithm.
3 Our Approach
3.1 Multiple View Computation
As the projective transformation induced by camera motion around a smooth surface can be locally approximated by an affine transformation, we locally model the apparent deformations arising from the camera motions by affine transformations. In order to deal with affine distortions, we propose to integrate local patch information under various affine transformations for feature description rather than estimating the local affine transformation (e.g., [21, 16]).
Since we employ scaleinvariant detector to select keypoints, we first extract a local patch at the given scale around each keypoint and then resize it to a uniform size of
. To deal with linear illumination changes, the local patch is usually normalized to have zero mean and unit variance. Here we skip this step since the subsequent computation of linear subspace is invariant to linear illumination changes.
The local patch is aligned by the local dominant orientation to achieve invariance to inplane rotation. In order to efficiently estimate such orientations, we sample some pattern points in the local patch similar to BRISK [10]. The dominant orientation is then estimated by the average gradient direction of all the sampling points:
(1) 
where is the number of sampling points, is the average gradients, and are the directional and directional gradients of sampling point respectively. Since there are only a few sample points, e.g., in our experiments, the orientation can be estimated very fast.
Let be the aligned reference patch around a keypoint at a given scale, the warped patch under an affine transformation is computed by:
(2) 
where is the warping function using transformation . To avoid the case that some parts of the warped patch may not visible in the reference patch, we take the reference patch a little larger in practice. Hence, Eq. (2) can be rewritten as:
(3) 
where is a function that extracts a small central region from the input matrix.
To encode the local information of each
, we propose to use a simple PCA based representation for its compactness and efficiency. By using PCA, the local patch is projected into the eigenspace and the largest
principal component coordinates are taken to represent the patch, i.e., the PCApatch vector. Mathematically, the PCApatch vector for can be computed as :(4) 
where is the learned PCA projection matrix, denotes vectorization of a matrix, and . By substituting Eq. (3), Eq. (4) can be rewritten as:
(5) 
The idea of using PCA for feature description is not novel, e.g., PCASIFT descriptor in [24] and GLOH descriptor in [17]. Here we only use such a technique to effectively generate a set of compact vectors as the intermediate representations. Further representation of the keypoint will be explored based on these intermediate representations.
3.2 Subspace Representation
Suppose there are parameterized affine transformations to warp a local patch, we can get a PCApatch vector set for a keypoint by the above approach. Inspired by Hassner et al.[8] who dealt with scale invariant matching by using a linear subspace representation of SIFT descriptors extracted on multiple scales, we proposed to construct a subspace model to represent the PCApatch vectors extracted on multiple views.
The key observation is that the PCApatch vectors extracted under various affine transformations of a same keypoint approximately lie on a lowdimensional linear subspace. To show this point, we conducted statistical analysis on the reconstruction loss rates^{1}^{1}1It is defined as the rate between reconstruction error and the original data, while the reconstruction error is the squared distance between the original data and its reconstruction by PCA. of PCApatch vectors for about 20,000 keypoints detected from images randomly downloaded from the Internet. For each keypoint, its PCApatch vector set is computed and used to estimate a subspace by PCA. Then the reconstruction loss rates of each set by using different numbers of subspace basis are recorded. Finally, the loss rates of all PCApatch vector sets are averaged. Fig. 3 shows how the averaged loss rate is changed with different subspace dimensions. It can be observed that a subspace of 8 dimensions is enough to approximate the 24 dimensional PCApatch vector set with 90% information kept in average. Therefore, we choose to use a dimensional linear subspace to represent . Mathematically,
(6) 
where are basis vectors spanning the subspace and are the coordinates in the subspace. By simulating enough affine transformations, the basis can be estimated by PCA.
Let and be the PCApatch vector sets of keypoints and respectively, the distance between and can be measured by the distance between corresponding subspaces and . As shown in [5], all the common distances between two subspaces are defined based on the principal angles. In our approach, we use the Projection Frobenius Norm defined as:
(7) 
where is a vector of sines of the principal angles between subspaces and , and are matrixes whose columns are basis vectors of subspaces and respectively.
To obtain a descriptor representation of the subspace, similar to [8] we employ the subspacetopoint mapping proposed by Basri et al. [2]. Let be the matrix composed of orthogonal basis of subspace , the proposed ASR descriptor can be obtained by mapping the projection matrix into vectors. Since is symmetric, the mapping can be defined as rearranging the entries of into a vector by taking the upper triangular portion of , with the diagonal entries scaled by . Mathematically, the ASR descriptor q is
(8) 
where are elements of , and is the dimension of the PCApatch vector. Thus the dimension of is .
By such mapping, it is worth noting that the Projection Frobenius Norm distance between subspaces and is equal to the Euclidean distance between the corresponding ASR descriptors and :
(9) 
It is worth noting that ASR is inherently invariant to linear illumination changes. Suppose is the set of PCApatch vectors for a keypoint, while is its corresponding set after linear illumination changes. For each element in , where and parameterize the linear illumination changes. Let and be their covariant matrixes, it is easy to verify that
. Therefore, they have the same eigenvectors. Since it is the eigenvectors used for ASR construction, the obtained ASR for
and will be identical.3.3 Fast Computation
Due to the high computational burden of warping patches, it would be very inefficient to compute a set of PCApatch vectors extracted under various affine transformations by utilizing Eq. (5) directly.
In [9], Hinterstoisser et al. proposed a method to speed up the computation of warped patches under different camera poses based on the linearity of warping function. We found that their method could be easily extended to speed up the computation of any linear descriptor of the warped patches. According to this observation, we develop a fast computation method of at the cost of a little accuracy degradation in this section.
Similar to [9], we firstly approximate by its principal components as:
(10) 
where is the number of principal components, and are the principal components and the projection coordinates respectively.
Note that the warping function is essentially a permutation of the pixel intensities between the reference patch and the warped patch. It implies that
is actually a linear transformation. Since
and are also linear functions, Eq. (11) can be rewritten as:Although the computation of and is still time consuming, it can be previously done in an offline learning stage. At run time, we simply compute the projection coordinates of the reference patch by
(14) 
where is the learned projection matrix consisting of . Then, can be computed by a linear combination of and .
Obviously, this approach combines the patch warping and representation into one step, and moves most of the computational cost to the offline learning stage. Compared to the naive way in Eq. (5), it significantly reduces the running time. We refer to ASR descriptor computed by such a fast approximate algorithm as ASRfast descriptor, while the original one is referred to as ASRnaive descriptor.
4 Notes on Implementation
4.1 Parameterization of Affine Transformation
As shown in [19], any 2D affine transformation with strictly positive determinant which is not a similarity has a unique decomposition:
(15) 
where , is a rotation matrix, , , is a diagonal matrix with .
Fig. 3 gives a geometric interpretation of this decomposition: is the object plane, is the image plane, (longitude) and (latitude) are the camera viewpoint angles, is the camera inplane rotation, and is the zoom parameter. The projective transformation from image plane to object plane can be approximated by the affine transformation in Eq. (15).
Since the scale parameter can be estimated by scaleinvariant detectors and the inplane rotation can be aligned by local dominant orientation, we only sample the longitude angle and the tilt .
For , the sampling range is as indicated by the decomposition in Eq. (15). The sampling step is determined by considering the overlap between the corresponding ellipses of adjacent samplings. More specifically, for an affine transformation with tilt and longitude , the corresponding ellipse is . Let denotes the overlap rate between and , it can be proved that is a decreasing function of when . We can choose the sampling step as the max value that satisfies where is a threshold that controls the minimal overlap rate required for the corresponding ellipses of adjacent samplings. The larger is, the more will be sampled.
For , the sampling range is set to to make the latitude angle range from to . Thus, the sampling step is where is the sampling number of .
Setting these sampling values is not a delicate matter. To show this point, we have investigated the influence of different sampling strategies for and on image pair of ’trees 12’ of the Oxford dataset [1]. Fig. 5(a) shows the performance of ASRnaive by varying (, , and ) when . It can be seen that , and are comparable and they are better than . Therefore, we choose since it leads to the least number of affine transformations. Under the choice of , we also test its performance on various (, , and ) and the result is shown in Fig. 5(b). Although performs the best, we choose to make a compromise between accuracy and sparsity (complexity). According to the above sampling strategy, we totally have simulated affine transformations. Note that the performance is robust to these values in a wide range. Similar observations can be obtained in other test image pairs.
4.2 Offline Training
From Section 3, it can be found there are three cases in which PCA is utilized:

PCA is used for raw image patch representation to obtain a PCApatch vector for each affinewarped image patches.

PCA is used to find subspace basis of a set of PCApatch vectors for constructing ASR descriptor.

PCA is used to find principal components to approximate a local image patch for fast computation (c.f. Eq. (10).
In cases of (1) and (3), several linear projections is required. More specifically, principal projections are used for PCApatch vector computation and principal components are used to approximate a local image patch. These PCA projections are learned in an offline training stage. In this stage, the PCA projection matrix in Eq. (4), and in Eq. (13) are computed by using about 2 patches detected on training images provided by PASCAL VOC 2012. Thus the training images are significantly different from those used for performance evaluation in Section 5.
5 Experiments
In this section, we conduct experiments to show the effectiveness of the proposed method. Firstly, we study the potential impact of different parameter settings on the performance of the proposed method. Then, we test on the widely used Oxford dataset [1] to show its superiority to the stateoftheart local descriptors. With the image pairs under viewpoint changes in this dataset, we also demonstrate that it is capable of dealing with affine distortion without an affine invariant detector, and is better than the traditional method, e.g., building SIFT descriptor on Harris Affine region. To further show its performance in dealing with affine distortion, we conduct experiments on a larger dataset (Caltech 3D Object Dataset [18]), containing a large amount of images of different 3D objects captured from different viewpoints. The detailed results are reported in the following subsections.
5.1 Parameters Selection
In addition to and for sampling affine transformations, our method has several other parameters listed in Table1. We have investigated the effect of different parameter settings on image pair of ’trees 12’ in the Oxford dataset [1]. We simply tried several combinations of these parameters and compared the matching performance among them. The result is shown in Fig. 6. Fig. 6(a) is obtained by computing ASRnaive under different (, and ) and (, and ). It is found that the configuration of obtains the best result. For a trade off between the performance and descriptor dimension, we choose , leading to ASR with dimensions. Under the choice of , we investigate the fast approximate algorithm by computing ASRfast under different (, , and ). Fig. 6(b) shows that obtains the best result. A typical setting of all parameters is given in Table 1 and kept unchanged in the subsequent experiments.
parameter  description  typical value 

pattern number for dominant orientation estimation  60  
number of orthogonal basis for approximating local patch  160  
size of local patch  21  
dimension of the PCApatch vector  24  
dimension of the subspace that PCApatch vector set lies on  8 
5.2 Evaluation on Oxford Dataset
To show the superiority of our method, we conduct evaluations on this benchmark dataset based on the standard protocol [17], using the nearest neighbor distance ratio (NNDR) matching strategy. For comparison, the proposed method is compared with SIFT [13] and DAISY [23] descriptors, which are the most popular ones representing the stateoftheart. The results of other popular descriptors (SURF, ORB, BRISK etc.) are not reported as they are inferior to that of DAISY. In this experiment, keypoints are detected by DoG [13] which is the most representative and widely used scale invariant detector. Due to space limit, only the results of two image pairs (the vs. the and the vs. the ) for each image sequence are shown, which represent small and large image transformations respectively.
As shown in Fig. 7, it is clear that ASRfast performs comparable to ASRnaive in all cases except ’graf 14’ (Fig. 7(f)). This demonstrates the fact that the proposed fast computation strategy in Eq. (12) can well approximate the naive computation of PCApatch vector set. The performance degradation in ’graf 14’ can be explained by the difference in patch alignment. Since ASRfast does not generate the warped patches directly, it simply aligns the reference patch before computing the PCApatch vector set. This strategy could be unreliable under large image distortions since all the PCApatch vectors extracted under various affine transformations depend on the orientation estimated on reference patch. ASRnaive avoids this by computing the dominant orientation on each warped patch and aligning it separately. In other words, the inferior performance of ASRfast is because that the PCApatch vector (i.e., the intermediate representation) relies on robust orientation estimation, but does not imply that ASRfast is not suitable for viewpoint changes. Therefore, if we can use an inherent rotation invariant intermediate representation (such as the one in similar spirit to the intensity order based methods [6, 7, 22]), ASRfast is expected to be as good as ASRnaive. We would leave this for our future work.
According to Fig. 7, both ASRnaive and ASRfast are consistently better than SIFT in all cases and outperform DAISY in most cases. The superior performance of the proposed method can be attributed to the effective use of local information under various affine transformation. For all cases of viewpoint changes especially in ’graf 14’, ASRnaive outperforms all competitors by a large margin, which demonstrates its ability of dealing with affine distortions.
To further show ASR’s ability in dealing with affine distortions without a dedicated affine detector, we use image pairs containing viewpoint changes to compare ASR with traditional methods, i.e., build local descriptor on top of affine invariant regions. In this experiment, HarrisAffine (HarAff) is used for interest region detection and SIFT/DAISY descriptors are constructed on these interest regions. For a fair comparison, ASR is build on top of HarrisLaplace (HarLap) detector since HarrisAffine regions are build up on HarrisLaplace regions by an additional affine adaptive procedure. Therefore, such a comparison ensures a fair evaluation for two types of affine invariant image matching methods, i.e., one based on affine invariant detectors, while the other based on affine robust descriptors.
The results are shown in Fig. 8. To show the affine adaptive procedure is necessary for dealing with affine distortions if the used descriptor does not account for this aspect, the results of HarLap:SIFT and HarLap:DAISY are also supplied. It is clear that HarAff:SIFT (HarAff:DAISY) is better than HarLap:SIFT (HarAff:DAISY). By using the same detector, HarLap:ASRnaive significantly outperforms HarLap:SIFT and HarLap:DAISY. It is also comparable to HarAff:DAISY and HarAff:SIFT in ’graf 14’, and even better than them in all other cases. This demonstrate that by considering affine distortions in feature description stage, ASR is capable of matching images with viewpoint changes without a dedicated affine invariant detector. The failure of HarLap:ASRfast is due to the unreliable orientation estimation as explained before.
Another excellent method to deal with affine invariant image matching problem is ASIFT. However, ASIFT can not be directly compared to the proposed method. This is because that ASIFT is an image matching framework while the propose method is a feature descriptor. Therefore, in order to give the reader a picture of how our method performs in image matching compared to ASIFT, we use ASR descriptor combined with DoG detector for image matching and the NNDR threshold is set to 0.8. The matching results are compared to those obtained by ASIFT when the matching threshold equals to 0.8. ASIFT is downloaded from the authors’ website. The average matching precisions of all the image pairs in this dataset are 64.4%, 80.8% and 75.6% for ASIFT, ASRnaive, and ASRfast respectively. Accordingly, the average matching times of these methods are 382.2s, 14.5s and 8.3s when tested on the ’wall’ sequence. We also note that the average matches are several hundreds when using ASR while they are one magnitude more when using ASIFT. Detailed matching results can be found in the supplemental material.
5.3 Evaluation on 3D Object Dataset
To obtain a more thoroughly study of dealing with affine distortions, we have also evaluated our method on the 3D object dataset [18], which has lots of images of 100 3D objects captured under various viewpoints. We use the same evaluation protocol as [18]. The ROC curves are obtained by varying the threshold on the quality of the appearance match , while the stability curves are obtained at fixed false alarm rate of .
As previous experimental setting, we use HarrisLaplace (HarLap) detector to produce scale invariant regions and then compute ASR descriptors for matching. For comparison, the corresponding HarrisAffine (HarAff) detector is used to produce affine invariant regions and SIFT/DAISY descriptors are computed based on them.
Fig. 9 shows the results averaged on all objects in the dataset when the viewing angle is varied from to . It can be observed that HarLap:ASRnaive performs best, and HarLap:ASRfast is comparable to HarAff:SIFT and HarAff:DAISY. This further demonstrates that the subspace representation of PCApatch vectors extracted under various affine transformations is capable of dealing with affine distortion.
5.4 Timing Result
In this section, we conduct time test on a desktop with an Intel Core2 Quad 2.83GHz CPU. We first test the time cost for each components of ASR, and the detailed results are given in Table 2. It can be found that most of construction time in ASR is spent on patch warping. It is worthy to note that by using the fast approximate algorithm, ASRfast does not compute the warped patch directly and so largely reduce its time by about . For comparison, we also report the time costs for SIFT and DAISY. Note that these timing results are averaged over runs, each of which computes about 1000 descriptors on image ’wall 1’. It is clear that ASRfast is faster than SIFT and DAISY, while ASRnaive is slower than SIFT but still comparable to DAISY.
ASRnaive  ASRfast  SIFT  DAISY  

patch warping[ms]      
patch representation[ms]      
subspace representation[ms]      
total time[ms]  2.09  3.8 
6 Conclusion
In this paper, we have proposed the Affine Subspace Representation (ASR) descriptor. The novelty lies in three aspects: dealing with affine distortion by integrating local information under multiple views, which avoids the inaccurate affine shape estimation, a fast approximate algorithm for efficiently computing the PCApatch vector of each warped patch, and the subspace representation of PCApatch vectors extracted under various affine transformations of the same keypoint.
Different from existing methods, ASR effectively exploits the local information of a keypoint by integrating the PCApatch vectors of all warped patches. The use of multiple views’ information makes it is capable of dealing with affine distortions to a certain degree while maintaining high distinctiveness. What is more, to speedup the computation, a fast approximate algorithm is proposed at a little cost of performance degradation. Extensive experimental evaluations have demonstrated the effectiveness of the proposed method.
7 Acknowledgment
This work is supported by the National Nature Science Foundation of China (No.91120012, 61203277, 61272394) and the Beijing Nature Science Foundation (No.4142057).
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