Robust Affine Invariant Descriptors

An approach is developed for the extraction of affine invariant descriptors by cutting object into slices. Gray values associated with every pixel in each slice are summed up to construct affine invariant descriptors. As a result, these descriptors are very robust to additive noise. In order to establish slices of correspondence between an object and its affine transformed version, general contour GC of the object is constructed by performing projection along lines with different polar angles. Consequently, affine in-variant division curves are derived. A slice is formed by points fall in the region enclosed by two adjacent division curves. To test and evaluate the proposedmethod, several experiments have been conducted. Experimental results show that the proposedmethod is very robust to noise.


Introduction
Object recognition is an important topic in the area of computer vision and has been found numerous applications in real world.One of the common difficulties in object recognition is that the object shape is often distorted for observing under various orientations which can be appropriately described by perspective transformation 1 .Furthermore, if the size of observed object is far less than the distance between object and the observing position, the change of the object's shape can be described by affine transform.
The extraction of affine features plays a very important role in pattern recognition and computer vision 2-4 .Many algorithms have been developed.Based on whether the features are extracted from the contour only or from the whole shape region, the approaches using invariants features can be classified into two main categories: region-based methods and contour-based methods 5 .

The GC and Its Characteristics
Any object can be converted to a closed curve general contour of the object by taking projection along lines from the centroid with different angles central projection transform .In this section, we devote to studying the characteristics of GC.

The GC of an Object
Suppose that an object is represented by I x, y in the 2D plane.Firstly, the origin of the reference system is transformed to the centroid of the object, as denoted by O x 0 , y 0 , which can be computed from the geometric moments as follows: x 0 xI x, y dx dy I x, y dx dy , y 0 yI x, y dx dy I x, y dx dy .

2.1
Let R max x,y ∈D x − x 0 2 y − y 0 2 be the longest distance from x 0 , y 0 to a point x, y on the pattern.
To derive the GC of an object, the Cartesian coordinate system should be converted to polar coordinate system.The conversion is based on the following relations: x r cos θ, y r sin θ.

2.2
Hence, the shape can be represented by a function f of r and θ, namely, where r ∈ 0, R , and θ ∈ 0, 2π .
After the conversion of the system, we perform CPT to the object by computing the following integral: where θ ∈ 0, 2π .The function g θ is, in fact, equal to the total mass as distributed along different angle from 0 to R. The CPT method has been used to extract rotation invariant signature by combining wavelet analysis and fractal theory in 21 .A satisfying classification rate has been achieved in the recognition of rotated English letters, Chinese characters, and handwritten signatures.For more details of CPT, refer to 20 .
From a practical point of view, the images to be analyzed by a recognition system are most often stored in discrete formats.Catering to such two-dimensional discrimination patterns, we should modify 2.4 into the following expressions: where θ k ∈ 0, 2π , k 0, 1, 2, . . ., N − 1.
Definition 2.1.For an angle θ ∈ R, g θ is given in 2.4 , then θ, g θ denotes a point in the plane of R 2 .Let θ go from 0 to 2π, then { θ, g θ | θ ∈ 0, 2π } forms a closed curve.We call this closed curve the general contour GC of the object.
For an object Þ, we denote the GC extracted from it as ∂Þ.By discrete form 2.5 , the discrete GC of the object can be derived.For example, Figure 1

The Properties of GC
The GC of an object has the following properties: single contour, affine invariant, and robust to noise.

Single Contour
By 2.4 , a single value is correspond to an angle θ ∈ R. Consequently, a single closed curve GC can be derived from any object.For instance, see the GCs of Figures 1 a and In real life, many objects consist of several separable components.Contour-based methods are unapplicable to these objects.By performing projection along lines with different polar angles, a single closed curve can be derived, and contour-based methods can be applied to any object.Consequently, shape representation based on GC of the object may provide better data reduction than some region-based methods.

Affine Invariant
Affine maps parallel lines onto parallel lines, intersecting lines into intersecting lines.Based on these facts, it can be proved that the GC extracted from the affine transformed object is also an affine transformed version of GC extracted from the original object.

Robustness to Noise
It is shown in 23 that Radon transform is quite robust to noise.We can similarity show that GC derived from the object is robust to additive noise as a result of summing pixel values to generate GC.

Features Extraction by Cutting Object into Slices
To extract affine invariant features, we cut the object into slices.These slices are regions enclosed by affine invariant closed curves which are derived based on the GC of the object.A slice derived from the affine transformed object is the same affine transformed version of slice derived from the original object.

Cutting Object into Slices
Prior cutting the object into slices, we should derive affine invariant closed curves which are called division curves of the object.
Definition 3.1.For an object F, suppose that ∂F is its GC, and O is the centroid of the object as defined in 2.1 .If O and C superpose each other, the point P is selected as the centroid O.
Otherwise, the point P is selected on the line segment connected the centroid O and point C on the GC such that the following equation is satisfied.That is, where τ is a constant.As C going along the GC, the locus of point P formed a closed curve.We denote this closed curve as ∂F τ , which is called the τ-division curve of the object.
As the constant τ varied, different division curves will be obtained.Figure 3 shows division curves of Figures 1 a and 2 a .We can observed that division curves extracted from the affine transformed object are also affine transformed version of division curves extracted from the original object.
We denote the region enclosed by two different division curves τ 1 -division curve and τ 2 -division curve as DF τ 1 τ 2 , which is called τ 1 τ 2 -slice of the object F. Figure 4 shows some slices of the object in Figure 1 a .

Affine Invariant Descriptors
By different division curves, the object can be cut into a number of slices.We can employ some well-known methods such as AMIs 13 and MSA 19 to extract affine invariant feature vectors from a piece of these slices.Consequently, the object is recognized by composing these feature vectors into a united vector.However, the moment-based method is very sensitive to noise, and MSA has large computational complexity.In this paper, we extract affine invariant features by summing up gray values associated with points in region of the derived slices.It will be shown that these features are very robust to noise.Choose a series of numbers {τ 0 , τ 1 , . . ., τ n } such that For an object F, we denote MDF τ i τ j as the mass of τ i τ j -slice of the object F; that is, We denote S i as follows:

3.3
We will prove that S i i 0, 1, . . .n are affine invariants.In the experiments of this paper, the objects are cut equally into N parts.If we set the maximum τ n M, then the numbers {τ 0 , τ 1 , . . .τ n } are set to In this paper, M is set to 4.
Proof.As aforementioned, GC derived from the affine transformed object is the same affine transformed version of GC derived from the original object.In addition, affine transform preserves the ratios of distances along a line.Consequently, the τ-division curve derived from the affine transformed object is the same affine transformed version of τ-division curve derived from the original object.As a result, the τ 1 τ 2 -slice derived from the affine transformed object F α is the same affine transformed version of τ 1 τ 2 -slice derived from the original object F.
Affine maps have mass relative invariance property, which states that the mass of an affinely transformed object is equal to the product of its original object mass times the determinant of the transformation matrix.In other word, the slice of the original object DF τ 1 τ 2 and the slice of the affinely transformed object DF τ 1 τ 2 α satisfy the following equation: I x, y dx dy.
We call S i i 0, 1, . . .n given in 3.3 as affine invariant descriptors.

Robustness to Noise
In this section, we study the noise robustness of the affine invariant descriptors given in 3.3 .Let I x, y be the original image whose intensity values are random variables with mean μ I and variance σ 2 I .Suppose that the image is noised by noise with zero mean and variance σ 2 n .Since the affine invariant descriptors defined in 3.3 are the integral of gray values of the slices for the continuous case, the integral of noise in the slice is constant and is equal to the mean value of the noise which is assumed to be zero.Therefore, zero-mean white noise has no effect on the descriptors of the image in this situation.
In practice, the image is stored by a finite number of pixels.As aforementioned, we add up intensity values of the pixels in a slice DF τ i τ i 1 to calculate the affine invariant descriptors.Assume that we add up R pixels of I x, y to calculate MDF τ i τ i 1 , where R is the number of pixels in DF τ i τ i 1 .Suppose that E denotes the expected values and that D denotes the variance.Therefore,

3.8
Then, the expected value of MDF τ i τ i 1 2 is 3.9 Equations 3.8 and 3.9 indicates the relations of descriptors value and the mean the variance of the original image.
After introducing the noise with zero mean and variance σ 2 to the image, the signalto-noise ratio SNR of the image is

3.10
It follows from 3.9 that the SNR of the affine invariant descriptors can be given as follows:

3.11
This means the SNR is increased by μ 2 I /σ 2 n .Due to the fact that in many practical situation σ 2 I ≤ μ 2 I , we may alternatively write

3.13
This shows that SNR has been increased by a factor of R, which is practically a large quantity.As a result, the affine invariant descriptors are very robust to additive noise.

Experiments
In this section, some experiments are carried out to illustrate the performance of the proposed method.The gray-scale images utilized in our experiments are taken from the well-known Columbia Coil-20 database 22 , which contains 20 different objects shown in Figure 5.The Coil-20 database includes some sets of similar objects, such as three toy cars, ANACIN, and TYLENOL packs.They can be easily misclassified due to their similarity.
In the experiments, affine transformations are generated by the following transformation matrix 7 : where k, θ denote the scaling, rotation transformation, respectively, and a, b denote the skewing transformation.To each image, the affine transformations are generated by setting the parameters in 4.1 as follows:

Implement Issues
In practice, the objects are not available as continuous functions.We only have some amount of discrete samples.The digital image are represented as N 1 × N 2 matrix.With the affine transforms, the position of each point changes, and it is possible that number of points in any region changes too.Hence, the GC should be parameterized to establish one-to-one points correspondence between GC and its affine transformed version.Several parameterizations have been reported.In this paper, we adopt a curve normalization approach proposed by Yang et al. 24 , which is called EAN.The EAN method mainly composes of the following steps.
i For the discrete GC { x θ k , y θ k : k 0, 1, 2, . . ., N − 1}, compute the total area of the GC by the following formula: Let the number of points on the contour after EAN be N too.Denote S part S/ N.
ii Select the starting point on GC as the starting point P 0 x θ 0 , y 0 θ 0 of the normalized curve.From P 0 x θ 0 , y θ 0 on GC, search a point P 1 x θ 1 , y θ 1 along GC, such that the area of each closed zone, namely the polygon P 0 OP 1 equals to S part , where O denotes the centroid of the object.
This normalization provides a one-to-one relation between the points of original GC and transformed GC.For more information of EAN, refer to 24 .
Consequently, the τ-division curve can be constructed to establish one-to-one points correspondence between τ-division curves and its affine transformed version.Finally, the region enclosed by two different division curves forms the slice of the object.

Comparison with AMIs and MSA
In this experiment, we compare the proposed method with MSA and AMI.The AMIs method is implemented as discussed in 13 , and 3 AMIs invariants are used.The MSA method is implemented as discussed in 19 , and 29 MSA invariants are used.As aforementioned, the Coil-20 database is employed.Each image is transformed 140 times.That is to say, the test is repeated 2800 times using every method.In this experiment, our method is performed for N 10.The classification accuracies of the proposed method, AMIs, and MSA are 98.59%, 100%, and 95.31%, respectively.The results indicate that AMIs perform best in this test, and the proposed method is a little outperforms over MSA.
We firstly add the salt and pepper noise with intensities varying from 0.005 to 0.03 to the transformed images.Figure 6 shows the classification accuracies of all methods in the corresponding noise degree.We can observe that the classification accuracy of AMIs decreases rapidly from noise free condition to small noise degree.The classification accuracy decreases from 100% to less than 50% when the noise intensity is 0.005.MSA performs much better than AMIs, but the results are not satisfying.The drop of classification associated with the proposed method is even less than three percents.To large noise degrees, the proposed method keeps high accuracies all the time.
We add the Gaussian noise with zero mean and different variance varying from 0.5 × 10 −3 to 4 × 10 −3 to the transformed images.Figure 7 plots the classification accuracies of all methods in the corresponding noise degree.The results indicate that AMIs and MSA are much more sensitive to Gaussian noise than salt and pepper noise.Their classification results fall quickly once the image is suffered from Gaussian noise.However, the classification accuracies of the the proposed method greatly outperform AMIs and MSA in every noise degree.

The Affection of Slice Size
The affine invariant descriptors are constructed by cutting the object into slices.We test the performance of the proposed method with different slice sizes in this experiment.The slice sizes is affected by N in 3.4 .With big N, the object is cut into small slices with large computational complexity.On the other hand, if N is small, the object is cut into big slices with low computational complexity.
The gray-scale images of Coil-20 database are employed.These images are transformed as aforementioned; that is to say, the test is repeated 2800 times.These transformed images are classified according to their affine invariant descriptors by comparing their distance Euclidean distance to that of the training images.The accuracies are shown in Table 1.As N increasing, fine details of the object can be carried by small size slices.Hence, high accuracy can be achieved e.g., 99.55% accuracy for N 50 .
The images are always noise for reasons in real-life situations.So, we also test the robustness of the proposed method in this experiment.Every test image is added Gaussian noise with different intensities.The intensity level σ 2 is set to 0.005, 0.015, 0.020, 0.025.The accuracies are also shown in Table 1.We can observe that the accuracies decreased with   increasing noise level.Furthermore, under noisy conditions, although fine details of the object can be carried by small size slices, discrete error will affect the accuracy.For instance, the accuracy of N 40 is lower than it of N 30 all the time.

Discussions
In this study, we cut object into slices, and the affine invariant features are derived by summing up gray value associated with every pixels in each slices.Experimental results show that the proposed method is very robust to conventional Gaussian noise.However, conventional Gaussian noise is never enough.Recently, people are more interested in fractional Gaussian noise fGn than conventional one.For more details, see 25-28 .The simulation of fGn was discussed in 25, 29 .Multiscaled fGn can be found in 25, 30-32 .Our future research is to consider fGn in the proposed scheme.

Conclusions
In this paper, we describe a novel approach for the extraction of affine invariant features by cutting object into slices.Firstly, the general contour GC is derived from the object by performing projection along lines with different polar angles.Consequently, some affine invariant curves, which is called division curves are derived from the object based on the derived GC.Then, a slice is formed by points fallen in the region between two adjacent division curves.The affine invariant features are derived by summing up gray value associated with every pixels in each slices.These features are very robust to additive noise as a result of summing pixel values to generate these features.In comparison with AMIs and MSA, the proposed method is more robust to noise in the background.As for our future work, some characteristics of slices associated with an object will be deeply studied, and more experimental results will be reported.

Figure 1 :
Figure 1: a A gray-level image taken from the Coil-20 database.b A binary image taken from the DPEG-7 database.c GC derived from a .c GC derived b .
a is a gray-scale image taken from the well-known Columbia Coil-20 database 22 , and Figure 1 b shows the image of a binary image taken from the MPEG-7 database.Figures 1 c and 1 d show the GCs of Figures 1 a and 1 b , respectively.

1 b
in Figures 1 c and 1 d .Those objects have been concentrated into a integral pattern.

Figure 2
a shows an affine transformed version of Figure 1 a , and Figure 2 c shows the GC derived from Figure 2 a .Observing GCs in Figures 1 c , 2 c , 1 d , and 2 d , we can see that GC of an object is affine invariant.

Figure 2 :
Figure 2: a An affine transformation version of Figure 1 a .b An affine transformation version of Figure 1 b .c GC derived from a .d GC derived from b .

Figure 3 :Figure 4 :
Figure 3: Division curves: a division curves of Figure 1 a , b division curves of Figure 2 a .

Figure 6 :
Figure 6: Classification accuracies of AMIs, MSA, and the proposed method in case of affine transformation and different intensities of salt and pepper noise.

Figure 7 :
Figure 7: Classification accuracies of AMIs, MSA, and the proposed method in case of affine transformation and different intensities of Gaussian noise.

Table 1 :
Classification rates under different degrees of Gaussian noise with different slice sizes.