Novel Color Orthogonal Moments-Based Image Representation and 1 Recognition

: Inspired by quaternion algebra and the idea of fractional-order transformation, we propose a 23 new set of quaternion fractional-order generalized Laguerre orthogonal moments (QFr-GLMs) based 24 on fractional-order generalized Laguerre polynomials. Firstly, the proposed QFr-GLMs are directly 25 constructed in Cartesian coordinate space, avoiding the need for conversion between Cartesian and 26 polar coordinates; therefore, they are better image descriptors than circularly orthogonal moments 27 constructed in polar coordinates. Moreover, unlike the latest Zernike moments based on quaternion and 28 fractional-order transformations, which extract only the global features from color images, our 29 proposed QFr-GLMs can extract both the global and local color features. This paper also derives a new 30 set of invariant color-image descriptors by QFr-GLMs, enabling geometric-invariant pattern 31 recognition in color images. Finally, the performances of our proposed QFr-GLMs and moment 32 invariants were evaluated in simulation experiments of correlated color images. Both theoretical 33 analysis and experimental results demonstrate the value of the proposed QFr-GLMs and their 34 geometric invariants in the representation and recognition of color images.


Introduction
In the last decade, image moments and geometric invariance of moments have emerged as effective methods of feature extraction from images [1,2].Both methods have made great progress in imagerelated fields.However, most of the existing algorithms extract the image moments only from grayscale images.Color images contain abundant multi-color information that is missing in greyscale images.Therefore, in recent years, research efforts have gradually shifted to the construction of colorimage moments [3,4].Color-image processing is traditionally performed by one of three main methods: (1) Select a single channel or component from the color space of a color image, such a channel from a red-green-blue (RGB) image, as a grayscale image and calculate its corresponding image moments; (2) directly gray a color image, and then calculate its image moments; (3) calculate the image moments of each monochromatic channel (R, G and B) in a RGB image, and average them to obtain the final result.
Although all three methods are relatively simple to implement, they discard some of the useful image information, and cannot determine the relationship among the different color channels of a RGB image.This common defect reduces the accuracy of color-image representation in image processing or recognition.Owing to loss of correlations among the different color channels and part of the colorimage information, the advantages of color images over grayscale images are not fully exploited in practical application [5].
Recently, quaternion algebra based color image representation has provided a new research direction in color model spaces [6,7]such as RGB, luma-chroma (YUV), and hue-saturation-lightness (HSV) [8].Quaternion algebra has made several achievements in color-image processing [9,10].The quaternion method represents an image as a three-dimensional vector describing the components of the color image, which effectively uses the color information of different channels of the color image.
Wang et al. [11,12] constructed a class of quaternion color orthogonal moments based on quaternion theory.In ref [11], they proposed quaternion polar harmonic Fourier moments (QPHFMs) in polar coordinate space, and applied them to color-image analysis.They also proposed a zero-watermarking method based on quaternion exponent Fourier moments (QEFMs) [12], which is applied to copyright protection of digital images.Xia et al. [13] combined Wang et al.'s method with chaos theory, and proposed an accurate quaternion polar harmonic transform for a medical image zero-watermarking algorithm.The above results on quaternion color-image moments provide theoretical support for exploring new-generation color-image moments.However, image-moment construction based on quaternion theory is complex and increases the time of the color-image calculation.Moreover, the performance of the existing quaternion image moments in color-image analysis is not significantly improved from multi-channel color-image processing [14].Most importantly, the quaternion colorimage moments constructed by the existing methods are similar to grayscale-image moments, and extract only the global features; therefore, they are powerless for local-image reconstruction and region-of-interest (ROI) detection.In conclusion, the new generation of quaternion color-image moment algorithms requires further research.The new fractional-order orthogonal moments [15] effectively improve the performance of orthogonal moments in image analysis, and can also improve the quaternion color-image moments.The basis function of fractional-order orthogonal moments comprises a set of fractional-order (or real-order) orthogonal polynomials rather than traditional integer-order polynomials [16,17].
Fractional-order image moments have been realized only in the past three years, and their research is incomplete.Accordingly, their applications are limited to image reconstruction and recognition.In addition, the technique of the existing fractional-order orthogonal moments is only an effective supplement and an extension of integer-order grayscale image moments.Few academic achievements and investigations of fractional-order orthogonal moments have been reported in image analysis.
Inspired by fractional-order Fourier transforms, Zhang et al. [15] introduced fractional-order orthogonal polynomials in 2016, and constructed fractional-order orthogonal Fourier-Mellin moments for character recognition in binary images.Yang et al. [18] proposed fractional-order Zernike radial orthogonal moments based on Zernike radial orthogonal polynomials, and conducted related imagereconstruction testing.In image reconstruction tasks, the fractional-order Zernike radial orthogonal moments with different parameters outperformed traditional Zernike radial orthogonal moments.Based on Legendre polynomials with radial translation, Xiao et al. [19] constructed fractional-order orthogonal moments in Cartesian and polar coordinate spaces.They showed how general fractionalorder orthogonal moments can be constructed from integer-order orthogonal moments in different coordinate systems.Benouini et al. [20] recently introduced a new set of fractional-order Chebyshev moments and moment invariant methods, and applied them to image analysis and pattern recognition.
Although the existing fractional-order image moments provide better image descriptions than traditional integer-order image moments, their application to computer vision and pattern recognition remains in the exploratory stage.An improved fractional-order polynomial that constructs a superior fractional-order image moment is an expected hotspot of future research.Combining fractional-order image moments with quaternion theory, Chen et al. [21] newly developed quaternion fractional-order Zernike moments (QFr-ZMs), which are mainly used in robust copy-move forgery detection in color images.This paper combines the quaternion method with fractional-order Laguerre orthogonal moments, and hence develops new class of quaternion fractional-order generalized Laguerre moments (QFr-GLMs) for color-image reconstruction, and geometric-invariant recognition.The main contributions of this paper are summarized below.
1.The kernel function of the proposed QFr-GLMs is composed of fractional-order generalized Laguerre orthogonal polynomials, and the orthogonal moments are directly established in Cartesian coordinate space, providing better image descriptions than image moments constructed in polar coordinate space.
2. The state-of-the-art QFr-ZMs extract only the global features from color images.In contrast, the proposed QFr-GLMs not only describes the global features, but also extracts arbitrary local features (namely, the ROI) from a color image.
3. Based on the relationship between the proposed QFr-GLMs and geometric moments, this paper proposes a new set of feature-extraction approach, namely, quaternion fractional-order generalized Laguerre moment invariants (QFr-GLMIs), which can handle geometric-invariant transformations (rotation, scaling, and translation) of color images.

Preliminaries
In this section, we first introduce the basic concepts of quaternion theory and fractional-order image moments.The quaternion is a generalized form of complex numbers, a systematic mathematical theory and method proposed by the British mathematician Hamilton in 1843 [22], also fractional-order orthogonal moments are defined in Cartesian and polar coordinate spaces, and we present the transformation relationship between fractional-order orthogonal polynomials in Cartesian coordinate space and those in polar coordinate space.Then, we introduce the related contents of generalized Laguerre polynomials.

Representation quaternion algebra and fractional-order image moments
The quaternion is a four-dimensional complex number, also known as a hypercomplex.It is composed of one real component and three imaginary part components, and is formally defined in [5]: . ( To obtain the fractional-order image moments (Fr-IMs), we introduce the parameter  and slightly modify the basis of traditional geometric moments [19] as follows: As evidenced in Eq. ( 3), the order of the fractional-order geometric moments is ; that is, the integer-order is extended to real-order (or fractional-order).
In Cartesian and polar coordinate spaces, the fractional-order orthogonal moments are respectively defined as: where are the fractional-order orthogonal polynomials, and , are the binomial coefficients of the orthogonal polynomials [23,24].
Similarly to traditional integer-order image moments [25][26][27][28] We now determine the interchangeable relationship between the fractional-order orthogonal polynomials in Cartesian coordinate space and those in polar coordinate space.First, if orthogonal polynomial in Cartesian coordinates, the fractional-order orthogonal polynomial is expressed as , and the corresponding fractional-order radial orthogonal polynomials in polar coordinates is expressed as is an integerorder orthogonal polynomial in polar coordinate space, the fractional-order radial orthogonal polynomial is given by . The corresponding fractional-order orthogonal polynomial in Cartesian coordinates is then given by

Generalized Laguerre polynomials
The generalized Laguerre polynomials (GLPs), also known as associated Laguerre polynomials [29], are expressed as , GLPs satisfy the following orthogonal relationship in the range For convenience, we let   be a weighted function, and where nm  is the Kronecker delta function.) (  is the Pochhammer expression, and Using Eq. ( 11), 10) is redefined as To facilitate the calculation, we compute x L n  by the following recursive algorithm: , and . For details, see [17] and [22].

Methods
This section introduces our proposed QFr-GLMs scheme, derived from quaternion algebra theory, fractional-order orthogonal moments, and GLPs.After developing the basic framework of QFr-GLMs, we analyze the relationship between the quaternion based method and the single-channel based approach.
As shown in Fig. 1, the components of an image The remainder of this section is organized as follows.Subsection 3.1 defines and constructs our fractional-order GLPs (Fr-GLPs) and normalized Fr-GLPs (NFr-GLPs), and subsection 3.2 defines the proposed QFr-GLMs, and relates them to the fractional-order generalized Laguerre moments (Fr-GLMs) of single channels in a traditional RGB color image, and the basic framework is shown in Fig. 1.The QFr-GLMs invariants (QFr-GLMIs) are constructed in subsection 3.3.

Calculation of Fr-GLPs and NFr-GLPs
Fr-GLPs [30] can be expressed as: where, , similarly to Eq. ( 9).The Fr-GLPs satisfy the following orthogonality where . The Fr-GLPs can be rewritten as the following binomial expansion [19,30]: where , similar to Eq. ( 12), the Fr-GLPs can be implemented by the following recursive algorithm: In order to enhance the stability of polynomials, normalized polynomials are generally used instead of conventional polynomials.Therefore, normalized fractional-order GLPs (NFr-GLPs) are defined as: Using Eq. ( 16), we further obtain: , which completes the proof of Theorem 1.To reduce the computational complexity and ensure numerical stability, the NFr-GLPs are recursively calculated as follows: where . The detailed proof of the recursive operation is given in Appendix A.
Fig. 2 shows the distribution curves of the NFr-GLPs under different parameter settings.Note that the parameter mainly affects the amplitudes of the NFr-GLPs of different orders and the distributions of the zero values along the x-axis.Thus, if an image is sampled with NFr-GLPs, the local-feature regions (ROI) are easily extracted from the images.In addition, the parameter  can extend the integer-order polynomials to real-order polynomials ( ).Therefore, traditional GLPs are a special case of Fr-GLPs with . Note also that changing  changes the width of the zero-value distributions of the Fr-GLPs along the x-axis (Fig. 2(c)-(e)), thus affecting the imagesampling result.

Definition and calculation of QFr-GLMs
Pan et al. [22] proposed the generalized Laguerre moments (GLMs) for grayscale images in Cartesian coordinates.Recalling the introduction, the corresponding Fr-GLMs can be defined as: where ) , ( j i f gray represents a grayscale digital image.For convenience, we map the original two- Here, Using Eq. ( 24) with the help of Eq. ( 14), the right-side QFr-GLMs of an original RGB color image in Cartesian coordinates are defined as: is the unit pure imaginary quaternion.The QFr-GLMs expressed in quaternion and the Fr-GLMs of single channels in traditional RGB color images are related as follows: where Accordingly, an original color image ) , ( q p f rgb can be reconstructed by finite-order QFr-GLMs.The reconstructed image is represented as:

Design of QFr-GLMIs
The authors of [31] proposed a geometric invariance analysis method based on Krawtchouk moments.
We considered that the Krawtchouk moments can be calculated as a linear combination of their corresponding geometric moments.Therefore, the geometric-invariant transformations (rotation, scaling, and translation) of the Krawtchouk moments can also be expressed as the linear combination of their corresponding geometric-invariant moments.Inspired by the Krawtchouk moment invariants, this subsection proposes a new set of QFr-GLMIs.After analyzing the relationship between the quaternion fractional-order geometric moment invariants (QFr-GMIs) and the proposed QFr-GLMIs, we provide a realization scheme of the QFr-GLMIs; specifically, we construct the QFr-GLMIs as a linear combination of QFr-GLMs.Finally, we obtain the invariant transformations (rotation, scaling, and translation) of the proposed QFr-GLMIs.

Translation invariance of QFr-GMIs
Extending the traditional integer-order geometric moments to real-order (fractional-order) moments, the quaternion fractional-order geometric moments (QFr-GMs) of an N × N digital color image can be expressed as: Similarly to the traditional centralized geometric moments of integer-order, the centralized moments of QFr-GMs, can be defined as: where the centroid of a digital color image ) , ( c c y x is defined as: Above, we mentioned that a quaternion color image can be expressed as a linear combination of the single channels of an original color image.Let ) represent the zeroth-order and first-order moment of the B component of the image, respectively.In this case, Eq. ( 29) satisfies the translation invariance of the original color image.

Rotation, scaling, and translation invariance of QFr-GMIs
Referring to Eq. ( 17) in [31], the rotation, scaling, and translation invariants of QFr-GMIs can be expressed as follows: where The calculation steps of the rotational, scaling, and translation invariants of QFr-GMIs are detailed in [31].

Let
) , ( j i f rgb be the following weighted color-image representation: Eq. ( 32) can then be rewritten as follows: where 17)) and using Eq. ( 28), the above formula becomes Eq. ( 35) is derived in Appendix B.
The invariant transformations (rotation, scaling, and translation) of the QFr-GLMIs are obtained by substituting

Results and Discussion
In this section, the experimental results and analysis are used to validate the theoretical framework .The color-image reconstruction performance was evaluated by the mean square error (MSE) and peak signal-to-noise ratio (PSNR), which are respectively calculated as follows: ) / 255 lg( 10 Here, MSE denote the MSE values of the grayscale image corresponding to the independent red, green, and blue components of the color image, respectively, which are defined as ( In Eq. ( 39 , and (III) . The performances of the proposed QFr-GLMs were compared with those of QFr-ZMs and other state-of-the-art color image moments.The comparative results are shown in Figs. 3 and 4. The reconstruction performance of the low-order QFr-GLMs (

,  m n
) was poorer under parameter setting (III) than under parameters settings (I) and (II) (Fig. 3).Under parameter setting (III), the low-order QFr-GLMs were also outperformed by other color image moments (QGLMs, QFr-ZMs, and QZMs).Note that QGLMs are a special case of QFr-GLMs with . However, when the order of each color-image moment was sufficiently high (

,  m n
), the QFr-GLMs achieved the best image-reconstruction performance under parameter setting (III).The image reconstruction results of the QFr-GLMs clearly differed between the low-and high-order moments.In the low-order moments, the zero-value distributions of the QFr-GLMs polynomials were concentrated at the image origin under the parameter settings , so the sampling neglected the edges and details of the image.Conversely, in the high-order moments, the zero-value distributions of the polynomials approximated a uniform distribution, so the image reconstruction was optimal.To intuitively show the visual effect of image reconstruction, Table 1 presents the visualization results of the reconstruction experiments with different color-image moments.At higher orders, the proposed QFr-GLMs provided a better visual effect of the image reconstruction than the other color image moments.
To further verify the robustness of the proposed QFr-GLMs in noise resistance and non-conventional signal processing, the features of color images infected with salt and pepper noise or subjected to smooth filtering were extracted by the proposed QFr-GLMs and other color-image moments.New color images were reconstructed using the extracted features, and the performances of the image reconstructions were evaluated by the PSNR.Fig. 4 shows the color images subjected to salt and pepper noise (noise density = 2%) and smooth filtering (with a 5 × 5 filter window), and Fig. 5 compares the color images reconstructed from the different image moments.Regardless of the parameter settings, increasing the order of the image moment (especially the high-order moments) reduced the sensitivity of the proposed QFr-GLMs to salt and pepper noise and smoothing.Comparing the PSNR values of the different image moments, we find that the 28-order QFr-GLMs outperformed the QFr-ZMs by 8 dB.In summary, the proposed QFr-GLMs can properly describe color images under noise-free, noisy and smoothed conditions, and also exhibit high global feature extraction performance.Consequently, the proposed QFr-GLMs show promising applicability to color image analysis.

Experiments on local reconstruction of color images
In recent years, local-feature-extraction or ROI detection have presented new challenges for the existing orthogonal moments.The existing image moments, especially most of the orthogonal moments, extract only the global features, and cannot describe the local features.The detection of arbitrary ROIs in images is especially challenging.Among the existing orthogonal moments, only a few discrete orthogonal moments based on Cartesian coordinate space, such as the Krawtchouk [32] and Hahn [33] moments, can perceive the local features in an image.Thus far, the application of such discrete orthogonal moments has been limited to local-feature detection in binary images.Xiao et al. [19] proposed fractional-order shifting Legendre orthogonal moments, which extract the local features of a grayscale image by changing the parameter values of the fractional order.However, the local image is not well reconstructed (see Fig. 5 in [19]); especially, the details of the ROI are insufficiently protected in the local-image reconstruction.In addition, local-feature-extraction from color images has been little reported in the literature on image moments.In this subsection, we meet the challenge of applying the proposed QFr-GLMs to local-feature extraction from color images.The test images were three typical "block" color images selected from the COIL-100 database.The local features in the color images at different positions of the three "block" color images were reconstructed using the features extracted by the QFr-GLMs with different parameters.The experimental results are summarized in Table 2.This table shows that under different parameter settings, the proposed QFr-GLMs provided good image reconstructions in different regions of the original color image (the target areas of ROI extraction from the original color images are enclosed in the red-edged boxes).Under the parameter setting  2), the proposed QFr-GLMs well described the local features at different positions of the block color images, implying their effectiveness as a localfeature-extraction descriptor.
To further verify their local-feature extraction capability, the proposed QFr-GLMs were tested on a medical image (a computed tomography (CT) image of the human ankle, CT image seems to be a grayscale image, however, it is composed of R, G, and B three components, thus in this experiment, it is regarded as a color image).In this experiment, the QFr-GLMs were required to detect the ROI (the lesion area) in the human-ankle CT image.As shown in Fig. 6, the proposed QFr-GLMs properly detected the lesion in the CT image.

Optimal parameter selection
As presented in subsection 4.2, the proposed QFr-GLMs with determined translation parameters requires the proper selection of the fractional-order parameter  , because this parameter mainly affects the quality of the local-image feature extraction and the detailed descriptions of the reconstructed image.Therefore, optimizing the parameter  is the key requirement of image reconstruction and classification by the proposed QFr-GLMs.The optimal  will guarantee the quality of the image reconstruction and the accuracy of image classification.
To study the influence of the parameters x  and y  on the performance of the proposed QFr-GLMs, we selected four color images ("cat", "piggybank", "tomato", and "block") from the COIL-100 database.
Referring to the different image reconstructions, an approach for selecting the parameter optimization method is proposed in this subsection.To elucidate how image size affects the parameters To determine the optimal parameters x  and y  in combination, this subsection computes the performance of the proposed QFr-GLMs by the average statistical normalized image reconstruction error (ASNIRE), which is defined as follows: Here, the number of testing images L was 4, c f is an original color image, and   8).Therefore, when selecting the optimal parameter combination for the proposed QFr-GLMs, we suggest seeking within the range [1.0, 1.5].

Geometric-invariant recognition in color images
This subsection tests and analyzes the recognition of geometric-invariant transformations (rotation, scaling, and translation) by the proposed QFr-GLMs, and their robustness to noise and smoothing filter operations.This experiment was performed on two sets of public color-image databases: (128 × 128)sized color images selected from COIL-100 (Fig. 9), and (128 × 128)-sized butterfly color images selected from [5] (Fig. 10).To verify that the proposed QFr-GLMs recognize geometric invariants, the QFr-GLMs were employed with three parameter settings: (I)  (high-order moment).Fig. 12 shows the classification experiment results after smoothing.The proposed QFr-GLMs were strongly robust to rotation, scaling, and translation transformations, and achieved higher classification accuracies than the QZMs, QFr-ZMs, and QGLMs.

Computational times
This experiment determined the computational times of the proposed QFr-GLMs (for notational simplicity, we express the QFr-GLMs with the three groups of parameter settings as QFr-GLMs (I), QFr-GLMs (II) and QFr-GLMs (III)).The results are compared with those of the latest QFr-ZMs and other quaternion orthogonal moments (such as QZMs).The simulations were conducted on a Microsoft Window 7 operating system with a 2. The computational time of all orthogonal moments increased with order.However, as the polynomial of the moment in our approach is calculated by a recursive algorithm, the proposed QFr-GLMs color image moments in all parameter settings were computed faster than the QZMs and QFr-ZMs, and the computational time approached that of QGLM.In addition, because the QZMs and QFr-ZMs are computed in polar coordinates, the color images must be converted from Cartesian coordinates to polar coordinates, whereas the proposed image moments are directly constructed in the Cartesian coordinate system, which further reduces the computational time.

Conclusions
This paper proposed a new set of quaternion fractional-order generalized Laguerre moments (QFr-GLMs) based on GLPs and quaternion algebra.As color-image feature descriptors, the proposed QFr-GLMs can be used for color-image reconstruction and feature extraction, and the image moments are available for global and local color image representations in the field of image analysis.More importantly, based on the local image representation characteristics of the proposed QFr-GLMs, the application of the proposed moments in the field of digital watermarking [34][35][36] can effectively solve the problem of resisting large-scale cropping and smearing attacks, which is also one of our future work directions.After establishing the relationship between QFr-GLMs and Fr-GLMs, it was found that QFr-GLMs can be represented as linear combinations of Fr-GLMs.We also presented a new set of rotation, scaling, and translation invariants for object recognition applications.In comparison experiments with other state-of-the-art moments, i.e., the performance tests included global and localfeature extraction from color images, and geometric-invariant classification of color images.The proposed QFr-GLMs demonstrated higher color-image reconstruction capability and invariant recognition accuracy under noise-free, noisy, and smooth filtering conditions.Thus, the proposed QFr-GLMs are potentially useful for color-image description and digital watermarking [37][38][39][40].However, the only deficiency is that the perfect geometric invariance [41,42] cannot be achieved directly for invariant image recognition since the derivation of these QFr-GLMs invariants are not based on generalized Laguerre polynomials themselves.In the future, another focus of our work is to construct a new set of generalized Laguerre moment invariants, namely, deriving an explicit generalized Laguerre moment invariants approach, which can be directly applied to the field of image recognition.

List of figure Legend
three imaginary components of a pure quaternion.Therefore, an image in RGB color space can be expressed by the following quaternion:

1  and 2  be 1
zeroth-order and first-order moments of the R component of the original color image, respectively.Similarly, let zeroth-order and firstorder moments of the G component of the image, respectively, and let

4 . 1
developed in the previous sections.The performances of the proposed QFr-GLMs and QFr-GLMIs in image processing were evaluated in five sets of typical experiments.In the first group of experiments, the global reconstruction performance of the color images was evaluated under noise-free, noisy, and smoothing-filter conditions.The second group of experiments evaluated the proposed QFr-GLMs on local-image reconstruction, ROI-feature extraction, and the influence of different parameter conditions on image reconstruction.To improve the reconstruction and classification performance of the proposed QFr-GLMs on color images, the parameters were optimized through image reconstruction in the third group of experiments.The fourth group of experiments tested the image classification of the proposed QFr-GLMIs under geometric transformation, noisy, and smoothing-filter conditions.These experiments were mainly performed on different color-image datasets that are openly accessible on the Internet.In the last group, the computational time consumption of the proposed QFr-GLMs was compared with those of the latest QFr-ZMs and other orthogonal moments.All experimental simulations were completed on a PC terminal with the following hardware configuration: Intel (R) core (IM) i5, 2.5 GHz CPU, 8 GB memory, Windows 7 operating system.The simulation software was MATLAB 2013a.Experiments on global reconstruction of color images This subsection evaluates the global feature-extraction performance of the proposed QFr-GLMs on color images.The evaluation was divided into two steps: image-reconstruction evaluation of the QFr-GLMs and other approaches on original color images (i.e., noise-free and unfiltered images), and imagereconstruction evaluation of color images superposed with salt and pepper noise or pre-processed by a conventional smoothing filter.The QFr-GLMs and other image moments are then applied to image feature extraction, and are finally subjected to color-image reconstruction experiments.The test image in this experiment was the colored 'cat' image selected from the well-known Columbia Object Image Library (COIL-100).The test image was sized 128 128  the original two-dimensional N × N grayscale image and its reconstructed image, respectively.To assess the global reconstruction performance of the proposed QFr-GLMs, -GLMs extracted the upper part of the original color image.Meanwhile, the QFr-GLMs with the right-upper part of the original color image.We conclude that the translation parameters of the proposed QFr-GLMs determine the position information of the local features in the original color images, whereas the fractional-order parameters mainly affect the quality of the local-image feature extraction and the details of the reconstructed color image.Specifically, the proposed QFr-GLMs with smaller and larger values of the translation parameter along the x-and y-axes, respectively, mainly extracted the bottom part of the color image; conversely, the proposed QFr-GLMs with smaller and larger  values along the x-and y-axes, respectively, extracted the upper part of the color image.If the parameter values of different fractional orders along the x-and y-axes are combined, the QFr-GLMs obtain the local information at different positions in the original color image.As shown in the local-image-reconstruction results (Table

32 
. The results are shown in Fig.7.
color image.The SNIRE is the statistical normalized image reconstruction error function proposed in[19], defined as the combined results of their optimal values.

Fig. 8
Fig. 8 shows the reference selection range of the optimal parameter values

.
The images sets were categorized by a KNN classifier.The amplitudes of the color-image moments were arranged into a feature vector for classification as follows:

NExperiment 1 :Experiment 2 :
of the different image moments were determined by a measure called the correct classification percents (CCPs), expressed as represent the number of correctly classified objects and the total number of all testing objects, respectively.The color image dataset for this experiment was extracted from COIL-100.First, 100 color images from the COIL-100 dataset were rotated by 0° and 180°, obtaining 200 images (100 × 2) as the training set.Each image in the training set was then translated by , and a scale factor  was defined for the scaling operation.Rotating 200 images by i 7200 (36 ×200) color images for testing.Finally, salt and pepper noise (with noise density ranging from 0 to 25% in 5% increments) was added to each image in the existing test set, forming a new noisy test set.In this experiment, the vectors nm V of the different image moments were obtained at 12  k (low-order moment) and 28  k (high-order moment).Fig. 11 compares the correct classification rates (CCPs) of the proposed QFr-GLMs and other orthogonal moments (QZMs, QFr-ZMs, and QGLMs).As seen in the figure, the proposed QFr-GLMs outperformed the other moments in both cases ( The dataset for this experiment was extracted from the Butterfly color image database.As described in Experiment 1, the 20 color images in the extracted dataset were rotated by 0°, 90° and 270°, obtaining 200 images (20 × 3) as the training set.Next, following the steps described in experiment 1, we obtained 2160 (36 ×60) color images as the test set.Finally, each image in the test set was passed through a smoothing filter with different window sizes (3, 5, 7, and 9), obtaining 2160 new color images as the filtered test set.Again, the vectors nm V of different image moments were obtained at 12  k (low-order moment) and 28  k

Figure 1 Figure 8 Figure 9
Figure 1 title: Block diagram of QFr-GLM and single-channel Fr-GLM calculations Figure 2 title: Distribution curves of the NFr-GLPs under different parameter settings Figure 3 title: PSNR versus moment-order curves of different color-image moments Figure 4 title: The original and processed color images: (a) the original color image, (b) the image infected with 2% salt and pepper noise, and (c) the color image after smoothing through a filter with a 5 5 window Figure 5 title: PSNR versus moment-order curves of different color-image moments after various types