#### 1. Introduction

With the development of fast algorithms, the discrete cosine transform (DCT) and the Hadamard transform have been widely used in signal processing and image processing, most prominently in the compressed representations of images . The DCT is a real-to-real and integer-to-real transform that has been considered as an alternative to the Fourier transform for spectral analysis. The DCT has the additional advantage that its forward and inverse forms are identical, thus, simplifying implementation. The Hadamard transform is highly practical for representing signals, images, and mobile communications, mainly because the elements of the Hadamard matrix are either or −1. Thus, the computation of the transform of a signal consists of additions and subtractions of the signal samples. The Hadamard transform and many of its variations, such as the sequency-ordered complex Hadamard transform (SCHT) [4, 5] and the Jacket transform , have been proposed, and their applications to image processing and communications have been reported .

In recent decades, various factional series and transforms have been introduced and have found application in many fields in engineering . In , Pei and Yeh developed a recursive method to obtain Hadamard eigenvectors, and normalised eigenvectors were used to define the discrete fractional Hadamard transform. As an alternative to the recursive method, Tseng  investigated the eigenstructure of the Hadamard transform for defining its fractional transform and found that the Kronecker product is more suitable for calculating the eigenvalues and the eigenvectors of the Hadamard kernel function.

Second, we represent a signal from a limited number of terms in the partial sum of the COSHAD/FrCOSHAD series with high accuracy. The number of terms in the expansion is known to be increased; the series approximation converges in the region where the function is smooth. However, the overshoot near the discontinuity is never reduced, no matter how many terms are included in the partial sum. Furthermore, the numerical requirements grow dramatically with the increase in the number of harmonics. The expectation, in practice, is that the signal could be reconstructed from much less data at a high level of quality. To solve this problem, this study follows the main description of the inverse polynomial reconstruction method (IPRM) for the continuous Fourier series expansion  and derives a mathematics framework of the IPRM for the discrete COSHAD/FrCOSHAD series expansion to accurately recover a piecewise smooth signal. The spectral convergence of the proposed IPRM strategy in the discrete COSHAD expansion is discussed and proven, both theoretically and numerically.

The remainder of this paper is organised as follows. Section 2 provides a brief review of the discrete COSHAD transform. Section 3 introduces the discrete FrCOSHAD transform and its main properties. The quaternion discrete COSHAD/FrCOSHAD transforms based on quaternion algebra are also defined in this section. The derivation of the COSHAD-based 1D IPRM and the convergence properties are presented in Section 4. Section 5 reports the simulation results for the standard testing images and provides the numerical verification of the convergence theorems. Finally, in Section 6, we summarise the results and conclude the paper.

#### 2. Review of the COSHAD Transform

This section is devoted to a review of the results about the discrete COSHAD transform defined in , which may not be widely known. We start by defining the 1D forward COSHAD transform of order in the following form: where , is the length of signal , is the data vector, and is the transform vector. The th order COSHAD kernel function (transform matrix) is defined as and the symbol represents the Kronecker product. [DCT()] and [HAD()] are DCT (of size ) and Hadamard kernel functions (of size ), respectively. In this paper, we rely heavily on Corollary 13.8 in  as follows.

Corollary 1. If is orthogonal and is orthogonal, then is orthogonal.

According to Corollary 1, the defined [] is an orthogonal kernel function that belongs to a family of discrete orthogonal matrices ranging from the Hadamard kernel function () to the DCT kernel function (). Thus, the inverse COSHAD transform is defined as where denotes the matrix transpose. The 2D forward COSHAD transform of an matrix is defined by The corresponding inverse COSHAD transform can be calculated as

#### 3. The Discrete FrCOSHAD Transform and Its Properties

##### 3.1. The Discrete FrCOSHAD Transform

According to the Kronecker product properties  and the definition of the Hadamard transform, one can rewrite the Hadamard kernel function as follows: If the lowest-order Hadamard kernel function [] is expressed as Substituting (6) into (2), the COSHAD kernel function can be rewritten as If the eigendecompositions of the matrices [DCT()] and [HAD(1)] are respectively, and are the matrices composed of eigenvectors. We also need to use the following properties of the Kronecker, which are 13.12 in .

Property 1. Let have eigenvalues , , and let have eigenvalues , . Then, the eigenvalues of are , .
Therefore, the eigendecomposition of [] can be written in the following form: where , and . After the eigenvalues of the [] are determined, we define the kernel function of the discrete FrCOSHAD transform by taking the fractional powers of the eigenvalue.
Consider Thus, the 1D forward FrCOSHAD transform with parameter is Equation (12) indicates that the FrCOSHAD transform reduces to the standard COSHAD transform if the fractional powers . The corresponding inverse FrCOSHAD transform is written as Similarly, the 2D forward FrCOSHAD transform with order (, ) is given by The corresponding inverse FrCOSHAD transform can be generated as

##### 3.2. Properties of the Discrete FrCOSHAD Kernel Function

This subsection presents some properties of the discrete FrCOSHAD kernel function.

Proof. From (11), we have The proof is complete.

Property 3. Unitarity:

Proof. Using (11), we have The proof is complete.

##### 3.3. Quaternion Discrete Orthogonal Transform

Traditional methods process colour images by processing each colour channel separately and then summing the three individual outputs to derive the final result. Ell and Sangwine  were the first to use quaternions to encode colour images as vector fields. Motivated by their work, this paper defines the so-called Q-COSHAD and Q-FrCOSHAD transforms. We consider here only RGB colour images. The generalisation to other types of colour images is not difficult. Let f be an RGB image; the three channels of the image can be represented using the three components of a pure quaternion as follows: where , , and represent the red, green, and blue components of the colour image, respectively. This representation effectively equates the RGB colour cube to the right-handed coordinate frame imposed by the imaginary part of the quaternion space and treats the colour image as a whole rather than as separate components. As an example, we now introduce the definition of the right-side form of the Q-FrCOSHAD transform of a function with two variables. Similar methods can be used to obtain the Q-COSHAD transform matrices.

Definition 2. Given a quaternion root of −1, , for an colour image , the right-side form of the forward Q-FrCOSHAD transform of the image is defined using the matrix form as where and . This choice corresponds to the single coloured line in RGB space, on which all three components are equal. Letting , , and be the discrete FrCOSHAD transforms from (14) for the red, green, and blue components, respectively, of the colour image. Equation (22) can be rewritten as where
This paper defines the left-side forward Q-FrCOSHAD transform similarly by interchanging the positions of the colour image and the imaginary part .

Definition 3. Given a quaternion root of −1, , for an colour image , the left-side form of the forward Q-FrCOSHAD transform of the image is defined as Corresponding to the Q-FrCOSHAD transform, there are two forms of the inverse Q-FrCOSHAD transform. The right-side inverse transform satisfies the following equations: where where , , and are the three colour components of the reconstructed image, and , , , and are inverse FrCOSHAD transforms from (15) with respect to , , , and , respectively. The left-side inverse transform is similarly defined as follows:

#### 4. COSHAD-Based IPRM Is Spectrally Convergent

Spectral approximations have been widely implemented in the approximation of signals [20, 21]. The kernel functions very commonly used in spectral partial sum approximations are Fourier, Cosine, Legendre, Chebyshev, and Gegenbauer, and so forth. Sometimes, nonclassical kernel functions have been proven to be very useful for specific purposes. This important reason leads us directly to discuss the possibility of representing a signal from a limited number of expansion coefficients of COSHAD/FrCOSHAD with high accuracy.

The overshoot of an approximation to a function near the discontinuity or the domain boundaries will occur for most approximations based on standard orthogonal expansions . Many effective algorithms have been developed to overcome the overshoot and to improve the spectral accuracy . For obtaining IPRM in the discrete COSHAD domain, let us first introduce the explicit expression of the COSHAD kernel function as follows: with In the above, the symbol represents the th row and th column elements of the Hadamard kernel function, “mod” represents a modulo operation, and “[·]” represents the ceiling function. Thus, applying (29), the COSHAD expansion of an unknown function using standard inner product can be obtained; that is where Using (32), the truncated series expansion of a signal using only the first terms of the COSHAD coefficients is given by where

Assume that the original function is analytic and this original function can be expanded in Gegenbauer orthogonal polynomials as follows: where are the Gegenbauer coefficients defined on the Gegenbauer transform domain with is the Gegenbauer polynomial of order with parameter   , and is mapped as to satisfy the domain of the Gegenbauer orthogonal polynomials. The purpose of the COSHAD-based IPRM method is to find a finite approximation for the unknown function as So the error defined in the following: is orthogonal to the COSHAD transform domain. That is, where By substituting in (32) with from (39), we have By defining the matrix as with matrix elements given by Equation (43) can be rewritten as Equation (46) can also be written in matrix form as where the vectors and are respectively. Under the constraint we can obtain the following Gegenbauer coefficients through the pseudo-inversion of by

To prove the convergence of the proposed COSHAD-based IPRM, we define the truncation error and the regularisation error as follows

Lemma 4. If satisfies , then

Proof. According to (40)-(41), one can obviously obtain Thus, Combining (52) and (56), one has Thus, The lemma is, thus, proven.

Lemma 5. Let be an analytic function. Then, there exists a constant that is independent of , so that the regularisation error function defined in (52) satisfies the following inequality:

Because the IPRM is independent on the basis function , we can choose the Chebyshev polynomial in (37) with . Thus, the proof of Lemma 5 can easily be obtained by the same procedure as in .

Theorem 6 (spectral convergence). 1D IPRM for the discrete COSHAD expansion is spectrally convergent.

Proof. By Lemmas 4 and 5, if , Let denote the COSHAD approximation of . Using , Therefore, with (31), we have with Due to , for any number , and , for , therefore, Theorem 6 is, thus, proven. Similar methods could be adapted to prove the spectral convergence of 1D FrCOSHAD-based IPRM without difficulty.

#### 5. Experimental Results

##### 5.1. Energy Compaction

The purpose of this subsection is to evaluate the coding efficiency of the proposed FrCOSHAD transforms and to compare them with the other transforms. The coding gain  of the 1D and 2D cases will be investigated in this subsection. As mentioned above, understanding whether parameters and have an important influence over the coding gain. We first examine the fractional order with two examples where and 3 shown in Figures 3(a) and 3(b), respectively. This figure shows that the higher fractional order would generate a greater coding gain.

In the following experiment, we discuss the influence of parameter on the energy compaction. According to the constraints imposed on the parameters , , , indicated by (1) and (2), we have systematically chosen the parameters as in this experiment where the value of fractional order is fixed at 0.95, and only parameter is adjustable. The plots of coding gain are depicted in Figure 3(c). From this figure, the fifth choice () gives the highest coding gain among all of the test cases. In contrast to fractional order , parameter does not have as obvious an impact on the energy compaction. Only a limited decrease of the coding gain can be observed to respond to the decrease of parameter .

Based on the above test results, we found that the energy compactions of the FrCOSHAD kernel matrix are significantly affected by order and are moderately affected by . This result can be explained because the smaller the value offractional order , the less the effect of the FrCOSHAD transform. The FrCOSHAD kernel function has not fully played a role in the process of transformation. So far as parameter is concerned, the choice of a relatively large corresponding to the case where the emphasis of the kernel function is the DCT properties rather than the Hadamard properties gives a high coding gain. Comparing the DCT and the Hadamard transforms, the DCT transform has a strong energy of compaction property, a well-known fact. Therefore, this is the main reason why we use such a choice of parameters and, as far as possible, big parameter for compressing the representations of images.

For comparison purposes, we calculated the coding gain with for the several transforms while correlation coefficient is varied from 0.6 to 0.95. Figure 4(a) show that the energy compaction capabilities from high to low in order is DCT, COSHAD, FrCOSHAD, DHT, and the Hadamard transforms. The fractional order of the FrCOSHAD is in this experiment.

A comparison of the results in terms of the 2D coding gains between the traditional 2D transforms (DCT, DHT, and Hadamard) and the proposed transforms as increases is shown in Figures 4(b) and 4(c). The block sizes chosen for the experiment were 16 × 16. Regardless of whether the covariance is separable, the DCT transform consistently presents high coding gains in comparison with all of the other transforms. Both COSHAD and FrCOSHAD transforms had a higher coding efficiency than DHT and Hadamard transforms. The Hadamard transform yields poorer performance than all of the other transforms.

##### 5.2. Applications of the Proposed Transforms to Colour Image Compression

In this subsection, we will compare the reconstructed results from the proposed transforms against Q-DCT , Q-DHT, and Q-Hadamard. The purpose is to confirm the theoretical analysis regarding the energy compaction. The experiments were conducted on two colour test images (size: 512 × 512) shown in Figures 1(b) and 1(c). We performed all simulations for these images within Matlab 7.11.0 (2010b) under Window environment on an Intel(R) Core(TM) i5 CPU 2.67 GHz processor with 3 GB RAM.

Test images are first divided into equal-sized blocks, and each block is then transformed by using the kernel functions. Consequently, we arrange the absolute value of all of the transform coefficients in downward order and take the most significant part according to compression ratio (Cr) to get a recovered image. The universal image quality index (UQI)  values of the reconstructed images with respect to the original images “Lena” and “Pepper” are listed in Tables 2 and 3, where denotes the block size used in experiment. According to the theoretical analysis of coding gain illustrated above, the fractional orders in the 2D FrCOSHAD transform were chosen as , . Tables 2 and 3 show that the qualities of the Q-DCT transform are overall higher than the qualities of the other transforms when Cr is 1 : 2 or 1 : 8. The reconstructed images using the proposed transform with perform not only better than the Q-COSHAD/FrCOSHAD with but also better than the Q-DHT and the Q-Hadamard transforms. We think this is because the effect of the DCT on the FrCOSHAD method with is more obvious because the DCT has nearly optimal properties compared with the statistically optimal K-L transform. The Q-Hadamard transform gives the worst reconstructions for all of the test images. Figure 5 shows that the images we cut out from the reconstructed images have no manipulation of any sort, only for viewing more clearly. In spite of having the same , Cr, and , the reconstructed images are quite different: the Q-DCT, the Q-COSHAD, and the Q-FrCOSHAD transforms produced good results.

The computation times required for reconstruction of colour image “Pepper” are listed in Figure 6. Because of the simplicity of the block-based strategy, these algorithms can produce fast reconstructions along with relatively good visual qualities. In this test, is chosen for the Q-COSHAD/Q-FrCOSHAD transforms. However, the FrCOSHAD has a moderately computational burden for large block size due to the complexity of the algorithm. Similar results will be obtained with all other choices of the parameters , , and . We omit discussing them, considering the space limit.

Generally speaking, the image representation quality can be severely affected by image noise. For the sake of evaluating the robustness of the proposed transforms against the negative effects of different types of noise, two experiments were conducted using noise-contaminated images. The first type of noise is the salt and pepper noise (5%), while the second type of noise is the Gaussian white noise with zero mean and variance 0.01. We conducted a few comparisons between the proposed approaches and the other schemes. The visual quality of the reconstructed image “Pepper” is measured using PSNR demonstrated in Table 4. Compared with the Q-DHT and the Q-Hadamard transforms, both the Q-COSHAD and the Q-FrCOSHAD produce a slightly higher PSNR under the same conditions. However, the Q-DCT produces the highest PSNR of the transforms in all cases with the presence of either Gaussian noise or salt and pepper noise.

In the following experiment, we evaluate the influence of and over the proposed Q-FrCOSHAD transform for the case of noise. Gaussian white noises with zero mean and variance 0.01 are taken into account in this test. Figure 7 shows the corresponding PSNR values when and vary from 0.5 up to 1. This figure shows that with the increase of fractional orders and , the values of PSNR tend to rise gradually as expected. This example also proves once again that the effectiveness of the FrCOSHAD is severely affected by the fractional order: relatively large and leads to much better reconstructed results.

##### 5.3. Local Representation by the IPRM

In this subsection, we attach three numerical examples that verify the accuracy of the proposed IPRM in the multi-intervals. Without loss of generality, we first consider a window function with length on the three subintervals depicted in Figure 8

In fact, the generalisation to more than three domains (or one axis) is also straightforward. If the length of signal is not a power of two, zero padding is needed to meet the constraint of the COSHAD kernel matrix (see (1) and (2)). In Figure 8, we show the original function in the solid black line, the approximation through the COSHAD/FrCOSHAD partial sum with in the square dashed line. The COSHAD/FrCOSHAD partial sum approximation is obviously not accurate.

Next, we implement the proposed FrCOSHAD-based IPRM method in the following way. The original function is first split into three pieces and solved for the smoother parts of the function in each small interval, and then the approximations are glued together to recover the original function in the full interval. Parameter of the Gegenbauer polynomials was fixed as 0.5 in all numerical examples. We get explicitly the approximate Gegenbauer coefficient , defined in (51) as follows.

For the left interval: , , , and : For the middle interval: , , , and : For the right interval: , , , and : Once the approximate Gegenbauer coefficient is obtained, one can compute the approximation to by directly summing by using (39). If we let , , and note the finite approximation for function in the three subintervals, respectively, we can obtain the following approximation: Similarly, using all of these parameters and only setting fraction order , the approximation to using the IPRM in the discrete COSHAD domain can be computed as follows:

Using the present methods, the solutions are obtained for three subintervals shown in Figure 8. The present strategies clearly provide an exact numerical solution rather than that obtained in the approximation through the COSHAD/FrCOSHAD partial sum methods. To understand the convergence behaviour of the current methods whether independent of the parameters and , the same calculation is conducted using other values. Here, the error, in the maximum norm, between the exact and the reconstructed solution as defined by is used to demonstrate the numerical efficiency and convergence. Table 5 illustrates the logarithmic pointwise errors using present methods with respect to parameters and . There is better convergence behaviour for all of the parameter selections.

The second example considers a more complex piecewise smooth signal with length given by and there are two subintervals of equal length. The solid black line in Figure 9 shows the original piecewise function. Function has the point of discontinuity at 128. The partial sums of the COSHAD/FrCOSHAD series with and can be found in Figure 9. The values of parameters and are chosen as for the COSHAD and and for the FrCOSHAD. Figure 9 shows that the approximations through the partial sum are inaccurate regardless of the COSHAD or the FrCOSHAD. However, the solution obtained by the proposed methods gives excellent results even for the small and , which corresponds to their respective interval. In this experiment, we let the reconstruction coefficient vectors and denote in the left interval [1, 128] and in the right interval [129, 255], respectively. Exact reconstruction coefficient vectors can be obtained in terms of (51). The results are listed in Table 6, and the COSHAD and the FrCOSHAD coefficient matrix and computed using (47) is illustrated in Table 7.

The study found that the FrCOSHAD-based IPRM is not very sensitive to the fractional order . Rather arbitrarily choosing and for the left and the right interval, the FrCOSHAD-based IPRM can also achieve accurate reconstructions of the discontinuous function from their own spectra with and (see Figure 9(c)). In contrast, the COSHAD/FrCOSHAD partial sum with and gives poor approximations throughout the domain. The parameters used for the FrCOSHAD are and , in this case.

In the last example, we consider another function of the same length, which is piecewise smooth, defined by

This function too has a jump discontinuity at value 128. Plots of this function and the COSHAD/FrCOSHAD series approximation to are shown in Figure 10. Again, this figure shows that the finite COSHAD/FrCOSHAD series expansion is not good enough for and . However, if the COSHAD/FrCOSHAD-based IPRM reconstructions are conducted by setting the same parameters used by the above example for the function , the exact reconstruction can be obtained, as expected. Similarly, the reconstruction coefficient vectors and series coefficient matrix are calculated and listed in Tables 8 and 9, respectively. These experimental results again demonstrate how the proposed approaches actually represent the piecewise smooth function .

We now return to the issue of the error analysis in the reconstruction process. Figure 11 illustrates the errors in the maximum norm, with a logarithmic scale. As might be expected, good convergence behaviour of the proposed methods can be observed because the maximum error is near zero. Furthermore, the convergence of this method is extremely rapid and requires only a few terms. However, beyond that, this figure again indicates that changing coefficients and has no obvious effect on the exponential convergence properties of the IPRM in the FrCOSHAD expansion.

#### 6. Conclusions

In this paper, the discrete FrCOSHAD transform was defined and developed using the eigendecomposition method. By means of adjusting the parameter , the developed FrCOSHAD transforms offer the advantages of both the DCT and the Hadamard by combining the optimal performance of the former with the ease of implementation of the latter. Several mathematical properties were discussed. Angle additivity and energy compaction capabilities were verified using the 1D and the 2D signals. The paper also proposed the quaternion COSHAD/FrCOSHAD transforms for colour images using the quaternion algebra. The main advantage of the quaternion strategy is that it provides the possibility of processing colour images by considering the three components of the colour RGB image as a whole, not separately. Block-based image reconstruction results demonstrated that the proposed transforms outperform some classical methods in qualitative and quantitative measurements.

Besides, a mathematical strategy for the 1D IPRM in the COSHAD/FrCOSHAD expansion was developed for the representation of the piecewise smooth functions from the information contained in a finite set of the COSHAD or the FrCOSHAD spectra. In this paper, we restricted ourselves to use the COSHAD/FrCOSHAD series expansion for description of the 1D signals because it is not difficult to extend to higher dimensions. Moreover, the paper provided a theoretical proof that the 1D COSHAD-based IPRM not only is spectrally convergent but also has independent parameters of its own.

Further investigations may include studying why, as fractional order increases, the trend of the PSNR values is almost a straight line under Gaussian white noise conditions (see Figure 7). Is it coincidence, or is there a fundamental principle involved? Another question that should be addressed in future work is how to obtain other fractional orthogonal transforms using the Kronecker product methods.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China under Grant nos. 61371150, 61271357, and 61370174.