Image analysis using modified exponent-Fourier moments
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Abstract
Classic exponent-Fourier moments (EFMs) have been popularly used for image reconstruction and invariant classification. However, EFMs lack natively the translation and scaling-invariant; in addition, they exhibit two types of drawbacks, namely numerical instability and reconstruction error, which in turn influence their reconstruction capability and image classification accuracy. This study considers the challenge of defining modified EFMs (MEFMs), which are based on modified exponent polynomials. In our methods, the basis function of traditional EFMs is appropriately modified, and these modified basis functions are used to replace the original ones. The basis function of the proposed moments is composed of piecewise modified exponent polynomials modulated by a variable parameter exponential envelope. Various types of optimal-order moments can be established by slightly adjusting the bandwidth of the modified basis functions. Finally, we extend the rotation-invariant feature of previous works and propose a new method of scaling and rotation-invariant image recognition using the proposed moments in a log-polar coordinate domain. The translation invariance can then be achieved by an image projection operation, which is substituted for the traditional approach based on the calculation of image geometric moments. The experimental results demonstrate that the MEFMs perform better than traditional EFMs and other classic orthogonal moments including the latest image moments in terms of the image reconstruction capability and the invariant recognition accuracy of smoothing filters, in both noise-free and noisy conditions.
Keywords
Orthogonal moments Modified exponent-Fourier moments Image classification Image reconstruction Rotation-invariantAbbreviations
- BFMs
Bessel–Fourier moments
- CCPs
Correct classification percentages
- EFMs
Exponent-Fourier moments
- Fr-LMs
Fractional order Legendre moments
- Fr-ZMs
Fractional order Zernike moments
- GHMs
Gaussian–Hermite moments
- LMs
Legendre moments
- MEFMs
Modified exponent-Fourier moments
- NaN
Not a number
- OFMs
Orthogonal Fourier–Mellin moments
- PCT
Polar cosine transform
- PHT
Polar harmonic transform
- PST
Polar sine transform
- RST
Rotation, scaling, and translation
- SNIRE
Statistical-normalization image-reconstruction error
- SOM
Semi-orthogonal moments
- ZMs
Zernike moments
1 Introduction
As mentioned earlier, the essence of image moments is the set of image transformations based on basis functions. The advantages and disadvantages of its basis functions will directly affect the performance of the constructed image moments. In light of whether the basis set satisfies orthogonal conditions, the image moments can be divided into orthogonal and non-orthogonal moments (similarly known as orthogonal and non-orthogonal transformations, respectively, for example, discrete cosine transform [17], Fourier transform [18], Haar-wavelet transform [19], and Walsh transform [20] belong to orthogonal transformations). Non-orthogonal moments like geometric moments [21], complex moments [22], and rotation moments [23] have made certain achievements in the field of moment applications. The basis functions of non-orthogonal moments are relatively simple with an image reconstruction that is difficult to realize. In addition, the non-orthogonal moments generally have information redundancy that is sensitive to noise. The orthogonal moments can overcome the disadvantages of the abovementioned non-orthogonal moments, thereby becoming a main focus area in the field of image moments in the recent years.
Orthogonal moments can be defined in different coordinate spaces. The basis functions of orthogonal moments defined in polar coordinates are composed of radial polynomials and Fourier complex exponential factors with angular variables (regarded as amplitude and phase coefficients as well); thus, they are called radial orthogonal moments. The radial orthogonal moments in Fig. 1 mainly include Zernike moments (ZMs) [13], pseudo-Zernike moments (PZMs) [14], orthogonal Fourier–Mellin moments (OFMMs) [6], Jacobi–Fourier moments (JFMs) [24], Tchebichef–Fourier moments (TFMs) [25], radial harmonic-Fourier moments (RHFMs) [26], Bessel–Fourier moments (BFMs) [18, 27], exponent-Fourier moments (EFMs) [7], and radial shifted Legendre moments (RSLMs) [28]. These radial orthogonal moments normally have the basic ability of image reconstruction. Moreover, their significant characteristic is that the radial polynomials satisfy orthogonal condition in the unit circle and natively possess a rotation-invariant feature. Thus, radial orthogonal moments have become the preferred descriptor for geometric invariant image recognition, especially for rotation-invariant recognition. Basis functions are regular polynomials defined in the Cartesian coordinates, which can be further divided into continuous orthogonal moments and discrete orthogonal moments, such as Legendre moments (LMs) [29] and Gaussian–Hermite moments (GHMs) [2] that belong to continuous orthogonal moments and Tchebichef moments (TMs) [30], Krawtchouk moments (KMs) [31], Hahn moments (HMs) [32], and Racha moments (RMs) [33] that belong to discrete orthogonal moments. Discrete orthogonal moments do not involve any numerical approximation operations; hence, their basis functions can accurately satisfy an orthogonal condition. Consequently, the image reconstruction performance is better than that of traditional continuous orthogonal moments. In addition, we can construct different moments in other spaces like Radon transform invariant moments and histogram invariant moments in the Radon transform space and histogram space, respectively.
Shortcomings still exist in the abovementioned traditional orthogonal moments. On the one hand, the order of the existing orthogonal moments can only be taken as an integer value, which makes the development of orthogonal moments encounter bottlenecks caused by this constraint. To solve this problem, Xiao et al. [34] and Yang et al. [35] proposed fractional orthogonal moments. The integer-order can be extended to a real-order (also known as fractional order) using their proposed models. Further experimental results showed that the fractional order orthogonal moments were better than the traditional orthogonal moments based on the integer order in image reconstruction, noise robustness, and image recognition. Chen et al. [36, 37] recently extended the ZMs and PZMs to a quaternion and a fractional framework for color image feature extraction. The application of image moments has also been further improved. On the other hand, for image sets with larger distinctions, the classification effect is preferable using the lower-order moments constructed using the basis functions of traditional orthogonal moments. However, for the classification effect of the image sets with smaller discrimination, numerical instability will occur when higher-order moments are adopted. The reason is that the basis functions of traditional orthogonal moments are fixed either in lower- or higher-order moments, which can result in poor classification results in pattern recognition. Wang et al. [38] proposed a circularly semi-orthogonal moment that can maintain a good numerical stability in higher-order moments and can obtain a better visual effect in image reconstruction. This method only performs a simple and fixed modulation on the orthogonal basis functions, and the basis functions of different-order moments are still fixed; hence, the method lacks generality.
Classic orthogonal moments (e.g., EFMs) have the defects of numerical instability and poor accuracy of image recognition in some image classifications, especially in texture image recognition. A modified exponent-Fourier moment (MEFM) is proposed herein based on the concept proposed in [34, 38]. We mainly make attempts in view of three aspects. First, we take on the challenge of studying the performance of semi-orthogonal basis functions at the intersections between the orthogonal and non-orthogonal moments for image reconstruction and pattern recognition. A general semi-orthogonal moment model suitable for different orders can also be established. Second, a new method of the theoretical analysis model of the image moments in the frequency domain is proposed, namely time–frequency correspondence analysis. Finally, a simple and useful algorithm for rotation, scaling, and translation (RST) of invariant image recognition using the proposed moments is introduced herein.
The remainder of this paper is organized as follows: Section 2 provides some preliminaries about the classic exponent-Fourier moments for the 2D images; Section 3 introduces the MEFMs in the polar coordinates and discusses some properties of the MEFMs; Section 4 describes the experiments on the computational complexities of the image moments, image reconstruction, optimal parameter selection, and RST invariant image recognition under both noisy and noise-free, smoothing filter conditions; and Section 5 presents the conclusions.
2 Preliminaries
This section briefly reviews the definition of the classic orthogonal exponent-Fourier moments (EFMs) [39] for an image along with some EFM properties.
2.1 Exponent-Fourier moments
2.2 Properties of EFMs and other radial orthogonal moments
For the existing radial orthogonal moments, the number of zeros of the orthogonal polynomials plays a significant role in describing the high-spatial-frequency components of an image. The real and imaginary parts of the radial polynomial of EFMs have 2n and 2n+1 zeros in the interval 0 ≤ r ≤ 1, respectively [39]. Meanwhile, the Bessel polynomials and the orthogonal Fourier–Mellion polynomials have n+2 and n zeros in the interval 0 ≤ r ≤ 1, respectively [6, 40]. Zernike polynomials only have (n − m)/2 zeros in the interval 0 ≤ r ≤ 1. Therefore, the degree n of EFMs required to represent an image is much lower than that in BFMs, OFMMs, and ZMs, thereby causing the EFMs to have a stronger capability in describing an image compared to the other orthogonal moments (e.g., BFMs, OFMMs, and ZMs) in the polar coordinates. Additionally, classic EFMs and other radial orthogonal moments have the property of rotation-invariance similar to geometric invariant recognition. The abovementioned properties show that the exponent-Fourier moments are potentially useful as feature descriptors for image analysis.
3 Methods
3.1 Analysis of the numerical instability involved in classic EFMs
3.2 Definition of MEFMs
where 2π is the normalization coefficient and δ_{nm} or δ_{pq} is the Kronecker delta function. Thus, the MEFMs can also be called semi-orthogonal EFMs.
3.3 Calculation of MEFMs
3.4 Computation complexity and stability analysis of MEFMs
3.5 Time–frequency analysis of MEFMs
4 Results and discussion
In this section, the experimental results are used to validate the theoretical framework developed in the previous sections. This section includes four subsections. In the first subsection, we discuss the computational complexities of MEFMs as compared to those of BFMs, ZMs, OFMs, PST, and PCT. In the second subsection, the question of how well an image can be represented using MEFMs is addressed, and the image reconstruction capability of MEFMs is compared with those of BFMs, ZMs, OFMs, SOMs, PST, and PCT. In the third subsection, the question of optimal parameter selection for image reconstruction and recognition is discussed. A new method for the RST invariant image recognition using the proposed moments and the experimental study on the RST recognition accuracy of MEFMs is provided in the last subsection.
4.1 Computational complexities
Basis functions computation time
Number | Image moments | Computation time (ms) |
---|---|---|
1 | ZMs | 113.9 |
2 | OFMs | 144.8 |
3 | BFMs | 158.3 |
4 | EFMs | 37.2 |
5 | PST | 30.6 |
6 | PCT | 42.9 |
7 | MEFMs | 40.3 |
4.2 Image reconstruction
Where f(x, y) is the original image and \( \overline{f}\left(x,y\right) \) is the reconstructed image.
Experiment 1
Experiment 2
Uppercase English letter “E” are reconstructed in parameters (α = 0 and α = 2) under different order moments
Order N | SNIRE | |
---|---|---|
MEFMs(α = 0) | MEFMs (α = 2) | |
5 | 0.5419 | 0.5197 |
7 | 0.4752 | 0.4062 |
9 | 0.4247 | 0.3860 |
11 | 0.3682 | 0.3168 |
13 | 0.3380 | 0.3368 |
15 | 0.3330 | 0.3647 |
17 | 0.3609 | 0.4173 |
19 | 0.3478 | 0.3847 |
21 | 0.3488 | 0.3924 |
25 | 0.2900 | 0.4871 |
30 | 0.2856 | 0.4849 |
35 | 0.2808 | 0.5094 |
40 | 0.1760 | 0.5080 |
45 | 0.1076 | 0.5271 |
50 | 0.0638 | 0.5160 |
55 | 0.0374 | 0.5276 |
60 | 0.0311 | 0.5269 |
65 | 0.0244 | 0.5251 |
Experiment 3
According to the characteristic analysis of the MEFMs’ radial function in frequency domain, we propose a method of image projection transformation for an original image using piecewise function (or polynomial), when an image is reconstructed at lower-moments and higher-moments, respectively (see Eq. (5) in Section 3.2). In order to verify the validity of the piecewise function in Eq. (5), the proposed image moments (MEFMs) are compared with the ZMs, OFMMs, BFMs, and EFMs in this study, and simulation experiments are performed by the reconstruction of the binary image of uppercase English letter “E.” From the experimental results of Fig. 11, it is known that the performance of the proposed image moments constructed by the basis functions, which consists of piecewise polynomials in Section 3.2 is superior to other classical orthogonal moments either in lower-order moments or higher-order moments. Especially with the increase in the order of moments, and when the order is N = 40, the reconstructed images by OFMMs is invalid. When the order is N = 50, the reconstructed images using ZMs is invalid, and the reconstructed images using BFMs and EFMs can maintain good numerical stability in higher-order moments, but those visual effect of image reconstruction are obviously worse than that of the proposed image moments (MEFMs).
4.3 Optimal parameter selection for image reconstruction and recognition
Based on the analysis theory of the time–frequency correspondence in Section 3.5, the selection of parameter value α in Eq. (5) is crucial for the proposed image moments (MEFMs) that will affect the image reconstruction accuracy and the image recognition rate. In other words, choosing the optimal parameter value α to obtain a better image description ability is a problem that needs to be solved at the present. Therefore, a selection method of parameter α must be selected for the proposed MEFMs, which could lead to desirable results in image reconstruction. The selection of parameter α is equivalent to an unconstrained optimization problem (i.e., \( \min \left\{\overline{\varepsilon^2}\left[f,\overline{f};\alpha, N\right]\right\} \)) if two variables α and N are limited based on α_{min} ≤ α ≤ α_{max}, N_{min} ≤ N ≤ N_{max}. For the unconstrained optimization problems, the genetic algorithm (GA) is the most popular and effective method in the recent years. Using GA computing in the proposed image moments, more precise values of parameters α and N can be obtained. However, considering the complexity of the GA implementation process, a simpler algorithm is adopted herein to realize the optimization of parameter α. If the order N of the proposed image moments is fixed, the unconstrained optimization problem of double variables is transformed into the unconstrained optimization problem of a single variable. The specific implementation process is presented below.
where g = 20 denotes the number of gray-level images from the Coil-20 database, and f_{n} and \( {\overline{f}}_n \) represent the nth original and reconstructed images, respectively. A lower value of D_{g} indicates a better performance of the proposed image moments in image reconstruction or recognition.
The search results of the D_{g} according to different parameter value α
D _{ g} | ||
---|---|---|
α | Order N = 10 | Order N = 60 |
− 3.5 | 0.6272 | 0.6458 |
− 3 | 0.5841 | 0.6012 |
− 2.5 | 0.5344 | 0.5283 |
− 2 | 0.4733 | 0.4942 |
− 1.5 | 0.4137 | 0.4090 |
− 1 | 0.3462 | 0.3126 |
− 0.5 | 0.2842 | 0.2081 |
0 | 0.2243 | 0.1431 |
0.5 | 0.1778 | 0.1535 |
1 | 0.1600 | 0.1942 |
1.5 | 0.1514 | 0.2575 |
2 | 0.1421 | 0.3302 |
2.5 | 0.2011 | 0.4567 |
3 | 0.2443 | 0.5649 |
3.5 | 0.3057 | 0.6529 |
4.4 Rotation, scaling, and translation invariant image recognition
In this section, a new RST invariant system for MEFMs that can be implemented in two steps is proposed: for translation invariance, the proposed image projection approach can be considered as a new alternative of the traditional algorithm (i.e., the method for the image translation invariant based on calculating the image geometric moments [28] and center moments [43]), followed by extending the basis functions of the MEFMs from the polar coordinate space to the log-polar space, such that the MEFMs have invariant properties of scaling and rotation at the same time.
4.4.1 Scaling and rotation invariance of MEFMs
Log-polar mapping
Thus, it is straightforward that \( \left|F\left(u,v\right){e}^{-2\tilde{\pi j}\left(u\ln \sigma + v\phi \right)}\left|=\right|F\left(u,v\right)\right| \).
MEFMs invariant computing method in the log-polar coordinate space
where w(α, r) = |T_{n}(α, r)|r can be regarded as a weighted function, and g(r, θ) is a weighted image in the polar coordinate system.
Equations (28) and (29) show that the scaling and rotation of an image by a scaling factor of σ and an angle of ϕ result in a shift of the MEFMs in the ρ-axis and θ-axis, respectively. This simple property leads to the conclusion that the magnitudes of the MEFMs of the scaled and rotated image function remain identical to those before scaling and rotation. Thus, the magnitudes \( \left|{M}_{nm}^{LPM}\right| \) of the MEFMs can be taken as a scaling and rotation invariant feature for image recognition. For the discretization calculation for scaling and rotation invariance of MEFMs, see Appendix.
4.4.2 Projection approach for the image translation invariance
- (1)
If the original image is a color image, the color image should first be gray-scale; otherwise, this step can be a default.
- (2)
Otsu’s algorithm [45] is used to determine the thresholds of the gray-scale image in the global region and then binarize the gray-scale image according to the thresholds.
- (3)
A high-quality binary image can be obtained via a simple image pre-processing operation for binary images (e.g., denoising, filtering, etc.).
- (4)
Calculating the projection image in the horizontal direction of the binary image and obtaining the position for the troughs of the projection image, segmentation is performed for the whole image according to the trough point.
- (5)
The projection operation in the vertical direction is the same as that in Step 4.
- (6)
Finally, according to the segmentation position of the binary image, the target image can be separated from the background in the original image. The experiments are performed on the selected cartoon cat color images (Fig. 15) from the Columbia University image library database [42]. Figure 15 shows (a) as the process of using the projection approach for the untranslated images and (b) as the process of using the projection approach for the translated images.
4.4.3 Test of classification results for the RST invariance
Datasets
Experiment 1
Comparative study of the CCPs by various methods (including classic ZMs, OFMs, BFMs, and ours) for D1 dataset contaminated by salt and pepper noise
CCPs (%) | KNN | |||||||
---|---|---|---|---|---|---|---|---|
N = 10 (lower-order moments) | N = 60 (higher-order moments) | |||||||
Noise densities | ZMs | OFMs | BFMs | Ours | ZMs | OFMs | BFMs | Ours |
0% | 65.8 | 71.6 | 72.2 | 71.9 | 91.7 | 98.8 | 99.3 | 100 |
5% | 61.6 | 68.5 | 70.6 | 71.2 | 90.2 | 95.8 | 94.9 | 99.8 |
10% | 58.8 | 62.9 | 67.2 | 68.1 | 88.6 | 90.6 | 89.8 | 95.3 |
15% | 55.7 | 60.3 | 62.8 | 64.7 | 86.3 | 89.8 | 89.4 | 94.6 |
20% | 50.3 | 57.2 | 60.7 | 60.9 | 81.5 | 85.5 | 86.1 | 91.4 |
Experiment 2
Comparison our approach results with the published results in recent years for D2 dataset
The number of moments | CCPs | |||
---|---|---|---|---|
EFMs [2014] | Fr-LMs [2017] | SOMs [2016] | Ours [proposed] | |
5 | 68.7% | 75.5% | 76.4% | 78.2% |
10 | 71.8% | 80.2% | 82.2% | 82.5% |
60 | 89.3% | 95.5% | 97.3% | 97.7% |
80 | 91.6% | 98.9% | 98.4% | 99.3% |
Experiment 3
Comparative results of the proposed approach and other published methods in recent years for D3 dataset corrupted by smoothing filter operation
CCPs (%) | KNN | |||||||
---|---|---|---|---|---|---|---|---|
N = 10 (lower-order moments) | N = 60 (higher-order moments) | |||||||
Smoothing windows | EFMs | Fr-LMs | SOMs | Ours | EFMs | Fr-LMs | SOMs | Ours |
[3] | 70.2 | 81.7 | 83.8 | 82.1 | 90.1 | 97.8 | 98.9 | 99.8 |
[5] | 68.7 | 80.9 | 82.4 | 81.7 | 89.7 | 96.5 | 98.2 | 99.7 |
[7] | 64.2 | 79.4 | 80.8 | 79.9 | 87.4 | 92.8 | 95.8 | 97.1 |
[9] | 59.9 | 76.8 | 78.9 | 79.0 | 83.5 | 89.9 | 93.2 | 95.3 |
5 Conclusions
- (1)
A new type of piecewise modified exponent polynomial, also known as the semi-orthogonal polynomial, was derived. The derived polynomial is the transformed versions of classical exponent polynomial.
- (2)
To build a series of numerically stable different-order image moments for image reconstruction and pattern recognition, a new method of time–frequency correspondence is proposed herein, which can improve the image reconstruction effect and accuracy of image recognition.
- (3)
We propose a new method for RST invariant recognition and compared it with the traditional moment invariant-based method. Our approach is more practical and effective for geometric invariant recognition. For the future work, we will search for superior semi-orthogonal image moments for local feature extraction in the image analysis because non-moment-based methods (e.g., [47, 48]) can effectively extract the local features of the image.
Notes
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions.
Authors’ contributions
JC developed the idea for the study modified exponent-Fourier moments and contributed the central idea in our manuscript; BH did the analyses for the properties of the proposed image moments, analyzed most of the data, and wrote the initial draft of the paper; and the remaining authors contributed to refining the ideas, carrying out additional analyses and finalizing this paper. All authors were involved in writing the manuscript. All authors read and approved the final manuscript.
Funding
This work was supported by National Natural Science Foundation of China (Grant No. 61472298, 61702403), the Fundamental Research Funds for the Central Universities (Grant No. JB170308, JBF180301), the Project funded by China Postdoctoral Science Foundation (Grant No. 2018 M633473), Basic research project of Weinan science and Technology Bureau (Grant No. ZDYF-JCYJ-17), and by project of Shaan xi Provincial supports discipline (mathematics).
Competing interests
The authors declare that they have no competing interests.
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