Abstract: As a generalization of Fourier transform, the Fractional Fourier Transform (FRFT) contains simultaneity the time-frequency information of the signal and is considered a new tool for time-frequency analysis. By way of simulation, this study analyses the energy distribution, the characteristic of the amplitude and phase in Fractional Fourier domain of an image. The present simulation results show that the FRFT can represent the time-frequency characteristic of an image and the distribution changes with the variety of the fractional powers. Moreover, some conclusions were also found from the reconstructed image using the amplitude and phase information of FRFT. All of the conclusions will help research how to use the FRFT in the area of image recognition and image edge extraction.
INTRODUCTION
Digital image processing technology is an emerging discipline to flourish and is widely used in the digital photogrammetry, remote sensing image processing, bio-image processing, geographical information systems and digital image processing technology transform domain has been a hot research field of image processing. The traditional two-dimensional Discrete Fourier Transform (DFT), a two-dimensional Discrete Cosine Transform (DCT), a two-dimensional Discrete Sine Transform (DST) and their transform ideas are that images are switched between the time-space domain and frequency domain, image information is extracted and image feature is analysed. Two dimensional Discrete Cosine Transform (DCT) with the highly energy accumulation has provided a theoretical basis for image compression and coding algorithms and still image and motion picture compression standard based on DCT transform has become a mainstream technology of image transmission and storage. However, it is well known that the above-mentioned transformation is the global transformation of time variable and is unable to extract local spectral features. Gabor and short-time Fourier transform has an attempt to compensate for the lack of transformation above but their window size does not change with frequency and these restrictions can not solve the resolution problem of the time and the frequency. Therefore, the merged wavelet transform has inherited and developed ideas of Gabor and short-time Fourier transform, as well as this has overcome that the window size does not change with frequency, the lack of a discrete orthogonal basis and other shortcomings. Wavelet transform time-frequency localization features make it ideally suited for image processing and has been widely used in the image compression standard in recent years, image segmentation, image reconstruction technique and become a hot research in image processing sector in recent years (Chen and Wu, 2002; Luo et al., 2005; Wang et al., 2005; Yan et al., 2006).
Fractional Fourier Transform (FRFT) is a new time-frequency analysis tool which is developed in recent years and it is the Fourier Transform of the generalized form. In essence, the signal makes the representation on the Fractional Fourier domain while this is the integration of signal information in the time domain and frequency domain. This new mathematical tool not only is closely linked with the Fourier Transform but also is very meaningful with other time-frequency analysis tools and one has been widely used in optical system analysis, filter design, signal analysis, solving differential equations, phase recovery and pattern recognition field (Pei and Ding, 2001). Most applied research of Fractional Fourier Transform in recent years is focused on the linear FM signal estimation, detection and filtering aspects, for its application is less in image processing. Image processing of Fractional Fourier Transform is only limited to the the chirp digital watermark detection of image (Tao et al., 2006; Zhang and Tao, 2008; Chen et al., 2014). Therefore, to explore and analyze the image characteristics of the Fractional Fourier domain and to tap the image value of the Fractional Fourier Transform is of great significance.
MATERIALS AND METHODS
Fractional fourier transform
FRFT Principle: The FRFT can be explained that the representation of the Fractional Fourier domain is formed after the signal does counterclockwise rotation any angle around origin in the time-frequency plane axes and this is a generalized form of Fourier transform.
FRFT of the signal x(t) is defined as (Tao et al., 2004):
| (1) |
where, FRFT transform kernel Kα (t,u).
| (2) |
The equation α = pπ/2 is the FRFT angle of rotation. For two-dimensional signal x(s,t), the two-dimensional FRFT can be expressed as:
| (3) |
By the discrete, FRFT can also be calculated by using digital methods. The most commonly used algorithm is the decomposition fast algorithm which is proposed by Ozaktas et al. (1994, 2000). The signals can be decomposed into FRFT convolution by the algorithm, the calculated results are compare similar with the continuous FRFT output. Decomposition type FRFT transformation matrix as follows:
| (4) |
where, D and J are, respectively the twice inside difference between the matrix and tthe matrix of the extraction operation, Kα is a discrete FRFT nuclear transforming matrix, namely:
| (5) |
A discrete dimensionless normalized Fractional Fourier transform is defined as:
| (6) |
Energy distribution of the image in Fractional Fourier domain: The discrete transform energy distribution of a two-dimensional image reflects the characteristics of the transforming image. Never lossy compression point of view, the transform purpose is that the energy as much as possible is focused to a small number of several coefficients after the image is transformed. After to be quantified and only a few coefficients is not zero which can get higher compression ratio. It is well known that DCT energy accumulation is superior to other transformation. The purpose of this section is the energy accumulation of the image FRFT domain. To this end, a normalized residual error factor ρ was defined first. For the MxN image, the coefficient of the FRFT is Fα(k,h). According to the qualitative analysis, the FRFT domain energy is also concentrated in the central region (Ozaktas et al., 1994, 2000), given by:
| (7) |
where, the r corresponding area is:
Image Fractional Fourier Transform amplitude and phase information: For image processing and image receiver, the phase information is more important than amplitude information and the phase information is the most vulnerable during transmission, the channel delay and Doppler shift will affect the phase information. It is necessary to study the phase and amplitude information and associated characteristics of the image by the FRF. In order to study the amplitude and phase characteristics of the image FRFT, the phase and amplitude characteristics are researched first by reviewing the traditional Fourier Transform.
Assumption F(k, h) is that the two-dimensional Fourier transform of the two-dimensional image f(x, y):
| (8) |
F(k, h) can be decomposed into the amplitude component and phase components, namely:
| (9) |
Among A(k, h) = |F(k, h)| is amplitude function, P(k, h) = F(k, h)/A(k, h) is phase function. Respectively, A(k, h) and P(k, h), s two-dimensional fourier inverse transform is to get α(k, h) and p(k, h).
RESULTS AND DISCUSSION
Simulation and performance analysis
Comparison of image information between FT domain and FRFT domain: The Saturn image is used as an example simulation, the FT simulation results is shown in Fig. 1.
The original image information is recovered from (k, h) in Fig. 1b which contains only the background information of the original image, its results are similar to the original image experienced a low-pass filter, the original image information is recovered from p(k, h) in Fig. 1c which can be edge information.
| Fig. 1(a-c): | (a) Saturn image, its (b) FT Magnitude part and (c) Phase section |
This shows that recovery image by the Fourier Transform phase contains only the edge information of the original image, the result is similar to the original image through the high-pass filter, so that the FT can be used for image edge detection and pattern recognition (Mi et al., 2012).
Use similar methods to study the amplitude and phase characteristics of the image in Fractional Fourier domain coefficient, p1 = p2 = 0.01, 0.1, 0.5, 0.8 were taken in the Fractional Fourier domain and the “Satrun” image two-dimensional Fractional Fourier Transform coefficients are divided into the amplitude function and phase function, respectively, by the Inverse transform in the Fractional Fourier Transform, the simulation results shown in Fig. 2 and 3.
It is shown in Fig. 2 and 3 that the different p is taken, from the IFRFT images of the FRFT amplitude function and the IFRFT images of the FRFT phase functions, the following conclusions can be obtained.
When the transform order P is changed from small to big, the image is restored only by the phase function, the revealing edge of the original image has become increasingly clear which is similar to the original image has gone through different cut-off frequency high pass filter. When p is small (0.01), it is correspond to the high-pass filter in the lower cutoff frequency and low frequency components emerge, so that the extraction of edge is blur, as shown in Fig. 2b. When p is large (0.8), it is correspond to the high-pass filter which cut-off frequency is higher, the majority of low-frequency components are filtered out and the extracted edge is clearer, as is shown in Fig. 3d, this time FRFT is degenerate into the basic FT.
| Fig. 2(a-d): | Saturn images recoved, respectively by the FRFT magnitude function and FRFT phase function. (a) Image recovered by FRFT magnitude function when p = 0.01, (b) Phase ↵ function when p = 0.01, (c) Magnitude function when p = 0.1 and (d) Phase 5 function when p = 0.1 |
| Fig. 3(a-d): | Saturn images recoved, respectively by the FRFT magnitude function and FRFT phase function, (a) Magnitude function when p = 0.5, (b) Phase ↵ function when p = 0.5, (c) Magnitude function when p = 0.8 and (d) Phase 5 function when p = 0.8 |
Similarly, when the order P of the transform is changed from small to big, only by the amplitude function to restore the image, it is closer and closer to the original image background which is similar to that the original image has experienced a low-pass filter in the different cut-off frequency. When p is small (0.01), it is correspond to the low-pass filter in the higher cutoff frequency, high-frequency component residue is more but also the outline of the original image is clearly seen, as it is shown in Fig. 2a, when p is large (for 0.8), it is correspond to the the low-pass filter in the lower cutoff frequency, most of the high-frequency components is filtered out, the background of the original image is only show, as it is shown in Fig. 3c.
In the transform, the order P is any other value, the restored image by FRFT phase function and amplitude function contains both the original image background and also includes the texture of the original image, as it is shown in Fig. 2b, c and 3a, b. In view of this situation, It can be deduced that this is similar to that the original image has gone through the FRFT time-frequency filtering and that is also about which the time-frequency plane is whirlled at an angle before filtering. If the cut-off frequency in frequency domain filter and bandwidth are fixed, when the rotation angle (order) is different, the projection is also different in the timeline and frequency axis, so, the output frequency components is also different in the frequency domain filter. Performance is to restore the image, that is the phase function and the amplitude function have contained the frequency component which change with the order.
In short, when the order of the transformation is small, the i-recovery image by the amplitude function and phase function of the FRFT shows a strong image information, they reflects the strong airspace characteristics (Fig. 2), when the transform order number is gradually increasing, the image which is restored by the FRFT of amplitude function has contained in the original image airspace characteristics which is gradually weakened until it disappears, by the FRFT phase function to restore, the image has the edge texture features of the original image which is gradually increased and it is compared according to Fig. 1, when the order increases in FRFT transform, the amplitude characteristics and phase one are closer and closer to one of the FT domain and that is frequency domain features (Fig. 3). These conclusions reflect the the two-domain characteristics of the time-frequency in FRFT domain.
Comparison of FRFT information between different images: In order to reflect the universality of the conclusion, the simulation selected seven images with different texture features and different sizes of standard gray-scale image (‘lena’ (512x512), ‘baboon’ (512x512), ‘bridge,’ (256x256), ‘cameraman’ (256x256), ‘rice’ (256x256), ‘moon’ (358x536), ‘saturn would’ (438x328)).
In order to simplify the simulation calculation, the transform order is p1 = p2 and after changing the order of p, the changes of the normalized residual error factor ρ is investigated, the simulation results shown in Fig. 4 and 5.
From the simulation results, it can be seen that the energy of the image in the fractional Fourier domain shows a certain regularity.
With increasing the order of the transformation (due to the symmetry of the FRFT order. changes in the interval (0,1) have general), the image share energy of a quarter of the coefficient increases in the middle of the FRFT domain. It can be seen that the energy of the image in Fractional Fourier domain shows the aggregation and its energy distribution tends to be the center FRFT domain of two-dimensional coordinate plane. Transform order p1 = p2≤0.5, with the increase of the transformation order, the aggregation degree of the image energy is also significantly improved. When the angle change is near 0.7, the energy aggregation in this FRFT domain area has reached more than 90%.
| Fig. 4: | Curve of the normalized residual error factor changes with the order number |
| Fig. 5: | FRF domain image energy peak curve with the change of the order |
Since, the Fractional Fourier Transform meets the Parseval criteria with the relationship o the energy conservation and FRFT image also contains an time-frequency information, so, the energy distribution in the space-frequency domain is also changing with change order. When the transform angle is the small (p1 = p2≤0.5), the rotative angle which is corresponding to the time-frequency plane is less than π/4, the energy distribution in FRFT does not reflect well the aggregation, almost half of the energy is distributed in the airspace or scattered throughout the NxN region, the changing trend is relatively flat, when p1 = p2>0.5, the energy accumulation changes are more obvious with the increasing trend of the transform angle, the corresponding time-frequency plane rotation angle is close to π/2, the energy distribution is close to Fourier transform in the frequency domain and the energy is concentrated in only a few coefficients. At that p1 = p2 = 1.0 time, FRFT degradation is on the Fourier Transform, the image energy is to achieve the greatest degree of aggregation.
From the above analysis, image FRFT has aggregation but their aggregation depend the global on the Fourier Transform and the energy distribution with the order changing characteristics also reflects the characteristics of dual-domain information on FRFT.
Comparison of single defocused image restoration: Figure 6a is the original image into focus, Fig. 6b is captured by the camera defocus blurred image, Fig. 6c-e are the FRFT recovery results with different order and the number of iterations, Fig. 6f is based on the results of Wiener filter (Gonzalez and Woods, 2003) recovery.
Table 1 shows the comparison of the parameter list in the different recovery modes, where in the Minimum Square Error (MSE) represents the mean square error of the original image and the restored image, MSE is smaller, the smaller the difference between the two. By comparison, it is found that a = 0.02 is in Fig. 6d, iterations T = 15, the effect of its minimum MSE value is best in the recovery image and it is better than Wiener filtering and Gaussian filtering.
| Fig. 6(a-f): | Single defocused image restoration (a) Original image, (b) Blurred image, (c) α = 0.01, T = 22, (d) α = 0.02, T = 15, (e) α = 0.05, T = 18 and (f) Wiener filter |
| Table 1: | Comparison of different parameters recovery mode |
This can be seen that FRFT defocused image restoration is better than the traditional filter recovery mode.
CONCLUSION
This study is that only from the perspective of qualitative simulation, the image is studied in the Fractional Fourier Transform energy distribution, the amplitude and phase characteristics and association with the original image. Simulation results show that: The image energy accumulation in the Fractional Fourier Transform is related to the transformation order, its aggregation is strongly dependent on and is close to the extent of its Fourier Transform, it is similar to the traditional Fourier Transform, the FRFT phase function contains the image texture information, the edge information which is contained in the phase function are not the same with the changing angle, this may be that the Fractional Fourier Transform can be more flexible for image edge extraction and recognition. How the FRFT features are researched by a more comprehensive, in-depth study and the image time-frequency characteristics is extracted full by the advantage of its dual-domain representation, above is the author further study.