Edge Detection from High Resolution Remote Sensing Images using Two-Dimensional log Gabor Filter in Frequency Domain

Edges are vital features to describe the structural information of images, especially high spatial resolution remote sensing images. Edge features can be used to define the boundaries between different ground objects in high spatial resolution remote sensing images. Thus edge detection is important in the remote sensing image processing. Even though many different edge detection algorithms have been proposed, it is difficult to extract the edge features from high spatial resolution remote sensing image including complex ground objects. This paper introduces a novel method to detect edges from the high spatial resolution remote sensing image based on frequency domain. Firstly, the high spatial resolution remote sensing images are Fourier transformed to obtain the magnitude spectrum image (frequency image) by FFT. Then, the frequency spectrum is analyzed by using the radius and angle sampling. Finally, two-dimensional log Gabor filter with optimal parameters is designed according to the result of spectrum analysis. Finally, dot product between the result of Fourier transform and the log Gabor filter is inverse Fourier transformed to obtain the detections. The experimental result shows that the proposed algorithm can detect edge features from the high resolution remote sensing image commendably.


Introduction
Edges in the high-spatial resolution remote sensing image, describe the structural information of ground objects i.e. road, buildings, river, and pond.Therefore, edge detection is important to image segmentation and land cover classification of remote sensing image.
There are many algorithms having been proposed for edge detection from images.Most edge detection methods belong on the gradient-based algorithms.Roberts [1], Prewitt [2] and Sobel [3] has developed Gradient-based features detection method.But there is great influence from image noises in the edge detection results.Thus, Marr and Hildreth [4] and Canny [5] introduced more systemic algorithms to retrieve more significant edges from images.Considering to the difference from different edge feature types, Morrone [6] [7] finds the theory of phase congruency in the research on Mach band.This theory is applied into image edge detection by using local energy model.In the later research, Kovesi [8] introduce Log Gabor filter to calculate the local energy, such that phase congruency model become more convenient for edge detections.Except for these edge detectors, there are different approaches having been explored, such as active contour models [9], level-set method [10].
Log Gabor filter is used into the image processing frequently.It is apply into detecting the texture and edge features [11], fingerprint image enhancement [12], weave defect detection [13] and human iris recognition [14].This paper use log Gabor filter in frequency domain to extract edge features.

Data
In the paper, the research data used throughout the research is a 512×512 pixel Quickbird image acquired in Noverber 21, 2004.This study area is a part landscape typical of Nanjing region, China.Quickbird image includes multispectral data in red, green, blue, near-infrared bands with 2.44m spatial resolution and a panchromatic band with 0.61m spatial resolution.The details are listed in Table 1.To get promising detection result and more details of the geographic objects in geo-spatial, this paper uses the panchromatic band as study data, which is shown in Figure 1.In this research data, there are different types of geographic objects including buildings, farmland, pond, and road.

A B
Because the speed of DFT is low, FFT was proposed to prompt the speed of transform.The magnitude spectrum presents the energy distribution of image, while the phase spectrum presents the location of image feature, and includes structure detail of image.After Fourier transform of image, the magnitude and phase spectrum can be retrieved, which are described as where |F(u,v)| and ϕ(u,v) are the magnitude and phase spectrum respectively; I(u,v) and R(u,v) are real and imaginary component respectively.This paper uses the magnitude spectrum of remote sensing image for edge detection.Before the processing of edge detection, some properties of magnitude spectrum are presented as follow: (1) DC (Direct Current) component of the magnitude spectrum reflects the mean of the image.
(2) The magnitude becomes higher with the distance being far away from DC component.
(3) The energy of magnitude spectrum mainly concentrates around DC component.
(4) The direction of the edge is orthometric with the direction of frequency energy of the edge.Because the difference between energy between low and high frequency is remarkable, the magnitude spectrum is transformed by the log function.Thus, the magnitude spectrum can be obtained by the definition as follow: The orientation of energy distribution is orthogonal with the extend direction of edge feature or texture in spatial domain, when the number of rows or columns of the original image are the same as each other.In addition, the distribution of frequency energy is symmetry to the DC component.Most energy is centralized around the center of the frequency spectrum image.According to the Equation 2 and 4, the magnitude spectrum of Figure 1 is shown in Figure 2. We can see that there is several visible line which correspond with the edges in original image.There are two line marked by A and B; line A is more distinct and successive.While some periodic bright spot show at line B, because the farmland in the original panchromatic band has a lot of periodic textures features.Furthermore, there the energy round DC component is much higher than other location.

Spectrum Analysis of edge features in remote sensing image
In this paper, radius sampling (Figure 3a) and angle sampling (Figure 3b) are used to analyze the magnitude spectrum.The curve of distribution of different frequency's energy can be retrieved by using radius and angle sampling which provide more details of original image.By using the radius and angle sampling, it is important to design a fit filter for edge detection.In Figure 4a, there are three peaks between the angles of 0° to 180° in angle sampling.The first peak appears at 2°; the second peak appears at 82°; the last one shows at 173°.Therefore, more details of edge features distribute around these peaks.As shown in Figure 4b, we can see that the whole curve of radius sampling descends extremely with the frequency becoming high.However, there is a little peak at 17. Thus, this frequency includes more information of original image.

Log Gabor Filter for Edge Detection
Daugman [15] developed two dimensions Gabor function according to the one dimensions Gabor function.The two dimensions Gabor function in spatial domain can be defined by where σ x and σ y decide the length of the Gabor filter in the direction of X and Y respectively, and (u 0 , v 0 ) decides the center frequency of the filter.While the expression of Gabor function in frequency domain can be defined by where f is the central frequency of the filter; θ is orientation of the sinusoidal plane wave; γ and η are the standard deviations of the Gaussian envelope along the primary and secondary direction respectively; . γ, η and θ decide the shape of Gabor filter in the frequency domain.G(u,v) is a Gaussian function that is shifted to the ƒ frequency units and rotated by an angle θ relative to the positive u-axis.
However, Gabor function with arbitrarily wide bandwidth and zero DC component can be constructed.Thus, log Gabor filter is proposed for more efficient applications.There is no DC component, which influence the output of edge detection, in the log Gabor filter.Furthermore, it has extended tails at the high frequency end, such that it can encode natural images more efficiently than ordinary Gabor filter.The definition of log Gabor function in frequency domain is described as follow: Figure 5 shows the presentations of Gabor and log Gabor filter in frequency domain.σ is kept to be the same, and f0 is locate at 3, 10, 30 and 50.We can see that the bandwidth of Gabor keeps the same, and the first Gabor function is not zero at DC component; while the bandwidth in log Gabor is different, and the DC component of log Gabor is zero.And it can be a combination of radial and angular components in polar coordinate system.Thus, the log Gabor function can be defined by Figure 6 shows a bank of log Gabor with 5 different central frequencies and the same bandwidth.Given f(i, j) be the original image, the edge detection can be implemented by Equation 8.
The procedure of edge detection from remote sensing image can be presented as follow: (1) Fourier transforms the image f(i, j) to obtain F(u, v) in frequency domain; (2) Frequency spectrum analysis for the design of log Gabor filters; (3) Dot product F(u, v) with the log Gabor filter; (4) Inverse Fourier transforms the result of (2); (5) Get the real component of the result of (3).

Result
According to the analysis of frequency spectrum, the main frequency energy distribute around the angle of 2°, 82° and 173°.Because the two angles of 2° and 173° are close, we design two log Gabor filter, the central frequencies of angular components of which are set to be 177.5°and 82°, to extract the edge features from the original image.Furthermore, the central frequency of radial component of the log Gabor filter is set to be 200 in order to retrieve more high frequency information.The bank of two log Gabor filters is shown in Figure 7. Finally, the edge detection result is retrieved by dot product between the log Gabor filter bank and the original image.The detection result is shown in Figure 8.It can be seen that there is good performance in the edge detection by using the proposed algorithm in this paper.Edges like ridges in farmland, boundary of workshop, boundary between water body and land and so on are detected readily.Therefore, we can conclude that the proposed algorithm can be applied into edge detection remarkably by choosing the suitable parameters in the design of the log Gabor filter.

Conclusion
In conclusion, we propose a new edge detection algorithm for high-spatial resolution remote sensing image.Firstly, the image is discrete Fourier transformed, and the magnitude spectrum is obtained.Secondly, the magnitude spectrum is analyzed by using radius and angle sampling to locate the edge distribution.Then, a filter bank with two log Gabor filters are designed, and then be multiplied with the frequency spectrum original high-spatial resolution remote sensing image.Finally, the result of product is transformed by using IFFT to retrieve the edge detection.
The detection result illustrates good performance of edge detection of high-spatial resolution remote sensing image.The experimental result proves that the edge detection can be realized in frequency domain by designing an appropriate log Gabor filter with suitable parameters.
TransformFourier transform proposed by Fourier in the research of heat transfer theory, has been applied into many fields extensively, especially in DSP and image processing.Because of image being discrete, Fourier transform is realized by two dimensions DFT (Discrete Fourier Transform) defined as

3 .
(a) Radius sampling (b) angle sampling Figure Radius sampling (a) and angle sampling (b) of the frequency spectrum.

35th
International Symposium on Remote Sensing of Environment (ISRSE35)

Figure 4 .
Figure 4. Radius distribution and angle distribution curves.

35thFigure 5 .
Figure 5.One dimensional Gabor filter and log Gabor filter with same parameters.Because of the limitation of log function at origin, log-Gabor filters are presented in the frequency domain.And it can be a combination of radial and angular components in polar coordinate system.Thus, the log Gabor function can be defined by

Table 1 .
Panchromatic and multispectral bands of QuickBird image