Independent component analysis approach to image sharpening in the presence of atmospheric turbulence
Introduction
The classical limit on performance of an imaging system is set by diffraction [1], [2], [3]. In accordance with the Rayleigh resolution criterion, the smallest linear dimension that can be resolved is given bywhere λ is the wavelength, f is the focal length of the lens, and B is the system aperture size. While resolution beyond the classical diffraction limit may be possible in certain cases [1], [4], [5] random fluctuations of the refractive index in space and time [14], [15] caused by atmospheric turbulence will generally degrade the performance much beyond the classical Rayleigh diffraction limit. One way to sharpen the image is to use the frame averaging technique described in [14]. In this paper we describe a new approach to image sharpening for incoherent imaging by incorporating the physical turbulence model into the processing algorithm. The image sequence is processed by using the ICA algorithm [6] and treating each frame of the video sequence as a sensor. Each turbulence spatial pattern is treated as one physical source and the diffraction limited image of the original object as another source. The physical justification of the adopted data model is based on propagation of the incoherent electromagnetic waves through turbid media, as described in Section 3. After a concise account of the ICA algorithm based on fourth-order (FO) statistics in Section 4, the proposed image processing technique is detailed in Section 5. The image performance evaluation employs the Laplacian operator and is carried out for the original frames, the ICA extracted source image of the object of interest, and for a frame obtained by averaging selected frames as well as for the average of all the frames present in the image sequence [15], [16]. The concluding remarks are set forth in Section 6.
Section snippets
Problem formulation
Inspired by the concept presented in [10], we have formulated a novel way of representing the image sequence in the ICA framework. The underlying hypothesis is similar to that used in the analysis of multispectral and/or hyperspectral data [11], [12], where ICA methods are applied to a multispectral/hyperspectral image cube under the assumption of hidden sources with different spectral signatures. In the ICA framework each spectral component is treated as a sensor, as illustrated in Fig. 1(a).
Derivation of the ICA data model
A quasi-monochromatic electromagnetic field in the object plane is transformed into its image (Fig. 2, Fig. 3) by the superposition integral (extended Huygens–Fresnel principle) [1], [3], [13], [14]where αi=ρi/L is the angular coordinate, L is the atmospheric path length, represents the electromagnetic field after the imaging lens, Uo is electromagnetic filed in the object plane and pl(α2,α1) is the system point spread function which may be represented by
The fourth-order statistics-based ICA algorithm
Strategy of ICA algorithms is to find a linear transformation Wsuch that the components of the vector are as statistically independent as possible. Based on the assumption that the source signals Ion are mutually statistically independent and non-Gaussian (except maybe one), the vector will represent the source signals within a permutation and a scale factor. Here, we are going to use the FO cumulant-based ICA algorithm JADE (joint approximate diagonalization
Experimental results
In order to verify the proposed image sharpening technique we have used an image sequence of the Washington monument. The image sequence contained 50 frames. Fig. 4 illustrates for three randomly selected frames how the ICA algorithm extracts one diffraction limited source image that corresponds to the original object and the two extracted source images that correspond to the spatial turbulence patterns. Fig. 4(a) shows three frames randomly selected from the 50 frames. In the ICA context these
Conclusion
A novel approach to the image sharpening in the presence of atmospheric turbulence is proposed in this paper which is based on the application of the ICA algorithm to randomly selected frames from the image sequence. The mutual information between the selected image frames was measured in order to ensure linearly independent measurements. In the ICA framework, the selected frames were used as sensors implying that the underlying sources are spatially independent. We have demonstrated the
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