CN105827249A - Fractional Fourier transform combining sparse optical sampling method - Google Patents

Abstract

The invention provides a fractional Fourier transform combining sparse optical sampling method. The method provides a novel optical remote sensing image sparse sampling model by being combined with fractional Fourier transform on the basis of a compressed sensing theory. The model utilizes a multi-order property and a characteristic of being applicable to processing non-stationary signals of fractional Fourier transform on the one hand, and carries out sparse sampling on image signals on the other hand so as to enable the observed data volume is far less than the data volume acquired by a traditional method, thereby reducing the development cost of an optical remote sensor, and reducing the storage and transmission cost.

Description

Sparse optical sampling method combined with fractional Fourier transform

Technical Field

The invention relates to the field of image processing and remote sensing information intelligent processing, in particular to a sparse optical sampling method combined with fractional order Fourier transform.

Background

In the field of digital signal processing, the problem of sampling signals has always been a fundamental and important research. The traditional remote sensor must satisfy Shannon (Shannon) sampling theorem in the process of signal acquisition, namely the sampling frequency must not be lower than 2 times of the highest frequency of the signal. With the development trend of high spatial resolution, high temporal resolution and high spectrum resolution of remote sensing images, a remote sensor designed according to the Shannon sampling theorem leads to massive sampling data, and the contradiction between storage, transmission and data processing is increasingly prominent.

In order to solve the contradiction, the compressed sensing theory comes from the beginning, and the basic idea is a signal compression and reconstruction technology based on sparse representation, which can also be called compressed sampling or sparse sampling. The compressed sensing causes substantial changes in the signal sampling and corresponding reconstruction modes, namely: the compression and sampling are combined, the non-adaptive linear dimension reduction projection (measured value) of the signal is firstly collected, and then the original signal is reconstructed from the measured value by a reconstruction algorithm. Candes, Tao and Romberg published a large number of papers to construct theoretical frameworks, and S.Mallat, D.Donoho, etc. deeply research on sparse representation methods of signals and images provides reliable mathematical proof and theoretical basis for compressed sensing. The sparse sampling mode breaks through the limitation of Shannon sampling theorem and provides a new opportunity for the research of the optical remote sensing technology.

Fractional Fourier transform (FrFT) has good time-frequency localization characteristic, is generalized Fourier transform and can be intuitively understood as the rotation of a signal on a time-frequency plane. Therefore, the fractional Fourier transform has time-frequency information of the signal at the same time, and the signal can be read from multiple angles. This flexibility of the fractional Fourier transform makes it of great research and application value in the field of signal processing, and has been widely used in the fields of image processing and pattern recognition.

Currently, signal sampling studies in the fractional order Fourier transform domain are mostly limited to the traditional shannon sampling theorem. However, researchers have increasingly conducted some useful trial studies on the combination of fractional order Fourier transforms and compressed sensing theory. Sultan Aldimaz et al propose that LFM signal obtains higher reconstruction probability in Fourier domain compressed sensing algorithm. The Jean minxu and the like combine fractional Fourier transform and a compressive sensing theory to be applied to UWB-LFM signal detection and estimation, and a better effect is achieved. The invention aims to research a sparse sampling mode of optical remote sensing image signals by combining the advantages of fractional Fourier transform.

Disclosure of Invention

The invention aims to solve the technical problems of reducing the development cost of an optical remote sensor and reducing the storage and transmission cost while ensuring that an original image signal is reconstructed with high quality.

Therefore, the invention aims to provide a sparse optical sampling method combined with fractional Fourier transform.

The above object of the invention is achieved by the features of the independent claims, the dependent claims developing the features of the independent claims in alternative or advantageous ways.

In order to achieve the purpose, the technical scheme adopted by the invention is as follows:

a sparse optical sampling method incorporating fractional order Fourier transforms, the method comprising: performing fractional Fourier transform on an original image signal by utilizing the multi-order property of the fractional Fourier transform; the original signal is then recovered by a fractional order Fourier inverse transform.

In a further embodiment, the method further comprises:

sparse sampling is carried out on the signals after fractional Fourier transform to obtain a linear measurement value Y, and the following are obtained:

Y＝ΦΨ^-1X＝ΦΘ

wherein X is a radicalA signal after a Fourier transform of a few orders, psi being a discrete fractional order Fourier transform matrix, psi^-1Inverse transformation of psi is carried out, phi represents that an observation matrix carries out linear projection on the signals;

according to the compressed sensing theory, solving the following expression to obtain X:

min||X||₁s.t.Y＝ΦΨ^-1X。

in a further embodiment, the original image signal is restored by inverse transformation of a fractional order Fourier transform, i.e. Θ ═ Ψ^-1X。

Compared with the prior art, the sparse optical sampling method combined with fractional Fourier transform can realize the dimensionality reduction of image signals by using the minimum observation times under the condition of not losing information required by the reconstructed original image, namely, the signals are directly subjected to less sampling to obtain effective representation of the signals, and meanwhile, the high-frequency information of remote sensing images is effectively represented by using the multi-order of the fractional Fourier transform and the characteristic of being suitable for processing unstable signals, so that the sampling and transmission cost is saved on the basis of reconstructing the original image signals with high quality.

It should be understood that all combinations of the foregoing concepts and additional concepts described in greater detail below can be considered as part of the inventive subject matter of this disclosure unless such concepts are mutually inconsistent. In addition, all combinations of claimed subject matter are considered a part of the presently disclosed subject matter.

The foregoing and other aspects, embodiments and features of the present teachings can be more fully understood from the following description taken in conjunction with the accompanying drawings. Additional aspects of the present invention, such as features and/or advantages of exemplary embodiments, will be apparent from the description which follows, or may be learned by practice of specific embodiments in accordance with the teachings of the present invention.

Drawings

The drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures may be represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing. Embodiments of various aspects of the present invention will now be described, by way of example, with reference to the accompanying drawings, in which:

fig. 1 is a general flow diagram of a sparse optical sampling method incorporating fractional Fourier transform.

Fig. 2 is a schematic diagram of a fractional Fourier transform time-frequency plane.

Fig. 3 is a diagram illustrating the energy of the fractional Fourier transform aggregate signal.

Fig. 4 is a schematic diagram of an optical implementation of fractional Fourier transform.

Detailed Description

In order to better understand the technical content of the present invention, specific embodiments are described below with reference to the accompanying drawings.

In this disclosure, aspects of the present invention are described with reference to the accompanying drawings, in which a number of illustrative embodiments are shown. Embodiments of the present disclosure are not necessarily intended to include all aspects of the invention. It should be appreciated that the various concepts and embodiments described above, as well as those described in greater detail below, may be implemented in any of numerous ways, and that the concepts and embodiments disclosed herein are not limited to any embodiment. In addition, some aspects of the present disclosure may be used alone, or in any suitable combination with other aspects of the present disclosure.

Fig. 1 is a schematic general flow chart of a sparse optical sampling method with a fractional Fourier transform according to an embodiment of the present invention, where 101 is an original image signal, and the present invention processes the original signal by using the multi-order of the fractional Fourier transform, that is, information of both time domain and frequency domain of the signal is included. Referring to fig. 2, fractional Fourier transform can be intuitively interpreted as that a signal rotates at any angle on a time-frequency plane, and when the rotation angle α is pi/2, the u-axis becomes a frequency axis in the conventional sense, i.e., the ω -axis; if α is 0, it is the time domain axis, i.e., the t axis.

In step 102, fractional Fourier transform is performed on the original image, the fractional Fourier domain is gradually transited from a time domain to a frequency domain along with the change of the transformation order, the fractional Fourier transform is gradually degenerated into the Fourier transform, and the energy distribution of the image in the fractional Fourier domain tends to the center of a two-dimensional coordinate plane. The fractional Fourier transform has an energy conservation relationship, with the energy of the entire image almost centered at one point and the energy in all other places almost equal to 0. In a sense, the fractional Fourier domain signal of the image can be regarded as sparse, and the sparsity just accords with the prior condition of signal compression perception. Referring to fig. 3, the invention uses fractional Fourier transform to project the time-frequency distribution of the image signal on the rotated frequency axis u, and selects a proper rotation angle to enable the signal to realize energy aggregation.

Inverse transformation formula according to fractional order Fourier transformIt can be seen that the signal x (t) is the inverse transform kernel K_-p(u, t) is the spread in space of the function of the basis, X_p(u) is x (t) is at K_-p(t, u) projection onto the basis, i.e. X (t) may be given by a set of weight coefficients X_pOrthogonal basis function K of (u)_-pAnd (t, u) and further constructing a fractional Fourier transform matrix from the discretization core matrix for signal sparsification. The invention obtains the transformation matrix by directly sampling continuous fractional Fourier transformation kernels.

Sampling an input function X (t) at intervals Δ t, f (a) X (a Δ t), and outputting the output function X_p(u) sampling at a sampling interval Δ u, F_p(b)＝X_p(b Δ u) wherein a ∈ [ -m, m]，b∈[-n,n]。

According to the definition of fractional order Fourier transform

$\begin{array}{c}{F}_p\left(b\right)=\sqrt{\frac{1-j\cot \mathrm{\α}}{2\mathrm{\π}}}\mathrm{\Δt\; exp}\left(j\frac{b^2{\mathrm{\Δu}}^2\cot \mathrm{\α}}{2}\right).\\ \underset{a=-m}{\overset{m}{\mathrm{\Σ}}}\exp \left[j(\frac{a^2{\mathrm{\Δt}}^2\cot \mathrm{\α}}{2}-\mathrm{ab\Δt\Δ}u\csc \mathrm{\α})\right]f\left(a\right)\end{array}---\left(1\right)$

Order to $F_p\left(b\right)=\underset{a=-m}{\overset{m}{\mathrm{\Σ}}}{K}_p(a,b)f\left(a\right),$ Then

$K_p(a,b)=\sqrt{\frac{1-j\cot \mathrm{\α}}{2\mathrm{\π}}}\mathrm{\Δt\; exp}\left(j\frac{b^2{\mathrm{\Δu}}^2\cot \mathrm{\α}}{2}\right)\·\exp [j(\frac{a^2{\mathrm{\Δt}}^2\cot \mathrm{\α}}{2}-\mathrm{ab\Δt\Δ}u\csc \mathrm{\α})---\left(2\right)$

Taking n to be more than or equal to m to ensure the inverse transformation of fractional Fourier transformWherein,is K_pThe conjugate transpose of (a, b) can be obtained

$f\left(a\right)=\underset{b=-n}{\overset{n}{\mathrm{\Σ}}}{K}_p^{*}(a,b)F_p\left(b\right)=\underset{b=-n}{\overset{n}{\mathrm{\Σ}}}\underset{c=-m}{\overset{m}{\mathrm{\Σ}}}{K}_p^{*}(a,b)K_p(b,c)f\left(c\right)---\left(3\right)$

By invertibility summing $\underset{b=-n}{\overset{n}{\mathrm{\Σ}}}\exp \left[\mathrm{jb}(a-c)\mathrm{\Δt\Δ}u\csc \mathrm{\α}\right]=\mathrm{\δ}(a-c),$ Thereby having

$\mathrm{\Δt\Δu}=\frac{2\mathrm{\π}N\sin \mathrm{\α}}{2n+1}---\left(4\right)$

Where N is an integer coprime with 2N + 1.

The normalization can be obtained by substituting the formula (4) into the formula (2)

$\begin{array}{c}{K}_p(a,b)=\sqrt{\frac{\mathrm{sgn}\left(\sin \mathrm{\α}\right)(\sin \mathrm{\α}-j\cos \mathrm{\α})}{2n+1}}.\\ \exp [j(\frac{1}{2}{b}^2{\mathrm{\Δu}}^2\cot \mathrm{\α}-\frac{2\mathrm{\πNab}}{2n+1}+\frac{1}{2}{a}^2{\mathrm{\Δt}}^2\cot \mathrm{\α})\end{array}---\left(5\right)$

When N is sgn (sin α) ± 1, sin α > 0 is substituted into the formula (1) to obtain

$K_p(a,b)=\sqrt{\frac{\sin \mathrm{\α}-j\cos \mathrm{\α}}{2n+1}}\·\exp \left[j(\frac{1}{2}{b}^2{\mathrm{\Δu}}^2\cot \mathrm{\α}-\frac{2\mathrm{\πab}}{2n+1}+\frac{1}{2}{a}^2{\mathrm{\Δt}}^2\cot \mathrm{\α})\right]---\left(6\right)$

Let m be n, the needed transformation matrix can be obtained

$K_{-p}=K_p^{*}={\left[K_p(a,b)\right]}^{*}---\left(7\right)$

Wherein a, b ∈ [ -m, m ].

Alternatively, the fractional order Fourier transform in this example can be implemented by combining free space and a lens.

Referring to the optical system shown in FIG. 4, let l be the distance between the incident surface and the lens and the distance between the lens and the output surface, and f be the focal length of the lens_αThe input plane is f (x, y) and the output plane is g (u, v) the output plane is a Fourier transform of order α of the input plane when:

$\begin{array}{cc}{f}_a=\frac{f_1}{\sin (\mathrm{\α\π}/2)}& l=f_1\tan (\mathrm{\α\π}/4)\end{array}x---\left(8\right)$

wherein f is₁Is a constant.

In step 103, sparse sampling is performed on the signal after fractional Fourier transform to obtain a linear measurement value Y, if any, a linear measurement value Y is obtained

Y＝ΦΨ^-1X＝ΦΘ(9)

Wherein, X is the signal after fractional Fourier transform, psi is discrete fractional Fourier transform matrix, psi^-1The inverse of Ψ, and Φ represents the linear projection of the signal by the observation matrix.

According to the compressed sensing theory, solving the following expression in step 104 yields X:

min||X||₀s.t.Y＝ΦΨ^-1X(10)

in the formula (10)Norm minimization is a non-convex optimization problem, which is an NP-hard problem requiring combinatorial search. However, non-convexThe norm minimization problem can equivalently be expressed as convex relaxationNorm minimization problem. Thus, X can be obtained by solving the following equation:

min||X||₁s.t.Y＝ΦΨ^-1X(11)

finally, the original signal is restored through the inverse transformation of fractional order Fourier transform, namely theta ═ psi^-1X。

Although the present invention has been described with reference to the preferred embodiments, it is not intended to be limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the invention. Therefore, the protection scope of the present invention should be determined by the appended claims.

Claims (5)

1. A sparse optical sampling method incorporating fractional Fourier transform, the method comprising: performing fractional Fourier transform on an original image signal by utilizing the multi-order property of the fractional Fourier transform; the original signal is then recovered by a fractional order Fourier inverse transform.

2. The sparse optical sampling method combining fractional Fourier transform of claim 1, wherein the time-frequency distribution of the image signal is projected on a rotated frequency axis u according to the fractional Fourier transform, and a suitable rotation angle is selected to enable the image signal to realize energy aggregation.

3. The method of sparse optical sampling in combination with fractional order Fourier transform of claim 2, wherein fractional order Fourier transform is implemented by combining free space and a lens.

4. The method of sparse optical sampling in conjunction with fractional Fourier transform of claim 1, 2, or 3, further comprising:

sparse sampling is carried out on the signals after fractional Fourier transform to obtain a linear measurement value Y, and the following are obtained:

Y＝ΦΨ^-1X＝ΦΘ

wherein X is a signal after fractional Fourier transform, psi is a discrete fractional Fourier transform matrix, psi^-1Inverse transformation of psi is carried out, phi represents that an observation matrix carries out linear projection on the signals;

according to the compressed sensing theory, solving the following expression to obtain X:

min||X||₁s.t.Y＝ΦΨ^-1X。

5. the method of claim 1, wherein an original image signal is recovered by an inverse of the fractional Fourier transform, i.e., Θ - Ψ^-1X。