CN103208098B - The Singularity spectrum function acquisition methods of high-definition picture restoration and system - Google Patents
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Abstract
The invention provides Singularity spectrum function acquisition methods and the system of a kind of high-definition picture restoration.Described method includes: obtain laterally or longitudinally pixel number and a width low-resolution image of high-definition picture, obtains the low-frequency spectra data of described high-definition picture according to described pixel number and low-resolution image；The zero padding method frequency spectrum data of high-definition picture described in the low-frequency spectra data acquisition according to described high-definition picture；Described zero padding method frequency spectrum data is made Fourier transformation to obtain the low-frequency spectra data padding method image of high-definition picture；According to the most preferably strange dissimilation operator of described low-frequency spectra data padding method Image Acquisition, singular function is obtained according to described optimal strange dissimilation operator, Singularity spectrum function is obtained according to described singular function, the Singularity spectrum function recovery for high-definition picture of high-definition picture restoration in the case of high frequency spectrum shortage of data, can be obtained quickly and efficiently.
Description
Technical Field
The invention relates to a singular spectral function acquisition method and system for high-resolution image restoration.
Background
The satellite remote sensing has the unique advantages of large coverage, long duration, strong real-time performance, no restriction of national boundaries and regions and the like, is widely applied to the fields of resource development, environmental monitoring, disaster research, global change analysis and the like, and is deeply and highly valued by various countries. The spatial resolution of the satellite image is a main index for measuring the satellite remote sensing capability and is also an important mark for measuring the national space remote sensing level. Improving the satellite observation spatial resolution has become a research hotspot of satellite engineering technology. In the process of acquiring images by a satellite, a plurality of factors can cause image quality reduction, atmospheric disturbance, motion, defocusing, transmission and noise can directly influence the reduction of image resolution, particularly, the cutoff frequency of an optical system is limited to be high by a satellite load-cut limit, and the size of a CCD chip pixel is limited to be small, so that the high-frequency component of a satellite image space is limited, and the image resolution is not high enough.
According to the optical Fourier spectrum theoryThe optical system having a cut-off frequency cfWhere D is the equivalent lens diameter, l is the CCD chip size, f is the lens focal length and λ is the wavelength of light. If the pixel size of the CCD chip is w, then according to the sampling theorem, the cut-off frequency is uw1/(2 w). In the subject, u is smaller than u only at the same timewAnd cfCan the spatial frequency components of c be acquired and imaged, if cf≠uwThis results in a waste of sampling resources or optical imaging resources. When the distance between the satellite and the subject is R, the resolvable distance Δ x of the satellite image is wR/f λ R/(D-l). If the pixel size of the CCD chip is reduced, u is increasedwAnd the optical cut-off frequency is increased accordinglyf＝uwThe spatial resolution of the image can be increased (currently minimum 50 μm)2) However, the pixel size w of the CCD chip is too small, and the signal-to-noise ratio is too low to be used normally. Therefore, the absence of high frequency components of satellite images is an irreconcilable scientific problem. Conventional high resolution (reference 1: J.L.Harris, Diffraction and resetting power, J.Opt.Soc.Amp., 54(7):931- & 1964 and reference 2: W.Lukosz, Optical systems with resetting power beyond the Optical system limit, J.Opt.Soc.Amp., 56(11):1463- & 1471,1966) means that the cut-off frequency c of the Optical system is exceededfAnd the lost image high frequency information is recovered, which is called high resolution restoration technology. Most believe that it is impossible to accurately recover spectral information outside the cutoff frequency, and this is called high resolution mystery.
When a sequence of multiple images of the same scene is available, a mathematical model g can be establishedi＝Hsi+ni1,2, k, wherein gi,si,niLow resolution images, high resolution images and noise images of the ith frame are respectively shown, and H represents various factors that cause low resolution of the images. By a multi-frame Interpolation method (see document 3: L. Zhang, X. Wu, An Edge-Guided Image Interpolation for direct Filtering and Datafusion, IEEE Transaction on Image processing,15(8):2226-2238,2006, document 4: D. Rajan D, S. Chaudhuri, Generalized Interpolation and bits applyidentification in super-resolution imaging, Image and Vision Computing,19(13):957-969,2001, reference 5: A.S a nchez-bed and G.Pajares, Non-iterative interpolation-based super-resolution analysing in the reconstructed image, IEEE trans. image Process, 17(10), pp.1817-1826,2008, document 6: S.RTRATATANATICH, N.K. BOSE, High Resolution Image format from low Resolution frames Using delay analysis, IEEETransmission on Image Processing,11(12): 1427-: gao, k.zhang, d.tao and x.li, Image Super-Resolution with sparse Neighbor Embedding, I IEEE trans. on Image Processing, vol.21, No.7, pp.3194-3205,2012, document 9: Z.M.Wang and W.W.Wang, Fast and Adaptive Method for SAR Superresolution Imaging Based on Point Scattering Model and Optimal BasisSelection, IEEE tran.on Image Processing,18(7):1477 and 1486,2009, document 10: marquina and S.J.Osher, Image super-resolution by TV regulation and Bdomain man iteration, Journal of Scientific Computing, vol.37, No.3, pp.367-382,2008 and document 11: j.yang, j.wright, t.s.huang and y.ma, "Image super-resolution video representation," IEEE trans.image process, vol.19, No.11, pp.2861-2873,2010), learning-based methods (see document 11, document 12: t.goto, Y.Kawamoto, Y.Sakuta, A.Tsutsui and M.Sakurai, Learning-based Super-resolution Image Reconstruction on Multi-core processor, IEEE Transactions on Consumer Electronics,58(3) 941-946,2012, reference 13: P.Purkait and B.Chanda, Super Resolution Image Reconstruction threading Bregman edition Using morphological Reconstruction, IEEE trans.on Image processing,21(9): 4029-4040,2012, document 14: P.P.Gajjar and M.V.Joshi, New learning based super-resolution, Use of DWT and IGMR-F primer, IEEE Trans.on image processing, Vol.19, No.5, pp.1201-1213,2010, document 15: (M.S. Crouse, R.D.Nowak, R.GBaraniuk, Wavelet-based statistical signal processing using Markov modelsIEEE Transactions on Signal Processing,46(4): 886-: m NDo, M.Vetterli, The constraint transform: An electronic direct multi-resolution Image representation, IEEE Transactions on Image Processing,14(12): 2091. sub. 2106,2005) and so on (see document 17: D.D. -Y Po and DO M.N.Do, directive multi-scale model of images using the conditional transform, IEEE Transactions on image Processing,15(6) 1610, 1620,2006 and 18: W.Dong, L.Zhang, G.Shi and X.Wu, Image subtraction and super resolution by adaptive sparse domain selection and adaptive reconstruction, IEEE trans.image processing, vol.20, No.7, pp.533-549, Jul.2011) obtains high-resolution images and improves Image quality degradation caused by undersampling. However, in satellite observation and shooting, multi-frame image acquisition in the same view field is extremely resource-wasting and difficult to achieve, and the single-frame image super-resolution restoration technology is the key technology for remote sensing image super-resolution restoration, but no substantial breakthrough exists so far. To study the high resolution restoration of a single frame image, we consider the following mathematical model of the low resolution image formation mechanism.
If no interference factor is considered, the imaging process for an imaging system can be described by the following equation:
g(x,y)＝p(x,y)*s(x,y)
here, g (x, y), p (x, y), s (x, y) respectively represent the remote sensing image, the field of view and the point spread function of the imaging system, and x represents the convolution operation. The spectral function for an image is: g (u, v) ═ P (u, v) S (u, v), the spectrum function P (u, v) ═ 1 of the point spread function, | u | < cf&|v|＜cfIs a rectangular window of limited bandwidth. When actual field cut-off frequency csGreater than the cut-off frequency c of the optical imaging systemfC of the time, field of viewfThe high frequency components other than the high frequency components are lost, and the low resolution image is obtained. Traditionally, c is considered to be outside the cut-off frequency of the optical imaging systemfThe spectrum is not recoverable. But according to the analytic extension theorem: if the analysis is known over a finite interval, the entire region can be extended uniquely. That is, "if two analytical functions are identical over any given area"they must be completely identical overall" is the same function (see, e.b. saff and a. d. snider, Fundamentals of Complex Analysis with Applications to engineering and Science,2003, Pearson edutation). The field of view may be viewed as a function on a bounded domain whose spectral function is an analytical function. Therefore, according to the analytic extension theorem, the image spectrum data G (u, v) ═ P (u, v) S (u, v), | u | < cf&|v|＜cfExtending to the whole spectral space, cut-off frequency cfInfinity. Early studies were carried out on high resolution restoration from a single frame Image by spectral extrapolation (see document 20: H.Greenspan, C.H.Anderson, S.Akber, Image Enhancement by nonlinear amplification in frequency space, IEEE transaction Processing, vol.9, no 6, pp.1035-1048,2000), long elliptic wave function method (see document 21: H.A.Brown, impact of localization on Image Enhancement by elliptical particle functions, Journal of Optical Society of America, Vol.59, no 2, pp.228-229,1969), superposition sinusoidal template method (see document 22: S.Wadaks, T.to, Superresolution sampling analysis of America, J.65, Journal of interpolation System of ultrasound analysis of 354, Journal of ultrasound resolution 3, Journal of ultrasound analysis of 3, and so on, and the like. However, these methods fully utilize the high resolution information of the image implied in the low resolution image, and do not understand and apply the mathematical principle of analytic extension theorem, so that the method for acquiring high frequency information from the low resolution image cannot or rarely studied, and thus the effect is very limited (see documents 23: S.C. park, M.K. park, M.G. kang, Super-resolution image retrieval: a technical overview, IEEESignal Processing Magazine, Vol.20, No.3, pp.21-36, May 2003).
Disclosure of Invention
The invention aims to provide a singular spectral function acquisition method and a singular spectral function acquisition system for high-resolution image restoration, which can quickly and efficiently acquire the singular spectral function for high-resolution image restoration to restore a high-resolution image under the condition of high-frequency spectrum data missing.
In order to solve the above problem, the present invention provides a singular spectral function acquisition method for high resolution image restoration, including:
acquiring the number of horizontal or longitudinal pixel points of a high-resolution image to be restored and a low-resolution image, and acquiring low-frequency spectrum data of the high-resolution image according to the number of the pixel points and the low-resolution image;
acquiring zero padding method frequency spectrum data of the high-resolution image according to the low-frequency spectrum data of the high-resolution image;
fourier transform is carried out on the zero padding method frequency spectrum data to obtain a low-frequency spectrum data zero padding method image of the high-resolution image;
and acquiring an optimal singular operator according to the low-frequency spectrum data zero padding method image, acquiring a singular function according to the optimal singular operator, and acquiring a singular spectrum function according to the singular function.
Further, in the above method, in the step of obtaining the number of horizontal or vertical pixel points of the high resolution image to be restored and a low resolution image, and obtaining the low frequency spectrum data of the high resolution image according to the number of pixel points and the low resolution image,
a low resolution image gl(i, j), i, j is 0, 1.. times, l is to be restored to a high resolution image G (i, j), i, j is 0, 1.. times, N > l, and the spectral data of the G (i, j) image is denoted as G (k) imagex,ky)，kx,ky∈ Ω, Ω is the spectrum space of the high resolution image, l represents the number of horizontal or vertical pixels of the low resolution image, N represents the number of horizontal or vertical pixels of the high resolution image to be restored, the spectrum data of the low resolution image is represented as Gl(kx,ky)，Wherein,denotes glThe Fourier transform of (i, j) shows that k is more than or equal to l/2 in the low-frequency range of g (i, j) imagex,kySpectral data of < l/2 are expressed as
(N/l)2Gl(kx,ky)。
Further, in the above method, in the step of acquiring the zero-padding method spectrum data of the high resolution image based on the low frequency spectrum data of the high resolution image,
the zero-filling method frequency spectrum data of the high-resolution image is represented as G (k)x,ky)P(kx,ky) Wherein
further, in the above method, in the step of performing fourier transform on the zero-padding method spectrum data to obtain the zero-padding method image of the low frequency spectrum data of the high resolution image,
the low-frequency spectrum data zero-filling method image of the high-resolution image is represented asWherein,represents G (k)x,ky)P(kx,ky) Inverse fourier transform of (d).
Further, in the above method, the step of obtaining an optimal singularization operator according to the low-frequency spectrum data zero padding method image includes:
initialization:phi (i, j) is (i, j), wherein,wherein "x" represents a convolution, (i, j) is a two-dimensional dirac function;
note that the four basic singularities are:
φ1(i,j)＝φi,j-(i,j)＝(i,j)-(i,j-1),φ2(i,j)＝φi-,j-(i,j)＝(i,j)-(i-1,j-1)，
φ3(i,j)＝φi+,j-(i,j)＝(i,j)-(i+1,j-1),φ4(i,j)＝φi-,j(i,j)＝(i,j)-(i-1,j)；
executeJudging whether to use
If so, the method will be usedIs assigned toAnd converting phi (i, j) phiI(i, j) assigning a value of phi (i, j), and repeating said executingAnd determining whether or notA step (2);
if not, outputting the optimal singularization operator phi (i, j).
Further, in the above method, the step of obtaining the singular function according to the optimal dissimilarity operator includes:
the singular function h (i, j) is obtained from a solution of the zero state of the difference equation phi (i, j) × h (i, j) ═ i, j.
Further, in the above method, in the step of obtaining the singular spectrum function according to the singular function, the singular spectrum function is
According to another aspect of the present invention, there is provided a singular spectral function acquisition system for high resolution image restoration, including:
the low-frequency spectrum data module is used for acquiring the number of transverse or longitudinal pixel points of a high-resolution image to be restored and a low-resolution image, and acquiring the low-frequency spectrum data of the high-resolution image according to the number of the pixel points and the low-resolution image;
the zero padding method frequency spectrum data module is used for acquiring zero padding method frequency spectrum data of the high-resolution image according to the low-frequency spectrum data of the high-resolution image;
the zero padding method image module is used for carrying out Fourier transform on the zero padding method frequency spectrum data to obtain a low-frequency spectrum data zero padding method image of the high-resolution image;
and the singular spectrum function module is used for acquiring an optimal singular operator according to the low-frequency spectrum data zero padding method image, acquiring a singular function according to the optimal singular operator, and acquiring a singular spectrum function according to the singular function.
Further, in the above system, the low frequency spectrum data module is configured to represent a low resolution image as gl(i, j), i, j being 0, 1.. times.l, the spectral data to be restored to a high resolution image is denoted G (i, j), i, j being 0, 1.. times.n, N > l, and the G (i, j) image is denoted G (k)x,ky)，kx,ky∈ Ω, Ω is the spectrum space of the high resolution image, l represents the number of horizontal or vertical pixels of the low resolution image, N represents the number of horizontal or vertical pixels of the high resolution image to be restored, the spectrum data of the low resolution image is represented as Gl(kx,ky)，Wherein,denotes glThe Fourier transform of (i, j) shows that k is more than or equal to l/2 in the low-frequency range of g (i, j) imagex,kySpectral data of < l/2 are expressed as
(N/l)2Gl(kx,ky)。
Further, in the above system, the zero-padding method spectrum data module may represent the zero-padding method spectrum data of the high resolution image as G (k)x,ky)P(kx,ky) Wherein
further, in the above system, the zero-filling image module represents the low-frequency spectrum data zero-filling image of the high-resolution image as a zero-filling imageWherein,represents G (k)x,ky)P(kx,ky) Inverse fourier transform of (d).
Further, in the above system, the singular spectral function module is configured to
Initialization:φ(i,j)＝(i,j)，wherein "x" represents a convolution, (i, j) is a two-dimensional dirac function;
note that the four basic singularities are:
φ1(i,j)＝φi,j-(i,j)＝(i,j)-(i,j-1),φ2(i,j)＝φi-,j-(i,j)＝(i,j)-(i-1,j-1)，
φ3(i,j)＝φi+,j-(i,j)＝(i,j)-(i+1,j-1),φ4(i,j)＝φi-,j(i,j)＝(i,j)-(i-1,j)；
executeJudging whether to use
If so, the method will be usedIs assigned toAnd converting phi (i, j) phiI(i, j) assigning a value of phi (i, j), and repeating said executingAnd determining whether or notA step (2);
if not, outputting the optimal singularization operator phi (i, j).
Further, in the above system, the singular spectrum function module obtains the singular function h (i, j) according to a solution of a zero state of a difference equation phi (i, j) × h (i, j) ═ i, j.
Further, in the above system, the singular spectral function module is based onAnd acquiring a singular spectral function.
Compared with the prior art, the method comprises the steps of obtaining the number of horizontal or longitudinal pixel points of a high-resolution image to be restored and a low-resolution image, and obtaining low-frequency spectrum data of the high-resolution image according to the number of the pixel points and the low-resolution image; acquiring zero padding method frequency spectrum data of the high-resolution image according to the low-frequency spectrum data of the high-resolution image; fourier transform is carried out on the zero padding method frequency spectrum data to obtain a low-frequency spectrum data zero padding method image of the high-resolution image; the method comprises the steps of obtaining an optimal singular operator according to a low-frequency spectrum data zero padding method image, obtaining a singular function according to the optimal singular operator, obtaining the singular spectral function according to the singular function, and obtaining the singular spectral function according to the singular function.
Drawings
FIG. 1a is a flow chart of high resolution image restoration according to an embodiment of the present invention;
FIG. 1b is a detailed flowchart of step S4 in FIG. 1 a;
FIG. 1c is a detailed flowchart of step S5 in FIG. 1 a;
FIG. 2 is a schematic diagram of a 256X256 low resolution image restored to a 512X512 high resolution image according to an embodiment of the present invention;
FIG. 3a is a reference image for simulation according to an embodiment of the present invention;
FIG. 3b is a diagram showing an error peak signal-to-noise ratio of an image restored by a Sinc interpolation, TV regularization and SSIT method, for a low resolution image with a cut-off frequency of 32-96 according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of an embodiment of the present invention for high resolution image restoration with a cut-off frequency of 64;
FIG. 5a is a graph showing the variation of the SNR of the peak of the restoration error with the noise level of the reference image of FIG. 3a
FIG. 5b is a graph showing the variation of the SNR of the peak of the restoration error with the noise after the noise is added;
FIG. 6a is a graph of a spectrum of a noise reference image according to an embodiment of the present invention;
FIG. 6b is a frequency spectrum diagram of an image restored by the Sinc method according to an embodiment of the present invention;
FIG. 6c is a spectral diagram of a TV method restored image according to an embodiment of the present invention;
fig. 6d is a frequency spectrum diagram of an image restored by the SSIT method according to an embodiment of the present invention;
FIG. 7a is a first reference image for testing in accordance with one embodiment of the present invention;
FIG. 7b is a second reference image for testing in accordance with one embodiment of the present invention;
FIG. 7c is a third reference image for testing in accordance with one embodiment of the present invention;
FIG. 7d is a fourth reference image for testing in accordance with one embodiment of the present invention;
FIG. 7e is a fifth reference image for testing in accordance with an embodiment of the present invention;
FIG. 7f is a sixth reference image for testing in accordance with an embodiment of the present invention;
FIG. 8a is a 128X128 low resolution image of an embodiment of the present invention;
FIG. 8b is a 256X256 image after high resolution restoration by the Sinc method according to one embodiment of the present invention;
FIG. 8c is a 256X256 image after high resolution restoration by the TV method according to one embodiment of the present invention;
FIG. 8d is a 256X256 image after high resolution restoration by the SSIT method according to one embodiment of the present invention;
FIG. 8e is a spectral diagram of FIG. 8 b;
FIG. 8f is a spectral diagram of FIG. 8 c;
FIG. 8g is a spectral diagram of 8 d;
FIG. 9 is a 512X512 image of a 128X128 low resolution image from the lower left corner that has been high resolution restored using SSTI method according to an embodiment of the present invention;
fig. 10 is a block diagram of an image acquisition system using a low-frequency spectrum data zero-filling method according to an embodiment of the present invention.
Detailed Description
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
As shown in fig. 1, the present invention provides a singular spectral function acquisition method for high resolution image restoration, including:
example one
As shown in fig. 1a, the present invention provides a singular spectral function acquisition method for high resolution image restoration, including:
step S1, acquiring the number of horizontal or vertical pixel points of the high-resolution image to be restored and a low-resolution image, and acquiring the low-frequency spectrum data of the high-resolution image according to the number of the pixel points and the low-resolution image.
Preferably, in step S1, a low resolution image g is obtainedl(i, j), i, j is 0, 1.. times, l is to be restored to a high resolution image G (i, j), i, j is 0, 1.. times, N > l, and the spectral data of the G (i, j) image is denoted as G (k) imagex,ky) Concretely, G (k)x,ky) May include low frequency spectral data and high frequency spectral data, kx,ky∈ Ω, Ω is the spectrum space of the high resolution image, l represents the number of horizontal or vertical pixels of the low resolution image, N represents the number of horizontal or vertical pixels of the high resolution image to be restored, the spectrum data of the low resolution image is represented as Gl(kx,ky)，Wherein,denotes glThe Fourier transform of (i, j) shows that k is more than or equal to l/2 in the low-frequency range of g (i, j) imagex,kySpectral data of < l/2 are expressed as
(N/l)2Gl(kx,ky) Specifically, the number of horizontal or vertical pixels of each image is equal, and the number of pixels of each image is the number of horizontal pixels multiplied by the number of vertical pixels, such as 256X256 and 512X512, i.e. l2Or N2。
And step S2, acquiring zero padding method spectrum data of the high-resolution image according to the low-frequency spectrum data of the high-resolution image.
Preferably, in step S2, the zero-padding method spectrum data of the high resolution image is represented as G (k)x,ky)P(kx,ky) Wherein
specifically, as shown in fig. 2, (b) 512 × 512 low-frequency spectrum data in the zero-padding spectrum data of the high-resolution image to be restored is derived from the 256 × 256 low-resolution image spectrum data in fig. 2 (a), and the high-frequency spectrum data portion in the zero-padding spectrum data of the high-resolution image is zero-padded zero-padding spectrum data.
And step S3, performing Fourier transform on the zero padding method frequency spectrum data to obtain a low-frequency spectrum data zero padding method image of the high-resolution image.
Preferably, in step S3, the low-frequency spectrum data zero-filling method image of the high-resolution image is represented asWherein,represents G (k)x,ky)P(kx,ky) Inverse fourier transform of (d).
And step S4, acquiring an optimal singular operator according to the low-frequency spectrum data zero padding method image, acquiring a singular function according to the optimal singular operator, and acquiring a singular spectrum function according to the singular function. Specifically, each image has its optimal singularization operator, which is unchanged even if the image is changed to a low-resolution image. The singular operator determines a singular spectral function, and the optimal singular operator can obtain the simplest singular information mathematical model of the image.
Preferably, as shown in fig. 1b, in step S4, the step of obtaining an optimal singularization operator according to the low-frequency spectrum data zero padding method image includes:
step S41, initializing:
φ(i,j)＝(i,j)，wherein "+ represents a convolution, (i, j) is a two-dimensional dirac function, as shown in fig. 2, and (c) is an image of the low-frequency spectrum data zero-filling image convolved by the optimal singular operator
Step S42, noting that the four basic singular operators are:
φ1(i,j)＝φi,j-(i,j)＝(i,j)-(i,j-1),φ2(i,j)＝φi-,j-(i,j)＝(i,j)-(i-1,j-1)，
φ3(i,j)＝φi+,j-(i,j)＝(i,j)-(i+1,j-1),φ4(i,j)＝φi-,j(i,j)＝(i,j)-(i-1,j)；
step S43, executeJudging whether to useIf yes, go to step S44, if no, thenGo to step S45.
Step S44, willIs assigned toNamely, it is And converting phi (i, j) phiI(i, j) is assigned to phi (i, j), i.eThereafter, the process proceeds to step S43.
And step S45, outputting the optimal singularization operator phi (i, j).
Preferably, in step S4, the step of obtaining the singular function according to the optimal dissimilarity operator includes:
the singular function h (i, j) is obtained from a solution of the difference equation phi (i, j) × h (i, j) ═ zero state (i, j), specifically, "+" represents convolution, (i, j) is a two-dimensional dirac function, and if the optimal singular operator is regarded as a system, the singular function h (i, j) is a unit impulse response of the optimal singular operator phi (i, j).
Preferably, in step S4, in the step of obtaining the singular spectrum function according to the singular function, the singular spectrum function isSpecifically, a singular spectral function H (k) is generated according to the optimal singular operators phi (i, j)x,ky) Make the parameters of the mathematical model of the singular information as small as possible
Wherein H (k)x,ky) Called singular spectral function, (a)c,ic,jc) Q is a parameter of the model to be determined, and q is less than N2The information quantity is required to be as small as possible; omega is the spectrum space of the high-resolution image, the singular function h (i, j) is the primary function of the singular spectrum function,
in the subsequent high-resolution image restoration method (singular information theory spectrum extension method, SSIT), step S5-step S7 may be performed according to a result obtained by the singular spectral function acquisition method for high-resolution image restoration.
And step S5, acquiring the coordinate parameters of the high-resolution image restoration by using a point spread function chromatography according to the optimal singularization operator. In particular, the point spread function is defined asWherein,where k isx,kyRepresenting spectral spatial points that can be estimated from the low resolution image.
Preferably, as shown in fig. 1c, step S5 includes:
step S51, initializing: c is equal to 1, and c is equal to 1,
step S52, calculating:will be provided withIs assigned toNamely, it isc ═ c +1, where (i)c,jc) Represents the coordinate parameters, c 1,2, q, the coordinate parameters being non-zero coordinates, the set of coordinate parameters being χ { (i)1,j1),(i2,j2)...,(iq,jq)}；
Step S53, determine whether toWherein | · | purple sweet2Indicating a quadratic norm, if so, go to step S52, otherwise, go toGo to step S54;
step S54, q being c, outputting the coordinate parameter { (i)c,jc),c＝1,2,...,q}。
And step S6, acquiring the weight parameters for restoring the high-resolution image according to the singular spectrum function and the coordinate parameters.
Preferably, in step S6, a singular information mathematical model is constructed according to the analytic extension theorem
Wherein e is 2.718281828459;
method for obtaining weighting parameter a of high-resolution image restoration by using pseudo-inverse matrix methodcQ, in particular, the weighting parameter acQ is a functionIs non-zero.
Step S7, obtaining the high frequency spectrum data of the high resolution image according to the weighting parameters and the singular spectrum function, obtaining the complete spectrum data according to the low frequency spectrum data and the high frequency spectrum data of the high resolution image, and outputting the high resolution image according to the complete spectrum data.
Preferably, in step S7, the step of obtaining the high frequency spectrum data of the high resolution image according to the weighting parameter and the singular spectral function, and the step of obtaining the complete spectrum data according to the low frequency spectrum data and the high frequency spectrum data of the high resolution image includes:
mathematical model based on singular informationExtending the high frequency spectrum data of the high resolution image, specifically, in FIG. 2, (d) obtaining the coordinate parameter (i) of the singular information by point spread function chromatographyc,jc) And a singular information graph obtained by solving the singular information mathematical model;
acquiring the complete spectrum data G (k) from the low frequency spectrum data and the high frequency spectrum data of the high resolution imagex,ky). Specifically, (e) in fig. 2 is an image of the complete spectrum data.
Preference is given toIn the step of outputting the high-resolution image based on the complete spectrum data in step S7, the high-resolution image is output based on the complete spectrum data G (k)x,ky) Outputting the high resolution image g (i, j),specifically, (f) in fig. 2 is a restored 512 × 512 high-resolution image.
In more detail, in order to verify the effectiveness of the high-resolution image restoration method, namely the singular information theory spectrum continuation method (SSIT), the method is researched by using a simulation experiment to determine the effectiveness of the method. The simulation experiment scheme is as follows: according to a mechanism of losing high-frequency spectrum components to cause a low-resolution image, high-frequency spectrum data outside cut-off frequency of the high-resolution image is removed to obtain the low-resolution image, and then a sinc interpolation method, a TV regularization method and the method (SSIT) of the invention are used for restoring the high-resolution image. The simulation experiment is as follows:
experiment I, the influence of the cut-off frequency on the algorithm is examined.
The remote sensing image with the size of 256X256 and the gray scale range (0-255) of the Shanghai Rainbow bridge airport in China as shown in FIG. 3a is used as a reference image, the cut-off frequency range is taken as (32-96), a low-resolution image with the size of 64X 64-192X 192 is generated, and high-resolution restoration is carried out by a sinc interpolation method, a TV regularization method and an SSIT method respectively. The error peak signal-to-noise ratio between each restored image and the reference image is shown in fig. 3 b. In general, the error peak signal-to-noise ratio (PSNR) of each method is improved along with the improvement of the Cut-off Frequency (Cut Frequency), and the image precision of the super-resolution restoration of the SSIT method is higher than that of the sinc and TV methods under various Cut-off frequencies. The SSIT method has cut-off frequency of about 45, and PSNR value reaches more than 30 dB.
Fig. 4 is a schematic diagram of an experiment for restoring a high-resolution image with a cutoff frequency of 64 according to an embodiment of the present invention, where when the cutoff frequency is 64, a 128X128 low-resolution image is shown in fig. 4 (a) as the 128X128 low-resolution image, and images restored by Sinc, TV, and SSIT methods are shown in fig. 4 (b) as the Sinc method restored image, (c) as the TV method restored image, and (d) as the SSIT method restored image, and their error peak signal-to-noise ratios are 32.2 db, 32.8 db, and 33.6 db, respectively. Although the PSNR is not much different, the image difference is significant. The images of the Sinc method are relatively blurred, and truncation artifacts can be clearly found beside two airplanes. The images of the TV method have a small amount of artifacts. The image of the SSIT method is closest to the reference image and has almost no artifacts. The difference between the error images of the three methods and the reference image can be seen more easily, as shown in fig. 4 (e) which is the error graph of fig. 4 (b) and fig. 3a, (f) which is the error graph of fig. 4 (c) and fig. 3a, and (g) which is the error graph of fig. 4 (d) and fig. 3a, the Sinc method has significant artifacts, the TV method is second to the first, the SSIT method has the least artifacts, and the displayed error ranges are all-31% to-31%.
Experiment two, investigating the influence of noise on the restoration algorithm
In order to examine the influence of noise on the high resolution restoration precision of the method, zero-mean white gaussian noise is added to a reference image 3a, the mean square deviations are respectively 1-10, a low resolution 128X128 image with a cut-off frequency of 64 is taken, high resolution restoration is carried out by respectively using Sinc interpolation, TV regularization and SSIT methods, restoration error peak-to-noise ratios are respectively calculated by using (a) in FIG. 2 and the image after noise addition as the reference image, and the results are shown in FIGS. 5a and 5 b. The PSNR values of the three methods all decrease with the increase of Noise (Noise Level), and the PSNR decreases more slowly than that of the image with reference to (a) in fig. 2, that is, the restored image is closer to the image before Noise addition, which means that the three methods all have the effect of removing Noise. PSNR decreases with increasing noise, meaning that the noise destroys the details of the image, affecting the accuracy of restoration by the three methods. Under the condition of noise at each level, the PSNR of the SSIT method is always higher than that of the Sinc and TV methods, and the recovery precision of the SSIT method is superior to that of the Sinc and TV methods in view of quantitative index PSNR.
It can also be seen from the image spectrogram that the SSIT method has a higher restoration accuracy than the Sinc and TV methods. Fig. 6a, 6b, 6c, and 6d are a spectrogram of a noise-added reference image, a spectrogram of a Sinc-method restored image, a spectrogram of a TV-method restored image, and a spectrogram of an SSIT-method restored image, respectively, under 10-level noise. In fig. 6b, the high frequency components other than the cut-off frequency are not observed in the spectrogram of the image restored by the Sinc method, which illustrates that the Sinc interpolation restores the image with the size of 256 × 256, and no high frequency information, i.e., image details, is added to the image. In fig. 6c, the spectrogram of the TV restored image has partial spectral components other than the cutoff frequency, but the spectral components other than the cutoff frequency are smaller than those of the spectrogram of the reference image, and the error is large. In fig. 6d, the spectrogram of the SSIT restored is quite close to the spectrogram of the reference image in both the spectrum outside the cutoff frequency and the spectrum inside the cutoff frequency, and it is explained that the corresponding image is also close to the reference image. This also illustrates from a spectral perspective that the accuracy of the SSIT method is higher than that of the TV and Sinc methods.
And thirdly, examining the influence of the image structure on the algorithm.
To examine the influence of the image structure and the gray distribution characteristics on the restoration algorithm, we select fig. 7a, 7b, 7c, 7d, 7e and 7f as reference images, all with a size of 256X256, and generate low-resolution 128X128 images with a cutoff frequency of 64, then perform high-resolution restoration by using sinc interpolation, TV regularization and SSIT methods, and calculate error peak signal-to-noise ratios PSNR of the restored images and fig. 7a, 7b, 7c, 7d, 7e and 7f, respectively, and the results are shown in the following table:
| image sequence number | a | b | c | d | e | f |
| Sinc | 34.0 | 33.4 | 30.1 | 32.2 | 29.1 | 27.1 |
| TV | 34.4 | 34.3 | 29.9 | 32.8 | 29.3 | 27.8 |
| SSIT | 34.7 | 35.5 | 31.4 | 33.6 | 29.7 | 28.2 |
According to the table, it can be found that the Sinc method does not have the capability of restoring the high frequency component, and the Sinc method is used for obtaining the image with high PSNR, which shows that the image has less high frequency component. It can be seen from the above table that the high frequency components in the six reference images sequentially increase in the order of 7a, 7b, 7d, 7c, 7e, and 7f. The TV method improves PSNR and improves image quality by restoring the reference images of 7a, 7b, 7d, 7e and 7f, but reduces PSNR by restoring the reference image of 7c, which shows that the TV method is not as good as Sinc interpolation for images of certain structures. The SSIT method can better recover the high-frequency components of the image under various image structure conditions.
Experiment four, actual high resolution recovery experiment
Through simulation experiments of the experiment I, the experiment II and the experiment III, the SSIT method can restore the high-frequency spectrum. In this experiment, two 128X128 images, such as the lower left corner and 8a of fig. 9, are taken from fig. 3a), the spectra are used as low-frequency spectrum data with a cutoff frequency of 64 for 256 restored images, and then the images are restored to 256X256 images by Sinc, TV and SSIT methods, respectively. The restored image and its spectrogram are shown in fig. 9, except for the lower left corner, and fig. 8b, 8c, 8d, 8e, 8f, and 8g, respectively. As shown in fig. 8b, the image restored by the Sinc method has artifacts, as shown in fig. 8c, the image restored by the TV method also has few artifacts, and as shown in fig. 8d, the image restored by the SSIT method has few artifacts. This is consistent with the results of simulation experiment one. From the spectral diagrams 8e, 8f and 8g of the three methods, the image of the Sinc method only has a low-frequency spectrum below the cut-off frequency, the image of the TV method has a few spectra beyond the cut-off frequency, the image of the SSIT method only has a relatively rich spectrum beyond the cut-off frequency, the spectral distribution within the cut-off frequency is morphologically consistent, and the spectrum beyond the cut-off frequency is an extension of the spectrum within the cut-off frequency, which indicates that the SSIT method really recovers the high-frequency components beyond the cut-off frequency. Similar results can also be found in fig. 9. The experimental result shows that the low-resolution image frequency spectrum can be regarded as the low-frequency spectrum of the high-resolution image, and the high-frequency spectrum of the high-resolution image can be extended by using the low-frequency spectrum, so that the restoration of the hyper-resolution image is achieved.
SSIT is a new efficient high resolution restoration method. The experimental results show that: the recovery accuracy of the SSIT method is higher than that of the Sinc and TV methods in the recovery images under various cut-off frequencies, various noises and various image structures. However, as in the Sinc and TV methods, the SSIT reduces the restoration accuracy as the loss of high-frequency information in the low-resolution image increases, and reduces the restoration accuracy as the noise in the low-resolution image increases. The former is because high-frequency information other than the cut-off frequency is lost too much, and the latter is because noise destroys the high-frequency information of the image, so that the singular information of the image is difficult to detect accurately, and the accuracy of the image restored by the SSIT method is reduced. The experimental result shows that the low-resolution image frequency spectrum can be regarded as the low-frequency spectrum of the high-resolution image, and the high-frequency spectrum of the high-resolution image can be extended by using the low-frequency spectrum, so that the restoration of the hyper-resolution image is achieved.
Example two
As shown in fig. 10, the present invention further provides another singular spectral function acquisition system for high resolution image restoration, which includes a low frequency spectrum data module 1, a zero padding method spectrum data module 2, a zero padding method image module 3, and a singular spectral function module 4.
The low-frequency spectrum data module 1 is used for acquiring the number of horizontal or longitudinal pixel points of a high-resolution image to be restored and a low-resolution image, and acquiring the low-frequency spectrum data of the high-resolution image according to the number of the pixel points and the low-resolution image.
Preferably, the low-frequency spectrum data module 1 is configured to represent a low-resolution image as gl(i, j), i, j being 0, 1.. times.l, the spectral data to be restored to a high resolution image is denoted G (i, j), i, j being 0, 1.. times.n, N > l, and the G (i, j) image is denoted G (k)x,ky)，kx,ky∈ Ω, Ω is the spectrum space of the high resolution image, l represents the number of horizontal or vertical pixels of the low resolution image, N represents the number of horizontal or vertical pixels of the high resolution image to be restored, the spectrum data of the low resolution image is represented as Gl(kx,ky)，Wherein,denotes gl(i, j) ofFourier transform, so that the low-frequency range-l/2 of the g (i, j) image is not less than kx,kySpectral data of < l/2 are expressed as
(N/l)2Gl(kx,ky)。
Preferably, the zero padding method spectrum data module 2 is configured to obtain zero padding method spectrum data of the high-resolution image according to the low-frequency spectrum data of the high-resolution image.
The zero-padding method spectrum data module represents the zero-padding method spectrum data of the high-resolution image as G (k)x,ky)P(kx,ky) Wherein
and the zero padding method image module 3 is used for performing Fourier transform on the zero padding method frequency spectrum data to obtain a low-frequency spectrum data zero padding method image of the high-resolution image.
Preferably, the zero-padding image module 3 represents the low-frequency spectrum data zero-padding image of the high-resolution image as a zero-padding imageWherein,represents G (k)x,ky)P(kx,ky) Inverse fourier transform of (d).
And the singular spectrum function module 4 is used for acquiring an optimal singular operator according to the low-frequency spectrum data zero padding method image, acquiring a singular function according to the optimal singular operator, and acquiring a singular spectrum function according to the singular function.
Preferably, the singular spectral function module 4 is used for
Initialization:φ(i,j)＝(i,j)，wherein "x" represents a convolution, (i, j) is a two-dimensional dirac function;
note that the four basic singularities are:
φ1(i,j)＝φi,j-(i,j)＝(i,j)-(i,j-1),φ2(i,j)＝φi-,j-(i,j)＝(i,j)-(i-1,j-1)，
φ3(i,j)＝φi+,j-(i,j)＝(i,j)-(i+1,j-1),φ4(i,j)＝φi-,j(i,j)＝(i,j)-(i-1,j)；
executeJudging whether to use
If so, the method will be usedIs assigned toAnd converting phi (i, j) phiI(i, j) assigning a value of phi (i, j), and repeating said executingAnd determining whether or notA step (2);
if not, outputting the optimal singularization operator phi (i, j).
Preferably, the singular spectrum function module 4 obtains the singular function h (i, j) according to a solution of the difference equation phi (i, j) × h (i, j) ═ zero state.
Preferably, the singular spectral function module 4 is based onAnd acquiring a singular spectral function.
And a subsequent coordinate parameter module 5 can obtain the restored coordinate parameters of the high-resolution image by using a point spread function chromatography according to the optimal singularity operator obtained by the singular spectrum function module 4.
Preferably, the coordinate parameter module 5 is used for
Initialization: c is equal to 1, and c is equal to 1,
and (3) calculating:will be provided withIs assigned toc ═ c +1, where (i)c,jc) Representing the coordinate parameters, c ═ 1, 2.., q, the coordinate parameters being non-zero coordinates;
judging whether to useWherein | · | purple sweet2The second-order norm is expressed,
if yes, repeating the calculating step;
if not, q is equal to c, and the coordinate parameter { (i) is outputc,jc),c＝1,2,...,q}。
And the weighting parameter module 6 is used for acquiring a weighting parameter for restoring the high-resolution image according to the singular spectral function and the coordinate parameter.
Preferably, the weighting parameter module 6 is configured to construct a singular information mathematical model according to the analytic extension theoremWherein e is 2.718281828459;
method for obtaining weighting parameter a of high-resolution image restoration by using pseudo-inverse matrix methodc,c＝1,2,...,q。
And the restoration module 7 is configured to obtain high-frequency spectrum data of the high-resolution image according to the weighting parameter and the singular spectrum function, obtain complete spectrum data according to the low-frequency spectrum data and the high-frequency spectrum data of the high-resolution image, and output the high-resolution image according to the complete spectrum data.
Preferably, the recovery module 7 is configured to mathematically model the singular information
Extending the high frequency spectrum data of the high resolution image;
acquiring the complete spectrum data G (k) from the low frequency spectrum data and the high frequency spectrum data of the high resolution imagex,ky)；
From the full spectrum data G (k)x,ky) Outputting the high resolution image g (i, j),
details of the second embodiment can specifically refer to corresponding parts in the first embodiment.
The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other. For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.
Those of skill would further appreciate that the various illustrative elements and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both, and that the various illustrative components and steps have been described above generally in terms of their functionality in order to clearly illustrate this interchangeability of hardware and software. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the implementation. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention.
It will be apparent to those skilled in the art that various changes and modifications may be made in the invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.
The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other. For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.
Those of skill would further appreciate that the various illustrative elements and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both, and that the various illustrative components and steps have been described above generally in terms of their functionality in order to clearly illustrate this interchangeability of hardware and software. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the implementation. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention.
It will be apparent to those skilled in the art that various changes and modifications may be made in the invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.
Claims (14)
1. A singular spectral function acquisition method for high-resolution image restoration is characterized by comprising the following steps:
acquiring the number of horizontal or longitudinal pixel points of a high-resolution image to be restored and a low-resolution image, and acquiring low-frequency spectrum data of the high-resolution image according to the number of the pixel points and the low-resolution image;
acquiring zero padding method frequency spectrum data of the high-resolution image according to the low-frequency spectrum data of the high-resolution image;
fourier transform is carried out on the zero padding method frequency spectrum data to obtain a low-frequency spectrum data zero padding method image of the high-resolution image;
and acquiring an optimal singular operator according to the low-frequency spectrum data zero padding method image, acquiring a singular function according to the optimal singular operator, and acquiring a singular spectrum function according to the singular function.
2. The singular spectral function acquisition method for high resolution image restoration according to claim 1, wherein in the step of acquiring the number of horizontal or vertical pixel points of the high resolution image to be restored and a low resolution image, and acquiring the low frequency spectrum data of the high resolution image based on the number of pixel points and the low resolution image,
a low resolution image gl(i, j), i, j being 0, 1.. and l being to be restored to the high resolution image g (i, j), i, j being 0, 1.. and N, N>>The spectral data of the l, G (i, j) image is denoted as G (k)x,ky)，kx,ky∈ Ω, Ω is the spectrum space of the high resolution image, l represents the number of horizontal or vertical pixels of the low resolution image, N represents the number of horizontal or vertical pixels of the high resolution image to be restored, the spectrum data of the low resolution image is represented as Gl(kx,ky)，Wherein,denotes glThe Fourier transform of (i, j) shows that k is more than or equal to l/2 in the low-frequency range of g (i, j) imagex,ky<The spectrum data of l/2 is expressed as
(N/l)2Gl(kx,ky)。
3. The singular spectral function acquisition method for high resolution image restoration according to claim 2, wherein in the step of acquiring zero-padding method spectral data of the high resolution image based on the low frequency spectral data of the high resolution image,
the zero-filling method frequency spectrum data of the high-resolution image is represented as G (k)x,ky)P(kx,ky) Wherein
4. the singular spectral function acquisition method for high resolution image restoration according to claim 3, wherein in the step of Fourier transforming the zero-padding method spectral data to acquire the zero-padding method image of the low frequency spectral data of the high resolution image,
the low-frequency spectrum data zero-filling method image of the high-resolution image is represented asWherein,represents G (k)x,ky)P(kx,ky) Inverse fourier transform of (d).
5. The singular spectral function acquisition method for high resolution image restoration according to claim 4, wherein the step of acquiring an optimal singularization operator from the low frequency spectrum data zero padding method image comprises:
initialization:phi (i, j) is (i, j), wherein,wherein "x" represents a convolution, (i, j) is a two-dimensional dirac function;
note that the four basic singularities are:
φ1(i,j)＝φi,j-(i,j)＝(i,j)-(i,j-1),φ2(i,j)＝φi-,j-(i,j)＝(i,j)-(i-1,j-1)，
φ3(i,j)＝φi+,j-(i,j)＝(i,j)-(i+1,j-1),φ4(i,j)＝φi-,j(i,j)＝(i,j)-(i-1,j)；
executeJudging whether to use
If so, the method will be usedIs assigned toAnd converting phi (i, j) phiI(i, j) assigning a value of phi (i, j), and repeating said executingAnd determining whether or notA step (2);
if not, outputting the optimal singularization operator phi (i, j).
6. The singular spectral function acquisition method for high resolution image restoration according to claim 5, wherein the step of acquiring a singular function according to the optimal dissimilarity operator comprises:
the singular function h (i, j) is obtained from a solution of the zero state of the difference equation phi (i, j) × h (i, j) ═ i, j.
7. The singular spectrum function acquisition method for high resolution image restoration according to claim 6, wherein a singular spectrum is acquired from the singular functionIn the step of function, the singular spectrum function is
8. A singular spectral function acquisition system for high resolution image restoration, comprising:
the low-frequency spectrum data module is used for acquiring the number of transverse or longitudinal pixel points of a high-resolution image to be restored and a low-resolution image, and acquiring the low-frequency spectrum data of the high-resolution image according to the number of the pixel points and the low-resolution image;
the zero padding method frequency spectrum data module is used for acquiring zero padding method frequency spectrum data of the high-resolution image according to the low-frequency spectrum data of the high-resolution image;
the zero padding method image module is used for carrying out Fourier transform on the zero padding method frequency spectrum data to obtain a low-frequency spectrum data zero padding method image of the high-resolution image;
and the singular spectrum function module is used for acquiring an optimal singular operator according to the low-frequency spectrum data zero padding method image, acquiring a singular function according to the optimal singular operator, and acquiring a singular spectrum function according to the singular function.
9. The singular spectral function acquisition system for high resolution image restoration according to claim 8, wherein said low frequency spectral data module is configured to represent a low resolution image as gl(i, j), i, j being 0, 1.. times.l, and the high resolution image to be restored is represented as g (i, j), i, j being 0, 1.. times.n, N>>The spectral data of the l, G (i, j) image is denoted as G (k)x,ky)，kx,ky∈ Ω, Ω is the spectrum space of the high resolution image, l represents the number of horizontal or vertical pixels of the low resolution image, N represents the number of horizontal or vertical pixels of the high resolution image to be restored, the spectrum data of the low resolution image is represented as Gl(kx,ky)，Wherein,denotes glThe Fourier transform of (i, j) shows that k is more than or equal to l/2 in the low-frequency range of g (i, j) imagex,ky<The spectrum data of l/2 is expressed as
(N/l)2Gl(kx,ky)。
10. The singular spectral function acquisition system for high resolution image restoration according to claim 9, wherein the zero-padding spectral data module represents the zero-padding spectral data of the high resolution image as G (k)x,ky)P(kx,ky) Wherein
11. the singular spectral function acquisition system for high resolution image restoration according to claim 10, wherein the zeroing image module represents the low frequency spectral data zeroing image of the high resolution image as a zero-filling imageWherein,represents G (k)x,ky)P(kx,ky) Inverse fourier transform of (d).
12. The singular spectral function acquisition system for high resolution image restoration according to claim 11, wherein said singular spectral function module is configured to perform said singular spectral function acquisition
Initialization:φ(i,j)＝(i,j)，wherein "x" represents a convolution, (i, j) is a two-dimensional dirac function;
note that the four basic singularities are:
φ1(i,j)＝φi,j-(i,j)＝(i,j)-(i,j-1),φ2(i,j)＝φi-,j-(i,j)＝(i,j)-(i-1,j-1)，
φ3(i,j)＝φi+,j-(i,j)＝(i,j)-(i+1,j-1),φ4(i,j)＝φi-,j(i,j)＝(i,j)-(i-1,j)；
executeJudging whether to use
If so, the method will be usedIs assigned toAnd converting phi (i, j) phiI(i, j) assigning a value of phi (i, j), and repeating said executingAnd determining whether or not
If not, outputting the optimal singularization operator phi (i, j).
13. The singular spectral function acquisition system for high resolution image reconstruction as claimed in claim 12, wherein said singular spectral function module acquires the singular function h (i, j) from a solution of a zero state of a difference equation Φ (i, j) × h (i, j) ═ i, j.
14. The singular spectral function acquisition system for high resolution image restoration according to claim 13, wherein the singular spectral function module is based onAnd acquiring a singular spectral function.
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