CN111786656B - Band-limited map signal sampling method based on filter - Google Patents

Band-limited map signal sampling method based on filter Download PDF

Info

Publication number
CN111786656B
CN111786656B CN202010474923.6A CN202010474923A CN111786656B CN 111786656 B CN111786656 B CN 111786656B CN 202010474923 A CN202010474923 A CN 202010474923A CN 111786656 B CN111786656 B CN 111786656B
Authority
CN
China
Prior art keywords
signal
sampling
graph
filter
space
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN202010474923.6A
Other languages
Chinese (zh)
Other versions
CN111786656A (en
Inventor
王保云
楚帆
李沛
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Nanjing University of Posts and Telecommunications
Original Assignee
Nanjing University of Posts and Telecommunications
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Nanjing University of Posts and Telecommunications filed Critical Nanjing University of Posts and Telecommunications
Priority to CN202010474923.6A priority Critical patent/CN111786656B/en
Publication of CN111786656A publication Critical patent/CN111786656A/en
Application granted granted Critical
Publication of CN111786656B publication Critical patent/CN111786656B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H17/00—Networks using digital techniques
    • H03H17/02—Frequency selective networks

Landscapes

  • Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • Computer Hardware Design (AREA)
  • Mathematical Physics (AREA)
  • Image Processing (AREA)

Abstract

The invention discloses a band-limited graph signal sampling method based on a filter, which comprises the following steps of firstly, establishing a mathematical model of graph signal sampling; then, the conditions that enable the reconstruction of the original signal by sampling the signal are determined:when the graph signal is a band-limited signal and the dimensionality of the sampling set S is not less than the signal bandwidth, reconstructing an original signal from the sampling signal; secondly, the space composed of all band-limited map signals with the same bandwidth is called Peley-wiener space PW ω (G) To PW ω (G) Reducing the computational complexity of the image signals in the space by using a filter-based method, and determining a sampling set; finally, a method for reconstructing the original signal from the sampled signal is determined: and (4) reconstruction is the inverse process of sampling, and a low-complexity reconstruction method based on a filter is obtained according to the step (3). The invention reduces the calculation complexity while ensuring equivalent or better reconstruction errors.

Description

Band-limited map signal sampling method based on filter
Technical Field
The invention belongs to the technical field of signal processing, and particularly relates to a band-limited graph signal sampling method based on a filter.
Background
With the rapid development of information and communication technology, more and more irregular structured signals are generated in people's production and life. Since these signals are not conventional time signals, they cannot be processed directly by classical signal processing techniques. In this scenario, graph Signal Processing (GSP) is a very effective technique for analyzing and Processing these complex signals.
In the GSP field, sampling is a very important technique that can reduce the signal transmission and storage dimensions, which is the basis for most other applications. The basic criterion for sampling is that the original signal can be recovered from a subset of its samples. At present, there are various researches on the sampling of the map signal, such as the relevant conditions for reconstructing the map signal, the cut-off frequency of the map signal, the sampling strategy based on the spectrum agent, and so on. Under Graph Fourier Transform (GFT), if the Graph signal is band-limited, i.e., the Fourier coefficients are not all non-zero terms, then the original signal can be perfectly reconstructed from a subset of samples. However, this sampling reconstruction method requires a large amount of computation for calculating the eigen decomposition of the graph shift matrix (usually, the adjacency matrix or laplacian matrix of the graph). In order to reduce the computational complexity of the sampling process, we propose a filter-based bandlimited map signal sampling method.
Disclosure of Invention
The invention aims to: the invention aims to solve the problem of high calculation cost in the process of sampling a diagram signal, provides a band-limited diagram signal sampling method based on a filter, and can effectively solve the problems of large calculation amount and complex calculation in the prior sampling technology.
The invention content is as follows: the invention provides a band-limited map signal sampling method based on a filter, which comprises the following steps:
(1) Establishing a mathematical model of the graph signal samples: mapping the complex signals with irregular structures to graph vertexes, and representing the relationship between the signals by using the topological structure of the graph;
(2) Determining the conditions under which the original signal can be reconstructed from the sampled signal: when the graph signal is a band-limited signal and the dimensionality of the sampling set S is not less than the signal bandwidth, reconstructing an original signal from the sampling signal;
(3) Analyzing the sampling strategy according to reconstruction conditions, and selecting a sampling set based on a filter: the space composed of all band-limited map signals with the same bandwidth is called Peley-wiener space PW ω (G) To PW ω (G) Reducing the computational complexity of the image signals in the space by using a filter-based method, and determining a sampling set;
(4) Determining a method for reconstructing an original signal from a sampled signal: and (4) reconstruction is the inverse process of sampling, and a low-complexity reconstruction method based on a filter is obtained according to the step (3).
Further, the mathematical model construction process of the map signal sampling in step (1) is as follows:
g = (v, epsilon, W) represents a graph where v = { ν = 1 ,ν 2 ,…,ν N Denotes the set of vertices, ε is the set of edges; w is the adjacency matrix of the graph, element W i,j The relationship between graph vertices i and j is represented; the graph signal is the function f: v →, defined over the set of graph vertices; the sampling vertex set is S; defining a sampling matrix Ψ ∈ m×n Sampled signal f obtained by sampling S Can be expressed as f S = Ψ f; the corresponding reconstructed signal is represented as
Figure BDA0002515572320000021
Where Φ is the reconstruction operator.
Further, the conditions for reconstructing the original signal in step (2) are as follows:
rebuilding Peltier-Venu space PW ω (G) In space PW ω (G) In (1), for all f, g ∈ PW ω (G) If f (S) = g (S) stands for f = g, then the subset S of graph vertices is one unique set.
Further, the step (3) comprises the steps of:
(31) Analyzing the relationship between the unique set and the Peley-Venu space
Figure BDA0002515572320000022
Represents PW ω (G) The complementary space of (2):
in the PW ω (G) In (1), the essential condition that S is the only set is that P (S) > 0,
Figure BDA0002515572320000023
is a high-pass filter whose frequency response is
Figure BDA0002515572320000024
Sets meeting the condition that P (S) > 0 are all unique sets, the unique set with the minimum dimension is the optimal unique set, and the following problem is converted to solve the optimal unique set:
Figure BDA0002515572320000025
s.t.P(S)＞0
wherein P (S) can be represented by the equation
Figure BDA0002515572320000031
The method comprises the steps of (1) obtaining,
Figure BDA0002515572320000032
to represent
Figure BDA0002515572320000033
The minimum eigenvalue of (d);
(34) The ideal filter is approximated through a matrix polynomial, and the minimum eigenvalue is estimated through the Geuer disk theorem to reduce the calculation complexity;
(35) And finally, obtaining an optimal sampling set through a greedy algorithm.
Further, the step (4) is realized as follows:
space PW ω (G) Signal of (1) satisfies Tf =0, translating the recovered signal into the following problem:
Figure BDA0002515572320000034
therein, Ψ l ＝Ψ(I-T)；
The solution to the problem can be expressed as:
Figure BDA0002515572320000035
has the advantages that: compared with the prior art, the invention has the beneficial effects that: the invention provides a sampling strategy based on a filter, and a Chebyshev matrix polynomial is used for approximating an ideal filter, so that the calculation complexity is reduced; the lower limit of the minimum characteristic value of the matrix is estimated by using the Geuer disc theorem, so that the complexity is further reduced; the reconstruction method with low complexity is obtained by using the approximate filter, and the calculation complexity is lower while the reconstruction error is equal to or better.
Drawings
FIG. 1 is a flow chart of an algorithm for filter-based sampling;
FIG. 2 is a flow chart of a reconstruction algorithm for reconstructing an original signal from a sampled signal;
FIG. 3 is a graph comparing the mean square error performance of the method of the present invention with other sampling reconstruction algorithms.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings. The invention provides a band-limited map signal sampling method based on a filter, which specifically comprises the following steps:
step 1: a mathematical model of the map signal samples is established.
G = (v, epsilon, W) represents a graph where v = { ν = 1 ,ν 2 ,…,ν N Denotes the set of vertices and ε is the set of edges. W is the adjacency matrix of the graph, whose element W i,j The relationship between graph vertices i and j is shown. The graph signal is a function f: v → defined on the set of graph vertices. For ease of explanation, consider the following undirected graph. Using a symmetric normalized Laplace matrix L = I-W -1/2 DW -1/2 As a change operator, where D = diag { D } 1 ,d 2 ,…,d n Is the degree matrix, the jth diagonal element d j ＝∑ j W i,j . Note that L is diagonalizable, so there is one orthonormal matrix U and one diagonal matrix Λ that satisfy L = U Λ U T 。
For the graph signal f e N Its Fourier transform of the Graph (GFT) is defined as
Figure BDA0002515572320000041
When a K E {1,2, \8230;, N } exists, the Fourier transform of the graph signal is satisfied
Figure BDA0002515572320000042
For all K ≧ K, then the graph signal is a band-limited graph signal, and the minimum K is the bandwidth of the graph signal f. The space composed of K bandlimited map signals is called the Peley-Wiener (Paley-Wiener) space, denoted as
PW ω (G)＝span(u i ,λ i ≤ω) (1)
For PW ω (G) And finding a sampling vertex set S. First, a sampling matrix psi ∈ is defined m×n The elements are as follows:
Figure BDA0002515572320000043
sampling signal f obtained by sampling S Can be expressed as f S = Ψ f. The corresponding reconstructed signal is represented as
Figure BDA0002515572320000044
Where Φ is the reconstruction operator.
And 2, step: determining the conditions for reconstructing the original signal: when the graph signal is a band-limited signal and the dimension of the sampling set S is not less than the signal bandwidth, the original signal can be perfectly reconstructed from the sampled signal.
In space PW ω (G) In (1), g ∈ PW for all f ω (G) If f (S) = g (S) is equivalent to f = g, then the subset of graph vertices, S, is a unique set. At this time, the original signal can be restored by the signal values in the subset S. Further, in the space PW ω (G) If and only if PW ω (G)∩L 2 (S C ) S is the only set when =0.
Further analyzing the relationship between the unique set and the Peley-Venu space
Figure BDA0002515572320000045
Represents PW ω (G) The complement space of (1). At PW ω (G) The essential condition for S to be the only set is P (S) > 0, where
Figure BDA0002515572320000046
Is a high-pass filter whose frequency response is:
Figure BDA0002515572320000051
first consider the sufficiency, when S is a unique set, for any graph signal f ∈ L 2 (S C ) And f ≠ 0, satisfies
Figure BDA0002515572320000052
By calculating the high frequency components of the map signal f, tf ≠ 0, which indicates P (S) > 0. Then considering the necessity, according to the condition that S is the only set: if and only if PW ω (G)∩L 2 (S C ) =0. When S is not the unique set, there is at least one graph signal f 0 ∈L 2 (S C ) And f 0 ∈PW ω (G) I.e. Tf 0 And =0. Graph signal f 0 Substitution into the definition of P (S) yields P (S) =0.
Thus, if and only if any non-zero pattern signalFrom L 2 (S C ) Is projected to
Figure BDA0002515572320000053
When not a zero vector, S is a unique set.
And step 3: the sampling method based on the filter comprises the following steps: analyzing the sampling strategy according to reconstruction conditions, and selecting a sampling set based on a filter: the space composed of all band-limited map signals with the same bandwidth is called Peley-wiener space PW ω (G) To PW ω (G) And (3) determining a sampling set by reducing the computational complexity of the image signals in the space by using a filter-based method.
Sets S satisfying P (S) > 0 are all unique sets, wherein the unique set with the minimum dimension is defined as the optimal unique set, so the following problem is formulated to solve the optimal unique set:
Figure BDA0002515572320000054
the expression of P (S) in the above problem is not intuitive by traversing the space L 2 (S C ) It is not practical to solve for P (S) in all combinations.
For any determined S, P (S) can be obtained by the following equation.
Figure BDA0002515572320000055
Wherein σ min (A) Represents the minimum eigenvalue of a.
Since T is a graph filter with ideal cut-off frequency in frequency response, the exact computation of P (S) requires eigenvalue decomposition of the laplacian matrix, which is very computationally expensive when the matrix is large. Thus, considering the approximation of an ideal filter by a chebyshev matrix polynomial, this only requires matrix-vector multiplication to be effectively implemented. Thus, the high-pass filter T is approximated by a Chebyshev polynomial Poly Instead of an ideal high-pass filter T, according to chebyshev polynomials:
Figure BDA0002515572320000061
substituting the sum into the solution, the problem was found to be NP-hard. Therefore consider using a greedy-based algorithm to solve. Most of the existing algorithms are from an empty set
Figure BDA0002515572320000062
Initially, in each iteration, an attempt is made to go from S C And selecting the optimal vertex and adding the optimal vertex into the S to form a sampling set. This procedure does not lead to a solution to the problem, since
Figure BDA0002515572320000063
The constraints of the problem are not satisfied. Therefore, consider the use of the complement S of the sample set C . In the initial condition of the process, the temperature of the reaction mixture,
Figure BDA0002515572320000064
s = v is undoubtedly the only set. Then, adding vertices from within S to S one after the other C Ensuring that the magnitude of the P (S) reduction is minimal in each iteration, i.e. selecting the vertex
Figure BDA0002515572320000065
Resulting in algorithm 1.
To further reduce the complexity of the algorithm, the lower bound of the minimum feature value is estimated using the Gershgorin Circle Theorem (GCT). According to GCT, each eigenvalue of the matrix T lies in the corresponding Gerr circle (T) i,i ,R i ) In which T is i,i Is the center of a circle, R i Is the radius, i.e.:
T i,i -R i ＜λ i ＜T i,i +R i ， (7)
where R is i ＝∑ j≠i |T i,j L. Definition of
Figure BDA0002515572320000066
Corresponding to a radius of
Figure BDA0002515572320000067
As can be seen,
Figure BDA0002515572320000068
is located at a radius of
Figure BDA0002515572320000069
In the space of (a). Thus, can obtain
Figure BDA00025155723200000610
The lower limit of (A) is:
Figure BDA00025155723200000611
each time a vertex is selected
Figure BDA00025155723200000612
Adding S C In (1).
The final algorithm flow chart is shown in fig. 1, and first, the initial parameters of the algorithm, i.e., the graph G = (v, epsilon), the laplacian matrix L, the sample set size M, and the initial parameters are determined
Figure BDA00025155723200000613
Then, the characteristic value lambda of L is calculated k Calculating the truncated Chebyshev polynomial approximation corresponding to h (lambda), and calculating
Figure BDA00025155723200000614
Then entering into circulation and adding S according to the selected non-sampling vertex C In (1). Until N-M non-sampled points are selected. The final sampling set is S = v \ S C 。
And 4, step 4: determining a method for reconstructing an original signal from a sampled signal: reconstruction is the inverse of sampling, resulting in a low complexity reconstruction method based on filters according to step 3.
Space PW ω (G) The signals in (1) are all band-limited map signals, when K ≧ K
Figure BDA0002515572320000071
The output through the high pass filter T is thus Tf =0. The sampled signal is defined as f S = Ψ f. The optimal estimate of the original signal f can be obtained as a vector by solving the following optimization problem:
Figure BDA0002515572320000072
therein, Ψ l ＝Ψ(I-T)。
To understand the problem, regularization is used and a penalty factor δ is introduced. By minimizing the weighted norm sum of squares, the problem translates into:
Figure BDA0002515572320000073
wherein g (f) = | | | f S -Ψ l f|| 2 +δ||Tf|| 2 . Obtaining a gradient vector:
Figure BDA0002515572320000074
the optimal solution to the problem can thus be expressed as:
Figure BDA0002515572320000075
in general, since a large number of operations are required to calculate the pseudo-inverse of the matrix, it is considered to approximate the pseudo-inverse using gradient descent. A specific reconstruction algorithm flowchart is shown in fig. 2. Input as a sampling signal f S Penalty factor delta, iteration step size mu, convergence coefficient eta, initial signal f 0]＝Ψ H f S . Circular computation
Figure BDA0002515572320000076
Until convergence.
The simulation results are shown in fig. 3, and the results show that the algorithm provided by the invention has lower complexity while the performance is equal to or better than that of the previous algorithm.

Claims (1)

1. A band-limited map signal sampling method based on a filter is characterized by comprising the following steps:
(1) Establishing a mathematical model of the graph signal samples: mapping the complex signals with irregular structures to graph vertexes, and representing the relationship between the signals by using the topological structure of the graph;
(2) Determining the conditions under which the original signal can be reconstructed from the sampled signal: when the graph signal is a band-limited signal and the dimensionality of the sampling set S is not less than the signal bandwidth, reconstructing an original signal from the sampling signal;
(3) Analyzing the sampling strategy according to reconstruction conditions, and selecting a sampling set based on a filter: the space composed of all band-limited map signals with the same bandwidth is called Peley-wiener space PW ω (G) To PW ω (G) Reducing the computational complexity of a graph signal in a space by using a filter-based method, and determining a sampling set;
(4) Determining a method for reconstructing an original signal from a sampled signal: the reconstruction is the inverse process of sampling, and a low-complexity reconstruction method based on the filter is obtained according to the step (3);
the mathematical model construction process of the map signal sampling in the step (1) is as follows:
g = (v, epsilon, W) represents a graph in which v = { ν 1 ,ν 2 ,…,ν N Denotes the set of vertices, ε is the set of edges; w is the adjacency matrix of the graph, element W i,j The relationship between graph vertices i and j is represented; graph signals are functions defined on a set of graph vertices
Figure FDA0003835678780000011
The sampling set is S; defining a sampling matrix
Figure FDA0003835678780000012
Sampling signal f obtained by sampling S Can be expressed as f S = Ψ f; the corresponding reconstructed signal is represented as
Figure FDA0003835678780000013
Where Φ is the reconstruction operator;
the conditions for reconstructing the original signal in the step (2) are as follows:
rebuilding Peli-Venu space PW ω (G) In space PW ω (G) In (1), g ∈ PW for all f ω (G) If f (S) = g (S) stands for f = g, then the sample set S is one unique set;
the step (3) comprises the following steps:
(31) Analyzing the relationship between the unique set and the Peley-Venu space
Figure FDA0003835678780000014
Represents PW ω (G) The complementary space of (2):
in the PW ω (G) In the above, the only requirement for the sampling set S is P (S)>0，
Figure FDA0003835678780000015
Is a high-pass filter having a frequency response of
Figure FDA0003835678780000016
Satisfy P (S)>The sets of 0 are all unique sets, the unique set of the minimum dimension is the optimal unique set, and the following problem is converted to solve the optimal unique set:
Figure FDA0003835678780000021
s.t.P(S)>0
wherein P (S) can be represented by the equation
Figure FDA0003835678780000022
The method comprises the steps of (1) obtaining,
Figure FDA0003835678780000023
to represent
Figure FDA0003835678780000024
The minimum eigenvalue of (d);
(32) The ideal filter is approximated through a matrix polynomial, and the minimum eigenvalue is estimated through the Gehr disk theorem to reduce the calculation complexity;
(33) Finally, obtaining an optimal sampling set through a greedy algorithm;
the step (4) is realized as follows:
space PW ω (G) Signal of (1) satisfies Tf =0, translating the recovered signal into the following problem:
Figure FDA0003835678780000025
therein, Ψ l ＝Ψ(I-T)；
The solution to the problem can be expressed as:
Figure FDA0003835678780000026
where δ is a penalty factor.
CN202010474923.6A 2020-05-29 2020-05-29 Band-limited map signal sampling method based on filter Active CN111786656B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202010474923.6A CN111786656B (en) 2020-05-29 2020-05-29 Band-limited map signal sampling method based on filter

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202010474923.6A CN111786656B (en) 2020-05-29 2020-05-29 Band-limited map signal sampling method based on filter

Publications (2)

Publication Number Publication Date
CN111786656A CN111786656A (en) 2020-10-16
CN111786656B true CN111786656B (en) 2022-11-08

Family

ID=72754544

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202010474923.6A Active CN111786656B (en) 2020-05-29 2020-05-29 Band-limited map signal sampling method based on filter

Country Status (1)

Country Link
CN (1) CN111786656B (en)

Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN106447716A (en) * 2016-09-08 2017-02-22 天津大学 Directional partial differential equation filtering method for striped projection images

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US9619755B2 (en) * 2013-10-23 2017-04-11 Mitsubishi Electric Research Laboratories, Inc. Method for estimating parameters of a graph spectral filter using training data
EP3617999B1 (en) * 2018-09-01 2023-04-19 Tata Consultancy Services Limited Systems and methods for dense surface reconstruction of an object using graph signal processing

Patent Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN106447716A (en) * 2016-09-08 2017-02-22 天津大学 Directional partial differential equation filtering method for striped projection images

Also Published As

Publication number Publication date
CN111786656A (en) 2020-10-16

Similar Documents

Publication Publication Date Title
Ainsworth et al. Multilevel techniques for compression and reduction of scientific data---the multivariate case
CN100490494C (en) Decoding apparatus, dequantizing method, distribution determining method, and program thereof
CN107918710B (en) Convex optimization-based design method of non-downsampling image filter bank
CN107241081B (en) Design method of sparse FIR prototype filter of cosine modulation filter bank
CN114581330A (en) Terahertz image denoising method based on multi-scale mixed attention
Li et al. An image denoising algorithm based on adaptive clustering and singular value decomposition
CN111310117B (en) Tensor train decomposition model-based traffic big data filling method
Emirov et al. Polynomial graph filters of multiple shifts and distributed implementation of inverse filtering
CN116797461A (en) Binocular image super-resolution reconstruction method based on multistage attention-strengthening mechanism
CN111786656B (en) Band-limited map signal sampling method based on filter
CN110675318A (en) Main structure separation-based sparse representation image super-resolution reconstruction method
CN111798531B (en) Image depth convolution compressed sensing reconstruction method applied to plant monitoring
CN111862256B (en) Wavelet sparse basis optimization method in compressed sensing image reconstruction
CN107239623B (en) Optimal design method of M-channel oversampling image filter bank based on convex optimization
CN110866344B (en) Design method of non-downsampled image filter bank based on lifting structure
CN107133921B (en) Image super-resolution reconstruction method and system based on multilevel neighborhood embedding
CN116862809A (en) Image enhancement method under low exposure condition
CN111243047A (en) Image compression sensing method based on self-adaptive nonlinear network and related product
CN114998107A (en) Image blind super-resolution network model, method, equipment and storage medium
CN108765350A (en) One kind is towards aerospace optical remote sensing image quantization filtering method
CN113962889A (en) Thin cloud removing method, device, equipment and medium for remote sensing image
Grohs et al. A shearlet-based fast thresholded Landweber algorithm for deconvolution
CN111161128A (en) Image transformation based on frequency domain direction filtering and application thereof in sparse decomposition
CN107315713B (en) One-dimensional signal denoising and enhancing method based on non-local similarity
Tanaka Graph spectral filtering

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant