CN106097275A - Quantum morphological filter building method - Google Patents

Abstract

The invention discloses a kind of quantum morphological filter building method, relate to the method and technology field of signal processing.Described method comprises the steps: to obtain the Entangled State structural element ρ needed for quantum morphological filter_n, local projection measurement operator M and average Structure Filter A, use ρ_n, M and A construct quantum morphological filter.The present invention constructs a kind of quantum morphological filter based on maximal entangled state Yu local projection measurement, overcomes the unstructured shortcoming on Complete Orthogonal base of superposition state structural element of quantum derivative morphological method.Processed with measured signal by emulation testing, demonstrate its filter capacity higher than original Mathematical Morphology Method, the adaptability of thundercloud ground electric field monitoring signal de-noising problem is then better than wavelet analysis method, lightning physics research and the needs of lightning protection engineering can be met.

Description

Quantum morphological filter building method

Technical field

The present invention relates to the method and technology field of signal processing, particularly relate to a kind of quantum morphological filter building method.

Background technology

Thundercloud ground electric field monitoring signal be Lightning Warning and the foundation of the charged structural research of thundercloud, to lightning physics with Lightning Prevention Technique is the most critically important.Due to background white noise and the impact of distant place weak discharge signal, obtained ground electric field Monitoring signal is the most seriously polluted, for obtaining can adapt to the ground electric field data that research needs, needs to carry out monitoring signal Noise reduction process.

As a kind of important non-linear filtering method, mathematical morphology is widely used in computer assisted image processing, letter Number feature extraction, computer vision and the field such as pattern recognition and signal de-noising, nowadays, have become as in Digital Signal Processing A hot research direction.But, although achieving immense success, due to the complexity that nonlinear filtering is theoretical, the most still The morphological filter method for designing not having set of system is come out.Because the usefulness of mathematical morphology filter is strongly depend on structural elements Element and the form of morphological transformation, major part morphological algorithms is for particular problem specific design, and lacks theory analysis and instruct.By The quantum information and technology of quantum calculation and quantum communications composition is the emerging research field developed rapidly.Make For theory of information and quantum-mechanical perfect adaptation, it shows powerful superiority all multi-direction of signal processing, because Unique information that qubit system is had carries and tupe.

As a key character of quantum information, tangle the source being referred to as quantum information technology, because many cores On the basis of quantum information technology is built upon it, such as Teleportation, dense coding and quantum secret communication etc..As quantum information The branch learned, Quantum signal processing technology is intended to utilize principle of quantum mechanics to construct new signal processing method or to existing Classical signal processing method is modified improving.

Summary of the invention

The technical problem to be solved is to provide a kind of quantum morphological filter building method, and described method builds Wave filter on the one hand can noise reduction effectively, on the other hand can preferably keep again signal characteristic, process sophisticated signal Aspect is more preferable than traditional form wave filter and wavelet filter effect.

For solving above-mentioned technical problem, the technical solution used in the present invention is: a kind of quantum morphological filter structure side Method, it is characterised in that comprise the steps:

Obtain the Entangled State structural element ρ needed for quantum morphological filter_n, local projection measurement operator M and average structure Wave filter A, uses ρ_n, M and A construct quantum morphological filter.

Further technical scheme is: use maximal entangled state to carry out structural texture element, it is contemplated that all maximums are tangled The contribution of state, constructs a mixed state

${\mathrm{\&rho;}}_n=\underset{j=+,-}{\underset{i=1,\mathrm{...},2^{n-1}}{\&Sigma;}}{P}_i^j|{\mathrm{\&psi;}}_i^j><{\mathrm{\&psi;}}_i^j|$

In formulaMeet normalizing conditionRepresent maximal entangled state, j=+ ,-sign The spin of qubit.

Further technical scheme is: local projection measurement operator M is defined as

$M=I_1\&CircleTimes;I_2\&CircleTimes;...\&CircleTimes;I_{i-1}\&CircleTimes;|\mathrm{\&psi;}><\mathrm{\&psi;}|\&CircleTimes;I_{i+1}\&CircleTimes;...\&CircleTimes;I_n$

In formula, i is Entangled State structural element ρ_nOrigin, | ψ > be the input signal state corresponding with i position, M is to ρ_n Effect be described as

Further technical scheme is: the expression formula of average structure wave filter A is A (x)=0.5 × [OP (x)+PO (x)], it is prepared by the following:

If f (x) and g (x) is defined in L respectively²Input signal spatially and structural element, the definition territory of f and g is respectively For D and G, then the corrosion of f (x) is respectively defined as by g (x) with expansion

${\mathrm{f\&Theta;g}}^s\left(x\right)=\underset{y\&Element;D\&cap;G}{inf}\{f\left(y\right)-g(y-x)\}$

$f\&CirclePlus;g^s\left(x\right)=\underset{y\&Element;D\&cap;G}{sup}\{f\left(y\right)+g(y-x)\}$

Wherein g^sX () represents the reflective operation to structural element, i.e. g^sX ()=g (-x), open and close computing is respectively defined as

$f\&CenterDot;g\left(x\right)=\&lsqb;(f\&CirclePlus;g^s)\mathrm{\&Theta;}g\&rsqb;\left(x\right)$

Average structure wave filter is defined as

A (x)=0.5 × [OP (x)+PO (x)]

Wherein, OP (x)=[f ο g (x)] g (x) refers to that opening and closing cascading filter, PO (x)=[f g (x)] ο g (x) are Refer to make and break cascading filter.

Use produced by technique scheme and have the beneficial effects that: the present invention construct a kind of based on maximal entangled state with The quantum morphological filter of local projection measurement, overcome the superposition state structural element of quantum derivative morphological method unstructured Shortcoming on Complete Orthogonal base.Processed by emulation testing and measured signal, it was demonstrated that its filter capacity is higher than original mathematics Morphological method, is then better than wavelet analysis method, Ke Yiman to the adaptability of thundercloud ground electric field monitoring signal de-noising problem Foot lightning physics research and the needs of lightning protection engineering.

Accompanying drawing explanation

Fig. 1 is original form wave filter noise reduction capability figure under different structure element；

Fig. 2 a-2d is wave filter of the present invention and wavelet de-noising contrast test (noisy sine wave)；

Fig. 3 a-3d is wave filter of the present invention and wavelet de-noising contrast test (noisy sine-wave superimposed Gauss ripple bag)；

Fig. 4 a-4c is QMF and wavelet de-noising contrast test figure (I) of thundercloud ground electric field monitoring signal；

The QMF of Fig. 5 a-5c thundercloud ground electric field monitoring signal and wavelet de-noising contrast test (II)；

Fig. 6 a-6c is the FFT spectrum of thundercloud ground electric field monitoring signal QMF and wavelet de-noising contrast test (II).

Detailed description of the invention

Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete Ground describes, it is clear that described embodiment is only a part of embodiment of the present invention rather than whole embodiments.Based on Embodiment in the present invention, it is every other that those of ordinary skill in the art are obtained under not making creative work premise Embodiment, broadly falls into the scope of protection of the invention.

Elaborate a lot of detail in the following description so that fully understanding the present invention, but the present invention is all right Using other to be different from alternate manner described here to implement, those skilled in the art can be without prejudice to intension of the present invention In the case of do similar popularization, therefore the present invention is not limited by following public specific embodiment.

Quantum inverting belongs to the category of quantum calculation, and quantum calculation and quantum communications together constitute quantum information science Main body.No matter it is to calculate or communication, is the most all a kind of transmission to information or process, in terms of physics's angle, letter Breath is known as the states of matter change generation of the physical system of information source, and the transmission of information is the transmission of the states of matter of coding, information Process is the evolution that physical system state algorithmically has control.Classical information based on classical states of matter coding information, When the states of matter of the information of coding is quantum electronics description, just become quantum information science.

In quantum information science, most important concept is exactly quantum bit (qubit).In classical theory of information, the list of quantity of information Position is the quantity of information that bit (bit), 1bit give classical two one value of valve system.In quantum information science, quantity of information Unit is qubit, and 1qubit is exactly a bifurcation quantized system, and the eigenstate of two Line independent is generally designated as | 0 > and | 1 >, such as two eigenstates of electron spin projection.Owing to the value of quantum information can be in | 0 > and | 1 > coherent superposition state On, therefore 1qubit is exactly a two-dimentional Hilbert space:

Quantum information value can be by a bit in unit sphere as can be seen from the above equation Representing, this is referred to as the Bloch coordinate representation of qubit.Body for n two-state system composition System, its state can be described by the vector in the direct product space in n two dimension Hilbert space, and such system is exactly a n- Qubit, it is one 2ⁿThe Hilbert space of dimension, its basic vector is made up of the direct product of the basic vector of former n qubit.

In quantum information science, another important concept is quantum door.It is that the quantum state to coding is carried out that quantum information processes A series of unitarys develop, and wherein the unitary transformation that qubit is most basic are referred to as quantum door, according to the number of acted on qubit, point It is a door, two doors or multi-position door.It is below the most frequently used quantum door:

Hadamard gate, the basic vector for Pauli-Z presentation to Pauli-X presentation converts, important when being to prepare Entangled State Operation

$H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$

Pauli door, respectively three Pauli matrixes, wherein Pauli-X door is also called quantum non-gate

$Pauli-X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),Pauli-Y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),Pauli-Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$

Quantum rotating gate, is widely used in quantum optimization algorithm

$U=\left(\begin{array}{cc}cos\mathrm{\&theta;}& -sin\mathrm{\&theta;}\\ sin\mathrm{\&theta;}& \cos \mathrm{\&theta;}\end{array}\right)$

Controlled not-gate, for two most basic doors, it acts on 2-qubit | a > | b > result beWhereinRepresenting that mould 2 adds, Cnot door is applied the widest in the complicated quantum door group network of structure

$Cnot=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$

To sum up, use quantum door that qubit is operated and constitute the content that quantum information science is most basic.Pure state and density Matrix a: if state can be with a ket | ψ > describe, then it is called a pure state.For pure state | ψ >, definable is linearly calculated SymbolBe readily seen it to act as arbitrary state to | ψ projection, thereforeBe referred to as pure state | ψ > projection operator.As Really quantized system is by many kets | ψ_iComponent system described by > is constituted, and the probability that each ket occurs in systems Determine, then claim this system to be in mixed state, be commonly described as

Wherein P_i＞ 0,Analogy pure state, for mixed state definable operatorReferred to as the density operator of mixed state, is exactly density matrix in concrete presentation.

Density matrixThere is hermiticity, orthotropicity and the one mark, system status has been fully described,Mechanical quantity A under state Meansigma methods beWhen each component state isEigenvector time,Expression formula be referred to as nature launch, now P_i For corresponding characteristic value, therefore the necessary and sufficient condition of pure state isThere is characteristic value 1.From description above it can be seen that pure state is Coherent superposition state and mixed state is non-coherent superposition state.

Entangled State (entangled state) is phenomenon the most charming in many scale of constructions subsystem.When two subsystems compositions System is in pure state | ψ >, if | ψ > (can obtain by taking part mark containing two or more than two in even state expansion Arrive), then | ψ > it is called Entangled State, otherwise | and ψ > it is called separable state, this definition may extend to mixed state, i.e. when each composition When state is all in non-Entangled State, title system is in the non-Entangled State of mixing.When the system being in Entangled State is made local measurement, system To unconditionally be collapsed to should determine that mutually state.Here we do not discuss the metric question tangled, and directly give maximal entangled state A kind of situation of (maximally entangled state), if system is in pure state | ψ >, and the reduced density square of each subsystem When battle array is the multiple of unit matrix, | ψ > it is called maximal entangled state.

Based on above theoretical, the invention discloses a kind of quantum morphological filter building method, overall, described method bag Include following steps: obtain the Entangled State structural element ρ needed for quantum morphological filter_n, local projection measurement operator M and averagely tying Structure wave filter A, uses ρ_n, M and A construct quantum morphological filter.

Concrete, described method obtains by the following method:

Mathematical morphology obtains image (signal) feature or right by the interaction of structural element with image (signal) It is filtered processing.This interaction is referred to as morphological transformation, altogether includes seven kinds of primitive forms, i.e. burn into expand, Open and close, hitting, refine and be roughened, the most front four kinds of conversion are usually used in one-dimensional signal and process.

Assume that f (x) and g (x) is defined in L respectively²(the definition territory of f and g is divided for input signal spatially and structural element Wei D and G), then the corrosion of f is respectively defined as by g with expansion

${\mathrm{f\&Theta;g}}^s\left(x\right)=\underset{y\&Element;D\&cap;G}{inf}\{f\left(y\right)-g(y-x)\}$

$f\&CirclePlus;g^s\left(x\right)=\underset{y\&Element;D\&cap;G}{sup}\{f\left(y\right)+g(y-x)\}$

G in formula^sX () represents the reflective operation to structural element, i.e. g^s(x)=g (-x).Erosion operation is permissible in simple terms The crest cutting low input-signal expands its trough simultaneously, and contrary dilation operation is then expanded the crest of input signal and filled and led up its ripple Paddy.The morphological operations of all complexity is all to be made up of with the various combination expanded corrosion, opens exactly for two kinds of most common of which And closed operation, they are respectively defined as

$f\&CenterDot;g\left(x\right)=\&lsqb;(f\&CirclePlus;g^s)\mathrm{\&Theta;}g\&rsqb;\left(x\right)$

Opening operation can effectively filter out negative polarity noise jamming and closed operation can effectively filter out positive polarity noise jamming.

Open and close computing is two kinds of most basic morphological filters, can construct more complicated by their various combination Filter construction.For eliminating opening operation and the statistical bias of closed operation, opening and closing (OP) cascade and make and break (PO) cascade two kinds of filtering Device application is the most universal.This method selects average structure wave filter to construct quantum morphological filter, is defined as

A (x)=0.5 × [OP (x)+PO (x)]

Wherein, OP (x)=[f ο g (x)] g (x) refers to that opening and closing cascading filter, PO (x)=[f g (x)] ο g (x) are Refer to make and break cascading filter.

In addition to filter construction, structural element is the most important factor affecting morphological filter effect.Conventional structure Unit have flat, linear, trigonometric sum is sinusoidal, each of the configurations element is suitable for several different waveform input signal, mainly Depend on the similarity of structural element shape and signal shape.In addition to shape, length is to affect structural element action effect Most important factor.

For obtaining the more superior wave filter of performance to meet the needs of ground electric field monitoring signal de-noising, this method throughput Son calculates and transforms traditional mathematical morphology.Transformation is primarily directed to what structural element was carried out, including the maximum of structural element Entangled State and corresponding local projection measurement.

Described method maximal entangled state carrys out structural texture element, and maximal entangled state is that the one of many scale of constructions subsystem is special State, reflects the quantum essential attributes such as relevant, probability and nonlocality, at quantum teleportation, quantum cryptology and quantum There is important application in the fields such as dense coding, tangle and can provide more information than conventional coded system.One n-qubit is exactly One 2ⁿDimension Hilbert space, this state space has a maximal entangled state basic vector group, as a example by 3-qubit system

$\begin{array}{cc}|{{\mathrm{\&psi;}}_1}^{\&PlusMinus;}>=\frac{1}{\sqrt{2}}\left(\right|000>\&PlusMinus;|111>)& |{{\mathrm{\&psi;}}_2}^{\&PlusMinus;}>=\frac{1}{\sqrt{2}}\left(\right|100>\&PlusMinus;|011>)\end{array}$

$\begin{array}{cc}|{{\mathrm{\&psi;}}_3}^{\&PlusMinus;}>=\frac{1}{\sqrt{2}}\left(\right|010>\&PlusMinus;|101>)& |{{\mathrm{\&psi;}}_4}^{\&PlusMinus;}>=\frac{1}{\sqrt{2}}\left(\right|001>\&PlusMinus;|110>)\end{array}$

It can easily be proven thatIt is that 3-qubit system Hilbert is empty Complete Orthogonal basic vector group between.When measuring the most a certain position qubit, remaining qubit will unconditionally be collapsed to Determine state accordingly.

In view of the contribution of all maximal entangled states, a mixed state can be constructed

${\mathrm{\&rho;}}_n=\underset{j=+,-}{\underset{i=1,\mathrm{...},2^{n-1}}{\&Sigma;}}{P}_i^j|{\mathrm{\&psi;}}_i^j><{\mathrm{\&psi;}}_i^j|$

In formulaUse ρ_nAs structural element, can be than basic structural element and their simple linear superposition More information is provided.It is true that ρ_nGather all structural element features in one, and be in each maximum equiprobably and tangle State.

For carrying out quantum morphology operations, in addition it is also necessary to measuring operator M for one, described method local projection operator is as M. Different from general evolution operator, M has non-unitarity, is defined here as

$M=I_1\&CircleTimes;I_2\&CircleTimes;...\&CircleTimes;I_{i-1}\&CircleTimes;|\mathrm{\&psi;}><\mathrm{\&psi;}|\&CircleTimes;I_{i+1}\&CircleTimes;...\&CircleTimes;I_n$

In formula, i is structural element ρ_nOrigin, | ψ > be the input signal state corresponding with i position.With 3-qubit As a example by system (initial point is second qubit), M is to ρ₃Effect be described as

Entangled State structural element ρ_n, local projection measurement operator M and average Structure Filter A together constitute one completely Quantum morphological filter (QMF).

Emulation testing

For testing the noise reduction capability of QMF, carry out a series of l-G simulation test.Test input signal selects the sine of 50Hz Ripple, amplitude is 1, and first phase is 0, and sample frequency is 5kHz.By signal to noise ratio (Signal Noise Ratio, SNR) as wave filter The assessment level of noise reduction capability.

First the impact on noise reduction of the structural element length is determined.Input signal is polluted with the white noise of standard deviation 0.3, Test signal SNR before noise reduction is 8.1461dB, enters test signal with four kinds of basic structural elements of a length of 1 to 20 respectively Row noise reduction process.Fig. 1 shows original form wave filter noise reduction capability under different structure element.It will be noted from fig. 1 that No matter for which kind of structural element, filtered SNR obeys and first improves the rule reduced afterwards along with the growth of structural element length Rule；For test signal, optimum SNR is 16.0581dB, comes across sinusoidal structured length of element when being 17；Flat-structure element Noise reduction efficacy the highest, use flat-structure element time optimum SNR can reach 15.0129dB, when coming across a length of 7.

For testing the noise reduction usefulness of QMF, carry out the noise reduction contrast test of small echo and QMF.Test signal and test above Identical (original SNR is 8.1461dB), the flat-structure element the highest with efficiency contrasts, and takes the structural element length of QMF Being 7, the small echo db5 small echo that selection effect is best after attempting, 4 layers of decomposition, Fig. 2 a-2d shows noise reduction result (Fig. 2 and Tu The data of 3 are the waveforms of emulation experiment, and vertical coordinate, it is merely meant that amplitude, does not has concrete meaning, therefore do not has unit, and Fig. 4 is vertical to be sat Target unit is: kV/m).

Fig. 2 c is the noise reduction waveform of QMF, and SNR is 16.9247dB；Fig. 2 d is wavelet de-noising waveform, and SNR is 19.6152dB. To test signal, QMF is better than original form wave filter, under equal length (a length of 7) structural element, flat element, linear The noise reduction SNR of element, triangle element and sinusoidal element be respectively 15.0129dB, 12.5541dB, 10.1398dB and 13.2822dB, the 16.9247dB of respectively less than QMF.But small echo is better than QMF for the noise reduction capability of harmonic signal.

Proceeding the noise reduction contrast test of small echo and QMF, test signal takes sine-wave superimposed Gauss ripple bag, then with marking Standard differs from the white noise of 0.3 and pollutes, and original SNR is 9.6036dB.Still taking the structural element a length of 7 of QMF, small echo selects db5 little Ripple, 4 layers of decomposition, Fig. 3 a-3d shows noise reduction result.

Fig. 3 c is the noise reduction waveform of QMF, and SNR is 12.9172dB；Fig. 3 d is wavelet de-noising waveform, and SNR is 13.0109dB. The noise reduction SNR of two kinds of wave filter is essentially identical, but the waveform after wavelet filtering has bigger distortion, this is because morphology is A kind of time domain approach, it can proportionally process different frequency components.To sum up, it is concluded that, QMF mono-aspect can be effective On the other hand ground noise reduction, can preferably keep again signal characteristic, therefore in terms of processing sophisticated signal than traditional form with Small echo is advantageously.

Thundercloud ground electric field monitoring signal de-noising:

For the test QMF adaptability to thundercloud ground electric field monitoring signal de-noising problem, carry out the noise reduction of it and small echo Contrast test.Stimulus is gathered by LHJDC blade sensing inversion type atmospheric electric field detector, and collecting location is Mount Taishan, sample frequency For 1Hz.The structural element length of QMF is taken as 5 after attempting, and small echo selects db5 small echo, 4 layers of decomposition after attempting.Fig. 4 a-4c Showing the noise reduction waveform that noise reduction result, Fig. 4 b are QMF, Fig. 4 c is the noise reduction waveform of small echo.QMF as we can clearly see from the figure Noise reduction waveform more smooth than the waveform that wavelet method obtains, and not distortion.Therefore, with QMF, ground electric field data is carried out Pretreatment is adapted to the needs of the charged model inversion of thundercloud.

Monitor the noise reduction process of signal for having the ground electric field of stronger discharge characteristic, while filtering noise, Keep the discharge characteristic in signal as far as possible.Signal acquiring system is constant with collecting location, and QMF used and small echo are constant, to putting by force Electrical feature signal processes, and Fig. 5 a-5c shows that noise reduction result, Fig. 6 a-6c are corresponding FFT spectrum.Fig. 5 b and Fig. 6 b is The noise reduction waveform of QMF and FFT spectrum, 5c and Fig. 6 c is wavelet de-noising waveform and corresponding FFT spectrum.From the figure, it can be seen that QMF successfully inhibits noise jamming and maintains strong discharge characteristic and do not have big change, and the noise reduction waveform of contrary small echo still contains There is more noise and there is serious wave distortion.Observe both FFT spectrum it is found that FFT spectrum after QMF noise reduction From low frequency range to high frequency region, weakening degree and strengthen with frequency non-linear, while filtering noise, entirety maintains original signal Feature；Although the FFT spectrum of wavelet de-noising weakens degree at band segment is better than QMF, but primary frequency zone has moderate finite deformation and high frequency Part is also obvious not as QMF effect, not as QMF.

Tested by two above, can make to draw a conclusion.Signal de-noising problem, QMF is monitored for thundercloud ground electric field Adaptability be better than wavelet analysis method.Through the ground electric field data of QMF pretreatment, Lightning Warning and thundercloud lotus can be met The needs of electricity research.

To sum up, the present invention constructs a kind of quantum morphological filter based on maximal entangled state Yu local projection measurement, gram Take the unstructured shortcoming on Complete Orthogonal base of superposition state structural element of quantum derivative morphological method.Pass through emulation testing Process with measured signal, it was demonstrated that its filter capacity is higher than original Mathematical Morphology Method, and monitors thundercloud ground electric field The adaptability of signal de-noising problem is then better than wavelet analysis method, can meet lightning physics research and lightning protection engineering Need.

Claims (4)

1. a quantum morphological filter building method, it is characterised in that comprise the steps:

Obtain the Entangled State structural element ρ needed for quantum morphological filter_n, local projection measurement operator M and average Structure Filter A, uses ρ_n, M and A construct quantum morphological filter.

2. quantum morphological filter building method as claimed in claim 1, it is characterised in that:

Use maximal entangled state to carry out structural texture element, it is contemplated that the contribution of all maximal entangled states, construct a mixed state

${\mathrm{\&rho;}}_n=\underset{j=+,-}{\underset{i=1,...,2^{n-1}}{\mathrm{\&Sigma;}}}{P_i}^j|{\mathrm{\&psi;}}_i^j><{\mathrm{\&psi;}}_i^j|$

P in formula_i ^j=0.5ⁿ, meet normalizing condition ∑ P_i ^j=1,Represent maximal entangled state, j=+ ,-characterize qubit from Rotation.

3. quantum morphological filter building method as claimed in claim 1, it is characterised in that:

Local projection measurement operator M is defined as

$M=I_1\&CircleTimes;I_2\&CircleTimes;...\&CircleTimes;I_{i-1}\&CircleTimes;|\mathrm{\&psi;}><\mathrm{\&psi;}|\&CircleTimes;I_{i+1}\&CircleTimes;...\&CircleTimes;I_n$

In formula, i is Entangled State structural element ρ_nOrigin, | ψ > be the input signal state corresponding with i position, M is to ρ_nWork With being described as

4. quantum morphological filter building method as claimed in claim 1, it is characterised in that the table of average structure wave filter A Reaching formula is A (x)=0.5 × [OP (x)+PO (x)], is prepared by the following:

If f (x) and g (x) is defined in L respectively²Input signal spatially and structural element, the definition territory of f and g be respectively D and G, then the corrosion of f (x) is respectively defined as by g (x) with expansion

${\mathrm{f\&Theta;g}}^s\left(x\right)=\underset{y\&Element;D\&cap;G}{inf}\{f\left(y\right)-g(y-x)\}$

$f\&CirclePlus;g^s\left(x\right)=\underset{y\&Element;D\&cap;G}{sup}\{f\left(y\right)+g(y-x)\}$

Wherein g^sX () represents the reflective operation to structural element, i.e. g^sX ()=g (-x), open and close computing is respectively defined as

$f\&CenterDot;g\left(x\right)=\&lsqb;(f\&CirclePlus;g^s)\mathrm{\&Theta;}g\&rsqb;\left(x\right)$

Average structure wave filter is defined as

A (x)=0.5 × [OP (x)+PO (x)]

Wherein, OP (x)=[f ο g (x)] g (x) refers to that opening and closing cascading filter, PO (x)=[f g (x)] ο g (x) refer to close Open cascading filter.