CN107704788A - A kind of calligraphic copying method based on RF technologies - Google Patents

A kind of calligraphic copying method based on RF technologies Download PDF

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CN107704788A
CN107704788A CN201710868373.4A CN201710868373A CN107704788A CN 107704788 A CN107704788 A CN 107704788A CN 201710868373 A CN201710868373 A CN 201710868373A CN 107704788 A CN107704788 A CN 107704788A
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mrow
mtd
label
antenna
distance
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CN107704788B (en
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郭军
胡雪婷
常俪琼
李欣怡
牛近平
王安文
王举
陈晓江
房鼎益
汤战勇
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Northwest University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06K—GRAPHICAL DATA READING; PRESENTATION OF DATA; RECORD CARRIERS; HANDLING RECORD CARRIERS
    • G06K7/00—Methods or arrangements for sensing record carriers, e.g. for reading patterns
    • G06K7/10—Methods or arrangements for sensing record carriers, e.g. for reading patterns by electromagnetic radiation, e.g. optical sensing; by corpuscular radiation
    • G06K7/10009—Methods or arrangements for sensing record carriers, e.g. for reading patterns by electromagnetic radiation, e.g. optical sensing; by corpuscular radiation sensing by radiation using wavelengths larger than 0.1 mm, e.g. radio-waves or microwaves
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/20—Instruments for performing navigational calculations
    • G01C21/206—Instruments for performing navigational calculations specially adapted for indoor navigation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06K—GRAPHICAL DATA READING; PRESENTATION OF DATA; RECORD CARRIERS; HANDLING RECORD CARRIERS
    • G06K17/00—Methods or arrangements for effecting co-operative working between equipments covered by two or more of main groups G06K1/00 - G06K15/00, e.g. automatic card files incorporating conveying and reading operations
    • G06K17/0022—Methods or arrangements for effecting co-operative working between equipments covered by two or more of main groups G06K1/00 - G06K15/00, e.g. automatic card files incorporating conveying and reading operations arrangements or provisious for transferring data to distant stations, e.g. from a sensing device
    • G—PHYSICS
    • G09—EDUCATION; CRYPTOGRAPHY; DISPLAY; ADVERTISING; SEALS
    • G09B—EDUCATIONAL OR DEMONSTRATION APPLIANCES; APPLIANCES FOR TEACHING, OR COMMUNICATING WITH, THE BLIND, DEAF OR MUTE; MODELS; PLANETARIA; GLOBES; MAPS; DIAGRAMS
    • G09B11/00—Teaching hand-writing, shorthand, drawing, or painting

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Abstract

The invention discloses a kind of calligraphic copying method based on RF technologies, by setting RFID label tag on writing brush, the RSS and phase information that antenna receives a moving process are disposed in three dimensions, split writing process by stroke, extraction influences the key feature of calligraphy, calculates the calligraphy copy of every unicursal；The difference of learner and calligraphist are found using calligraphy copy, is improved so as to which guidance learning person improves, the present invention can estimate label antenna distance in grade, so as to realize redrawing to the writing process of calligraphy.

Description

A kind of calligraphic copying method based on RF technologies
Technical field
The present invention relates to indoor positioning technologies field, more particularly to a kind of calligraphic copying method based on RF technologies.
Background technology
Brush calligraphy calligraphy is a Chinese-character writing method and visual art, is the marrow of traditional Chinese culture.In recent years with Come, with concern of the people to traditional Chinese culture, calligraphy art, Sinitic method is appreciated in the study best as one Also followed extensively by learner.The grasp of calligraphy skill, indispensable is the imitation contrast to great master of calligraphy's works, often The exercise accumulation and correctly main points guidance in year.Existing certain methods are to go to track learner and calligraphist based on pixel Difference between works, but in Brush calligraphy calligraphy, its three indispensable fundamental is i.e. with a mode, scheme structure and stroke Feature.Difference could be reduced by how improving, and this, which still needs learner and expends considerable time and effort to repeatedly practise, tries to figure out, It can be lifted.
The content of the invention
In order to solve the above-mentioned problems of the prior art, it is an object of the present invention to provide a kind of based on RF technologies Calligraphic copying method, the writing process of calligraphy can be redrawn out with millimeter class precision, biography is provided for the imitation exercise of calligraphy learner The inaccessiable pinpoint guidance of system method institute.
To realize above-mentioned task, the present invention takes following technical solution：
A kind of calligraphic copying method based on RF technologies, comprises the following steps：
Step 1, calligraphy is write with pen, utilize the label being arranged in an antenna acquisition moving process on the penholder of pen RSS values Sa,j,iAnd phase information (t)Wherein, a represents the label of antenna, and a=1,2,3, j represent the mark of RFID label tag Number, j=1,2, i represent channels label, i=1,2 ..., N, N represent channel number, t represent a moving process in when Carve；
Step 2, to phase informationPre-processed to obtain pretreated phase informationPretreatment Including the processing of phase shift calibration, noise suppressed and multipath, pretreated phase informationCorresponding RSS values are Sa,j,m (t), wherein, m represents the label of clean channel, m=1,2 ..., M, M represent the number of clean channel；
Step 3, according to phase informationWith RSS values Sa,j,m(t) any instant t, is sought, antenna a corresponding to channel m With label j distance da,j,m(t)；The formula of use is as follows：
Wherein, fmWhat is represented is the frequency of m-th of clean channel, and c represents the light velocity；K is integer；
Step 4, according to linear relationship characteristic existing for the phase information that adjacent frequency channels receive, and triangle All distance values for being obtained to step 3 of length of side relational expression screen；For the distance value after screening, when calculating any Carve under t, the average of the distance of a-th of antenna and j-th of label corresponding to all channels, using the average value tried to achieve as the moment The distance between a-th of antenna and j-th label value da,j(t)；
Step 5, utilize distance value da,j(t) use in trigonometric function method calculating pen moving process, under any instant t, pen The vertical height h (t) of bar point midway；According to changing rules of the vertical height h (t) of penholder point midway between stroke, It is segmented at the time of in writing process, one stroke of each section of correspondence；
Step 6, for each moment in each stroke, ask multiple key feature amounts corresponding to it, multiple key features Amount forms template characteristic matrix corresponding to the stroke；Key feature amount include the elevation angle, nib length change, nib translational speed and Pen tip position；
Step 7, the method for repeat step one to step 6, obtain copying eigenmatrix corresponding to each stroke, by its with Template characteristic matrix is contrasted, and obtained difference result is used for instructing calligraphy learner and the gap of calligraphist.
Specifically, the method that the phase shift calibration in the step 2 uses is as follows：
With writing in calligraphy identical environment, the actual distance between label and antenna is measured；According to label and antenna Between actual distance, using distance and phase relation formula, try to achieve the real phase value of label；
Obtain the measurement phase value of label；
The difference of the measurement phase value of the real phase value and label of label is sought, as needs the phase shift removed
To the phase information of RFID label tagThe process of phase shift calibration is carried out, is exactly in phase informationIn Except phase shift
Specifically, the noise suppressed in the step 2, the method for use are as follows：
Given threshold value, if the phase deviation in phase information between the phase value at two continuous moment exceedes given threshold, It is equal with previous phase value then to adjust latter phase value；Phase information is carried out using kalman filter method again smooth.
Specifically, the multipath processing in the step 2, the method for use are as follows：
In phase informationIn select and belong to phase information corresponding to clean channel, as after multipath is handled Phase information
Wherein, the determination method of clean channel is：In the environment of calligraphy is write, RFID reader skips different frequency Channel received data, the phase information sequence of label is obtained in each channel, asks being averaged for all phase informations in each channel Value, i.e. phase average；With channel marked as abscissa, using phase average as ordinate, all phase averages connect forming curves, By observing the trend of curve, retain linear phase average, channel corresponding to these phase averages is clean letter Road.
Specifically, the determination method of the k in the step 3 is as follows：
In the environment of calligraphy is write, a label and an antenna are set, change the distance between label and antenna, profit The phase information and RSS information of label are obtained with RFID reader；Obtain what RSS values changed with the distance between label and antenna Curve map；When the distance between label and antenna scope are (0, λ/2), k=1, when the increase of the distance between label and antenna During λ/2, k value increase by 1, according to each distance range section between label and antenna, it may be determined that the scope area of RSS information Between；
According to RSS values, its affiliated range intervals is determined, you can determine k value.
Specifically, distance value d is utilized in the step 5a,j(t) use in trigonometric function method calculating pen moving process, Under any instant t, the vertical height h (t) of penholder point midway, the formula of use is as follows：
Wherein, h1(t) vertical height of label 1 is represented；h2(t) vertical height of label 2 is represented；d11Represent antenna A1 with The distance between label 1；d21Represent the distance between antenna A2 and label 1；d12Represent the distance between antenna A1 and label 2； d22Represent the distance between antenna A2 and label 2；A represents the distance between antenna A1 and antenna A2；
Described splits according to the vertical height h (t) of penholder point midway changes to the stroke in writing process；Adopt Method is as follows：
Threshold value h is setpThe mark that+η terminates as stroke, if h (te) it is more than the threshold value, show that pen is lifted off writing Plane, a stroke are completed, moment teAt the time of as a stroke terminates；Threshold value h is setpThe mark that-η starts as stroke, If h (ts) it is less than the threshold value, show that pen is in writing plane, channel stroke initially forms, moment teAs a stroke terminates Moment；hpFor penholder midpoint to nib distance；η is constant；
According to moment teWith moment ts, it is segmented at the time of in writing process, one stroke of each section of correspondence.
Specifically, in the step 6, it is as follows to calculate the formula that the elevation angle uses：
Wherein, d11Represent the distance between antenna A1 and label 1；d21Represent the distance between antenna A2 and label 1；d12 Represent the distance between antenna A1 and label 2；d22Represent the distance between antenna A2 and label 2；A represents antenna A1 and antenna A2 The distance between；B represents the distance between label 1 and label 2；
Nib length change is calculated, the formula of use is as follows：
Wherein, L is the distance that label 1 arrives nib on penholder；
Nib translational speed is calculated, the formula of use is as follows：
Distances of the antenna A1 to nib T0：
Wherein, T1' and T2' is the projection of label T1 and T2 on the penholder of pen respectively；d11'It is antenna A1 to T1' distance； d12'It is antenna A1 to T2' distance；
Nib translational speed：
Wherein, Δ t represents the t+1 moment to the time interval of t；
Pen tip position T0 (x (t), y (t)) is calculated, the formula of use is as follows：
X (t)=d11'(t)sinβ1+Lcosθ(t)sin(β2+β1)
Y (t)=d11'(t)cosβ1-Lcosθ(t)sin(β2+β1)
Wherein, β1Represent the angle between A1A3 and A1T1', β2Represent the angle between A1T1' and T1'T2'；
Four key feature amounts at above-mentioned all moment (t ... t+i...t+s) form the eigenmatrix of the stroke, i.e.,：
Compared with prior art, the present invention has following technique effect：
By setting RFID label tag on pen, the RSS and phase that antenna receives a moving process are disposed in three dimensions Information, split writing process by stroke, extraction influences the key feature of calligraphy, calculates the calligraphy copy of every unicursal；Utilize book Method copy finds the difference of learner and calligraphist, is improved so as to which guidance learning person improves, the present invention can estimate in grade Label-antenna distance, so as to realize redrawing to the writing process of calligraphy.
Brief description of the drawings
Fig. 1 is three different experiments environment position figures；
Fig. 2 is detailed deployment diagram of the system in office environment；
Fig. 3 is phase information by phase shift calibration and the result figure of noise suppressed；Wherein, after (a) represents phase shift calibration Result figure, (b) represent the result figure of noise suppressed；
Fig. 4 is the phase data figure that 16 different frequency channels receive in hall environment；
Fig. 5 is the phase data figure that 16 different frequency channels receive in library environment；
Fig. 6 is the relation of RSS values and antenna-tag distance.
Fig. 7 is writing brush Level Change and stroke segmentation schematic diagram in writing process.
Fig. 8 is the geometrical relationship figure for obtaining the writing brush elevation angle；
Fig. 9 is that writing brush nib length changes schematic diagram；
Figure 10 is geometrical relationship figure when estimating pen tip position；
Figure 11 is the coordinate diagram of pen tip position；
Figure 12 is the schematic diagram screened with length of side relation pair antenna A1, A2 of triangle and the distance of label 1,2；
Figure 13 is the tracking effect figure horizontal to different calligraphies；
Figure 14 is the schematic diagram screened with the length of side relation pair antenna A3 of triangle and the distance of label 1,2；
Figure 15 is the calculating signal of antenna A1 to T1 ' distance, antenna A1 to T2 ' distance, antenna A2 to T1 ' distance Figure；
Below in conjunction with drawings and examples, the present invention is described in further detail.
Embodiment
Referring to Fig. 1, the calligraphic copying method of the invention based on RF technologies, used system include RFID reader, Two Alien labels and the directional aerial of three 5dBi gains, two of which Alien labels are bonded at the penholder upper and lower ends of pen, Distance b=16cm, three directional aerials are connected with reader, and antenna 1 is located at the origin of coordinates, and antenna 2 is located on Z axis, antenna 3 In Y-axis, the distance of antenna 1 to antenna 2 is equal to distance a, a=40cm that antenna 1 arrives antenna 3.Deployment system on the table, shape The monitor area for being 50cm × 50cm into a size, the writing region of calligraphy is 40cm × 40cm rice paper, and the pen of use can Think common writing brush or there is soft written pen, in the present embodiment, sample writing brush, total length 29.8cm, pen head part Long 5.5cm, when pen enter monitor area just perpendicular contact is to paper when, height h=of the penholder point midway apart from paper 16cm；In writing process, the maximum change value of nib length is about 4cm.Pen is moved with 1-2cm/s speed.
Wherein, described RFID reader compatibility EPC Gen2 standards, the legal uhf band of China is operated in, i.e., 920.625MHz to 924.375MHz scopes；In data-gathering process, reader 16 different frequency channels of jump are adjacent Inter-channel frequency interval 500KHz.
The calligraphic copying method based on RF technologies of the present invention, comprises the following steps：
Step 1, calligraphist write calligraphy with writing brush, and in writing brush moving process, multiple channels of RFID reader pass through 3 Individual antenna (A1, A2 and A3) inscribes the RSS values S of RFID label tag when obtaining each in writing brush moving processa,j,iAnd phase information (t)Wherein, a represents the label of antenna, and a=1,2,3, j represent the label of RFID label tag, and j=1,2, i represent the mark of channel Number, i=1,2 ..., N, N represent the number of channel, in this embodiment, at the time of N=16, t are represented in writing brush moving process.
Step 2, the phase information of the RFID label tag obtained to step 1Pre-processed, after obtaining pretreatment Phase informationWherein, m represents the label of clean channel, m=1,2 ..., M, M represent the number of clean channel； Pretreatment includes phase shift calibration, noise suppressed and multipath processing.
Wherein, the phase shift method that uses of calibration for：
One label and an antenna are placed on calligraphist and write in the environment of calligraphy, close to day when label is placed The position of line, to reduce the quantity of multipath；With the actual distance between laser instrumentation amount label and antenna；Pass through the mark of measurement Sign the actual distance between antenna,, can be with as shown in formula (1) according to the relational expression between actual distance and true phase The real phase value of label is calculated.RFID reader can obtain the measurement phase value of label by antenna.Look for the truth reality The difference of the phase value of label that obtains of phase value and RFID reader, as need the phase shift removed
To the phase information of RFID label tagThe process of phase shift calibration is carried out, is exactly in phase informationIn Except phase shift
The method that noise suppressed uses for：
After constant phase shift is removed, due to random phase noise, still observe in phase information big peak value be presentFor same channel, the phase information under the same antenna, same label, given threshold value=2, if two in phase information Phase deviation between the phase value at continuous moment exceedes given threshold, then it is equal with previous phase value to adjust latter phase value； Phase information is carried out using kalman filter method again smooth.
Multipath processing procedure realizes that reason is as follows based on multi channel method：(1) although commercial RFID reader uses Directional aerial is transmitted, but compared with Wi-Fi, the much smaller number of its multipath；(2) because the tracing area of writing is smaller, Label-antenna distance is also smaller.Therefore directapath signal seizes ascendancy very much, reduces the effect of multipath；(3) it is commercial RFID reader can skip the channel of different frequency in the normal operation period, by using considerable available channel (16 It is individual), just necessarily it can recognize that indivedual hardly by the channel (being dominated by directapath signal) of multi-path jamming.If not by multipath Interference, give a label position, it is known that the center frequency difference of two channels is Δ f, their phase differenceIt can represent For：
Wherein, c is the light velocity, and Δ δ is phase noise, and mod is mod function, and the phase value of different channels is linear. Once which channel is dominated by multipath signal, it is broken with the linear relationship of other channels.But due to directional aerial use with And the scope of system deployment is smaller, so most of channel still suffers from this linear relationship, we are referred to as this channel " clean Channel ".That is, only directapath signal, the corresponding phase place reading propagated in these clean channels may be used for Range estimation.
The determination method of the clean channel is, with the present embodiment identical environment, setting 1 antenna and 1 RFID label tag, RFID reader skip the channel received data of 16 different frequencies, and the phase letter of label is obtained in each channel Sequence is ceased, asks the average value of all phase informations in each channel, i.e. phase average；With channel marked as abscissa, with phase Average is ordinate, and all phase averages connect forming curves, by observing the trend of curve, retain linear phase Average, channel corresponding to these phase averages are clean channel.
Multipath processing implementation method be：In phase informationIn select and belong to corresponding to clean channel phase and believe Breath, the as phase information after multipath is handledPhase information after multipath processingCorresponding RSS values For Sa,j,m(t)；
Step 3, according to pretreated phase informationAnd and phase informationCorresponding RSS values Sa,j,m(t) the distance d in moment t, antenna a corresponding to channel m and label j, is soughta,j,m(t)；The formula of use is as follows：
Wherein, fmWhat is represented is the frequency of m-th of clean channel, and c represents the light velocity；K is integer；
K determination method is as follows：
In the environment of the present embodiment, a label and an antenna are set, change the distance between label and antenna, profit The phase information and RSS information of label are obtained with RFID reader；Obtain what RSS values changed with the distance between label and antenna Curve map；Experiment shows that signal strength values can reduce with the increase of antenna-tag distance.When distance changes λ/2, λ is represented Wavelength, corresponding one 2 π cycle of phase place but reading is constant, but the change of RSS values will be easily detected.We are institute Possible phase place number k sets corresponding RSS value changes scope, as shown in fig. 6, the corresponding relation of RSS values and distance；When When the distance between label and antenna scope are (0, λ/2), k=1；When the distance between label and antenna scope are (λ/2, λ) When, k=2；By above-mentioned relation, can obtain when RSS values are in a certain scope, k value.In the present embodiment, k value It is probably 1 or 2.
Step 4, by different frequency channels, obtain multiple estimated values of distance, these estimated values are in theory each other It is similar, therefore we exclude not meeting the value of similarity relation.Then according to antenna, label position between existing geometrical relationship Limitation, all antennas and the distance d of label obtained using the length of side relational expression of triangle to step 3a,j,m(t) sieved Choosing, the distance of antenna and label after being screened；For the antenna and the distance of label after screening, calculate under a certain moment t, The average of the distance of a-th of antenna and j-th of label corresponding to all channels, it regard the average value tried to achieve as a-th of moment The distance between antenna and j-th label value da,j(t)。
Specific screening technique is as follows：
Referring to Figure 12, using the length of side relation of triangle, for the t1 moment, antenna A2 corresponding to channel i and label T1 it Between distance d2,1,i(t1), according to triangle A2, T1 and T2,1 group of relational expression is obtained：
Fixed d2,1,i(t1), d2,2,m(t1) represent in the distance value corresponding to all channels between antenna A2 and label T2 In any one, choose d one by one2,2,m(t1) substitute into above formula to be calculated, if the d more than half2,2,m(t1) result of calculation Meet above-mentioned formula, then retain the d2,1,i(t1)；Otherwise, d is removed2,1,i(t1), i.e., the value is improper value；
For t1 moment, the distance between antenna A2 corresponding to channel i and label T2 d2,2,i(t1), according to triangle A2, T2 and T2, obtain 1 group of relational expression：
Fixed d2,2,i(t1), d1,2,m(t1) represent in the distance value corresponding to all channels between antenna A1 and label T2 In any one, choose d one by one1,2,m(t1) substitute into above formula to be calculated, if the d more than half1,2,m(t1) result of calculation Meet above-mentioned formula, then retain the d2,2,i(t1)；Otherwise, d is removed2,2,i(t1), i.e., the value is improper value；
For t1 moment, the distance between antenna A1 corresponding to channel i and label T2 d1,2,i(t1), according to triangle A1, T1 and T2, obtain 1 group of relational expression：
Fixed d1,2,i(t1), d1,1,m(t1) represent in the distance value corresponding to all channels between antenna A1 and label T1 In any one, choose d one by one1,1,m(t1) substitute into above formula to be calculated, if the d more than half1,1,m(t1) result of calculation Meet above-mentioned formula, then retain the d1,2,i(t1)；Otherwise, d is removed1,2,i(t1), i.e., the value is improper value；
For t1 moment, the distance between antenna A1 corresponding to channel i and label T1 d1,1,i(t1), according to triangle A1, A2 and T1, obtain 1 group of relational expression：
Fixed d1,1,i(t1), d2,1,m(t1) represent in the distance value corresponding to all channels between antenna A2 and label T1 In any one, choose d one by one2,1,m(t1) substitute into above formula to be calculated, if the d more than half2,1,m(t1) result of calculation Meet above-mentioned formula, then retain the d1,1,i(t1)；Otherwise, d is removed1,1,i(t1), i.e., the value is improper value；
Referring to Figure 14, for t1 moment, the distance between antenna A3 corresponding to channel i and label T1 d3,1,i(t1), according to Triangle A1, A3 and T1, obtain 1 group of relational expression：
Fixed d3,1,i(t1), d1,1,m(t1) represent in the distance value corresponding to all channels between antenna A2 and label T1 In any one, choose d one by one1,1,m(t1) substitute into above formula to be calculated, if the d more than half1,1,m(t1) result of calculation Meet above-mentioned formula, then retain the d3,1,i(t1)；Otherwise, d is removed3,1,i(t1), i.e., the value is improper value；
It is worth for the distance between all a certain antennas of a certain moment and a certain label and is averaging, the average value that will be tried to achieve As the distance between the moment antenna and label value da,j(t)。
Step 5, calculate the vertical height h (t) of each moment penholder point midway in writing brush moving process；According to vertical Changing rules of the height h (t) between stroke, it is segmented at the time of in writing process, one stroke of each section of correspondence；
The vertical height of label 1 in writing brush moving process on each moment penholder is h1(t) it is, every in writing brush moving process The vertical height of label 2 on individual moment penholder is h2(t)；
Wherein, d11Represent the distance between antenna A1 and label 1；d21Represent the distance between antenna A2 and label 1；d12 Represent the distance between antenna A1 and label 2；d22Represent the distance between antenna A2 and label 2；A represents antenna A1 and antenna A2 The distance between.
Stroke segmentation is carried out to writing process according to writing brush vertical height h change.In theory, each pen in writing process Writing brush movement between picture is all related to height h：H becomes big when starting writing, and h is held essentially constant when mobile, and h reduces when starting to write.Cause For because the flexible nature of nib causes to start writing, the amplitude started to write is all larger, and the length of nib may change when with brush writing 2-3cm, therefore h amplitude of variation is obvious, can detect for splitting stroke
Referring to Fig. 7, the writing brush movement in writing process between each stroke is all related to height h：H becomes big when starting writing, and moves H is held essentially constant when dynamic, and h reduces when starting to write.As shown in fig. 7, find out the timestamp t that each stroke terminateseStart with stroke Timestamp ts, phase sequence is divided into segment in these positions.
Threshold value h is setpThe mark that+η terminates as stroke, if h (te) it is more than the threshold value, show that writing brush is lifted off book Plane is write, a stroke is completed, moment teAt the time of as a stroke terminates；Threshold value h is setpThe mark that-η starts as stroke Will, if h (ts) it is less than the threshold value, show that writing brush is in writing plane, channel stroke initially forms, moment teAn as stroke At the time of end；hpFor penholder midpoint to nib distance；η is constant；Setting parameter η=5mm in the present embodiment；Thus find each The timestamp that stroke terminates and started, the phase sequence with timestamp is segmented, each section obtained after segmentation is independent The writing process of one stroke.
Step 6, for each moment in each stroke, ask multiple key feature amounts corresponding to it, multiple key features Amount forms template characteristic matrix corresponding to the stroke.
The nib of writing brush is different from general pen, its soft texture, and black writing is dipped in by nib, therefore writing effect can be by The significant impact of dynamics of wieling the pen and speed, such as：(1) differing tilt angles of writing brush and the depression degree of nib cause person's handwriting wide The change of degree；(2) the person's handwriting difference in appearance that nib is write with friction speed is very big；(3) pen-down position and moving direction of nib Determine the space layout of each in a word.If the relative position per unicursal is incorrect, then even if each pen is all That writes is very beautiful, and whole word also appears to and unsightly.Therefore, these characteristic quantities have been quantified.
(1) elevation angle
The elevation angle is the angle of writing brush and writing plane, represents the inclined degree of writing brush in writing process.We are the elevation angle θ is expressed as, θ ∈ (0,90 °), when θ=90 °, writing brush is perpendicular to writing plane.The elevation angle is bigger, and the width of ink marks is smaller.Stroke In each moment elevation angle theta (t) calculation formula it is as follows：
(2) nib length change Δ h
The depression degree of nib when nib length change illustrates writing, it is another important indicator of person's handwriting width.Pen Sharp length varying value is bigger, illustrate writing brush point touch writing region area it is bigger, also just explanation person's handwriting it is wider；When writing brush with Elevation angle theta gently touches writing region, and L is the distance that label 1 arrives nib T0 on penholder, then the situation that nib does not push Under, writing brush point midway is apart from the vertical range of paperTherefore, as long as calculating the vertical height h of writing brush, With regard to nib length change amount Δ h can be calculated；
The Δ h (t) at each moment in stroke calculation formula is as follows：
Wherein, L is the distance that label 1 arrives nib T0 on penholder；
(3) nib translational speed v
Shown in Figure 10, T1' and T2' are the projection of label T1 and T2 on writing brush penholder respectively.When writing brush with elevation angle theta gently Touch writing region, L be on penholder label 1 arrive nib T0 distance, antenna A1 to nib T0 distance d10(t), calculate public Formula is as follows：
d11'It is exactly antenna A1 to T1' distance.In triangle A1T1'T2',
So the distance of nib T0 to the antenna 1 under t is expressed as：
Wherein, d11'Acquiring method is：Referring to Figure 15, in triangle A1A2T1, the length on three sides is all, it is known that can use Inverse function calculates α1Size, then calculate d11', in triangle A2A1T2, the length on three sides is all, it is known that inverse function can be used Calculate α2Size, you can calculate d12'。
So, the translational speed of nib is calculated as：
Wherein, Δ t represents the t+1 moment to the time interval of t.
(4) pen tip position T0
Referring to Figure 11, the pen tip position T0 (x (t), y (t)) at each moment in stroke calculation formula is as follows：
X (t)=d11'(t)sinβ1+Lcosθ(t)sin(β2+β1)
Y (t)=d11'(t)cosβ1-Lcosθ(t)sin(β2+β1) (15)
Wherein, β1Represent the angle between A1A3 and A1T1', β2Represent the angle between A1T1' and T1'T2'；
β1Acquiring method it is as follows：In triangle A1A3T1', A1A3, A1T1' are, it is known that A3T1' method for solving is： A1 in Figure 15 is replaced with A3, in triangle A3A2T1, the length on three sides all, it is known that can be calculated with inverse function A2A3 with Angle α between A3T13Size, d31'=d31·sinα3；Angle beta can be calculated with inverse function1。
β2Acquiring method it is as follows：In triangle A1T1'T2', A1T1', A1T2' are, it is known that the distance between T1'T2' It can try to achieve, be bcos θ；Angle beta can be calculated with inverse function2。
Four key feature amounts at above-mentioned all moment form the eigenmatrix of the stroke, i.e.,：
Wherein, in the stroke it is t at the time of ... t+i...t+s.
Step 7, calligraphy learner writing calligraphy, according to the method for above-mentioned steps one to step 6, obtains calligraphy learner Each stroke imitation eigenmatrix, the eigenmatrix of itself and calligraphy expert are contrasted, obtained difference result is used for Guidance learning person finds the gap with calligraphist, improves and copies, and it is horizontal to improve calligraphy.
It is required that expert and learner write an identical Chinese character " big ".Then application system monitors ablation process and extracted Eigenmatrix.By comparing the eigenmatrix of expert and learner, the place that learner performs poor can be obtained.The rail of write-in Mark is as shown in figure 13.Compared with the track of expert, the track of learner is coarse and incorrect.First, learner's writing brush word In different strokes between relative position it is incorrect, we can be by correcting according to the difference between learner and expert The pen tip position T0 (x, y) of each stroke and improved.Secondly as non-smooth writing speed, stroke are uneven. Learner can imitate the translational speed v of expert to improve.Finally, stroke width is different that we are with horizontal stroke " horizontal stroke " Exemplified by.For the stroke write by expert, width is moderate, is from left to right broadened along stroke, but the stroke that learner writes is wide Degree is to change at random.Therefore, we can compare the length change Δ h of elevation angle theta between two works and nib.These differences Meeting guidance learning person reduces the gradient of writing brush, and mitigates the pressure applied on a writing surface, so as to improve stroke width Problem.

Claims (7)

  1. A kind of 1. calligraphic copying method based on RF technologies, it is characterised in that comprise the following steps：
    Step 1, calligraphy is write with pen, the RSS values for the label being arranged in a moving process on the penholder of pen are obtained using antenna Sa,j,iAnd phase information (t)Wherein, a represents the label of antenna, and a=1,2,3, j represent the label of RFID label tag, j= 1,2, i represent channel label, i=1,2 ..., N, N represent channel number, t represent a moving process at the time of；
    Step 2, to phase informationPre-processed to obtain pretreated phase informationPretreatment includes Phase shift calibration, noise suppressed and multipath processing, pretreated phase informationCorresponding RSS values are Sa,j,m(t), its In, m represents the label of clean channel, m=1,2 ..., M, M represent the number of clean channel；
    Step 3, according to phase informationWith RSS values Sa,j,m(t) any instant t, is sought, antenna a corresponding to channel m and mark Sign j distance da,j,m(t)；The formula of use is as follows：
    Wherein, fmWhat is represented is the frequency of m-th of clean channel, and c represents the light velocity；K is integer；
    Step 4, according to linear relationship characteristic existing for the phase information that adjacent frequency channels receive, and the side of triangle All distance values that long relational expression obtains to step 3 screen；For the distance value after screening, any instant t is calculated Under, the average of the distance of a-th of antenna and j-th of label corresponding to all channels, using the average value tried to achieve as moment a The distance between individual antenna and j-th label value da,j(t)；
    Step 5, utilize distance value da,j(t) use in trigonometric function method calculating pen moving process, under any instant t, in penholder The vertical height h (t) of point position；According to changing rules of the vertical height h (t) of penholder point midway between stroke, to book It is segmented at the time of during writing, one stroke of each section of correspondence；
    Step 6, for each moment in each stroke, ask multiple key feature amounts corresponding to it, multiple key feature amount shapes Into template characteristic matrix corresponding to the stroke；Key feature amount includes the elevation angle, nib length change, nib translational speed and nib Position；
    Step 7, the method for repeat step one to step 6, obtain copying eigenmatrix corresponding to each stroke, by itself and template Eigenmatrix is contrasted, and obtained difference result is used for instructing calligraphy learner and the gap of calligraphist.
  2. 2. the calligraphic copying method based on RF technologies as claimed in claim 1, it is characterised in that the phase shift in the step 2 It is as follows to calibrate the method used：
    With writing in calligraphy identical environment, the actual distance between label and antenna is measured；According between label and antenna Actual distance, using distance and phase relation formula, try to achieve the real phase value of label；
    Obtain the measurement phase value of label；
    The difference of the measurement phase value of the real phase value and label of label is sought, as needs the phase shift removed
    To the phase information of RFID label tagThe process of phase shift calibration is carried out, is exactly in phase informationMiddle removal phase Move
  3. 3. the calligraphic copying method based on RF technologies as claimed in claim 1, it is characterised in that the noise in the step 2 Suppress, the method for use is as follows：
    Given threshold value, if the phase deviation in phase information between the phase value at two continuous moment exceedes given threshold, adjust Whole latter phase value is equal with previous phase value；Phase information is carried out using kalman filter method again smooth.
  4. 4. the calligraphic copying method based on RF technologies as claimed in claim 1, it is characterised in that the multipath in the step 2 Processing, the method for use are as follows：
    In phase informationIn select and belong to phase information corresponding to clean channel, the as phase after multipath is handled Information
    Wherein, the determination method of clean channel is：In the environment of calligraphy is write, RFID reader skips the channel of different frequency Data are received, the phase information sequence of label is obtained in each channel, seeks the average value of all phase informations in each channel, i.e., Phase average；With channel marked as abscissa, using phase average as ordinate, all phase averages connect forming curves, pass through The trend of curve is observed, retains linear phase average, channel corresponding to these phase averages is clean channel.
  5. 5. the calligraphic copying method based on RF technologies as claimed in claim 1, it is characterised in that k's in the step 3 Determine that method is as follows：
    In the environment of calligraphy is write, a label and an antenna are set, change the distance between label and antenna, is utilized RFID reader obtains the phase information and RSS information of label；Obtain the song that RSS values change with the distance between label and antenna Line chart；When the distance between label and antenna scope are (0, λ/2), k=1, when the distance between label and antenna increase λ/2 When, k value increase by 1, according to each distance range section between label and antenna, it may be determined that the range intervals of RSS information；
    According to RSS values, its affiliated range intervals is determined, you can determine k value.
  6. 6. the calligraphic copying method based on RF technologies as claimed in claim 1, it is characterised in that the utilization in the step 5 Distance value da,j(t) use in trigonometric function method calculating pen moving process, under any instant t, the vertical height of penholder point midway H (t), the formula of use are as follows：
    <mrow> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>d</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;times;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi> </mi> <msub> <mi>cos&amp;alpha;</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>d</mi> <mn>11</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>d</mi> <mn>21</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow>
    <mrow> <msub> <mi>h</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>d</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;times;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi> </mi> <msub> <mi>cos&amp;alpha;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>d</mi> <mn>12</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>d</mi> <mn>22</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow>
    <mrow> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow>
    Wherein, h1(t) vertical height of label 1 is represented；h2(t) vertical height of label 2 is represented；d11Represent antenna A1 and label 1 The distance between；d21Represent the distance between antenna A2 and label 1；d12Represent the distance between antenna A1 and label 2；d22Table Show the distance between antenna A2 and label 2；A represents the distance between antenna A1 and antenna A2；
    Described splits according to the vertical height h (t) of penholder point midway changes to the stroke in writing process；Use Method is as follows：
    Threshold value h is setpThe mark that+η terminates as stroke, if h (te) it is more than the threshold value, show that pen is lifted off writing plane, One stroke is completed, moment teAt the time of as a stroke terminates；Threshold value h is setpThe mark that-η starts as stroke, if h (ts) it is less than the threshold value, show that pen is in writing plane, channel stroke initially forms, moment teAn as stroke terminate when Carve；hpFor penholder midpoint to nib distance；η is constant；
    According to moment teWith moment ts, it is segmented at the time of in writing process, one stroke of each section of correspondence.
  7. 7. the calligraphic copying method based on RF technologies as claimed in claim 1, it is characterised in that in the step 6, calculate The formula that the elevation angle uses is as follows：
    <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mrow> <mo>|</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mi>b</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mrow> <msubsup> <mi>d</mi> <mn>12</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>d</mi> <mn>21</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>d</mi> <mn>11</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>d</mi> <mn>22</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
    Wherein, d11Represent the distance between antenna A1 and label 1；d21Represent the distance between antenna A2 and label 1；d12Represent The distance between antenna A1 and label 2；d22Represent the distance between antenna A2 and label 2；A is represented between antenna A1 and antenna A2 Distance；B represents the distance between label 1 and label 2；
    Nib length change is calculated, the formula of use is as follows：
    <mrow> <mi>&amp;Delta;</mi> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mfrac> <mi>b</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mi>L</mi> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow>
    Wherein, L is the distance that label 1 arrives nib on penholder；
    Nib translational speed is calculated, the formula of use is as follows：
    Distances of the antenna A1 to nib T0：
    <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>d</mi> <mn>10</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mi>L</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>d</mi> <msup> <mn>11</mn> <mo>&amp;prime;</mo> </msup> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <mi>L</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>d</mi> <msup> <mn>11</mn> <mo>&amp;prime;</mo> </msup> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;times;</mo> <mfrac> <mrow> <msubsup> <mi>d</mi> <msup> <mn>11</mn> <mo>&amp;prime;</mo> </msup> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>b</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msubsup> <mi>d</mi> <msup> <mn>12</mn> <mo>&amp;prime;</mo> </msup> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <msub> <mi>d</mi> <msup> <mn>11</mn> <mo>&amp;prime;</mo> </msup> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>b</mi> <mi> </mi> <mi>cos</mi> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msqrt> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mi>L</mi> <mi>b</mi> </mfrac> <mo>)</mo> <msubsup> <mi>d</mi> <msup> <mn>11</mn> <mo>&amp;prime;</mo> </msup> <mn>2</mn> </msubsup> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>+</mo> <mo>(</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>b</mi> <mi>L</mi> <mo>)</mo> <msup> <mi>cos&amp;theta;</mi> <mn>2</mn> </msup> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <mfrac> <mi>L</mi> <mn>2</mn> </mfrac> <msubsup> <mi>d</mi> <msup> <mn>12</mn> <mo>&amp;prime;</mo> </msup> <mn>2</mn> </msubsup> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msqrt> </mrow> </mtd> </mtr> </mtable> </mfenced>
    Wherein, T1' and T2' is the projection of label T1 and T2 on the penholder of pen respectively；d11'It is antenna A1 to T1' distance；d12' It is antenna A1 to T2' distance；
    Nib translational speed：
    Wherein, Δ t represents the t+1 moment to the time interval of t；
    Pen tip position T0 (x (t), y (t)) is calculated, the formula of use is as follows：
    X (t)=d11'(t)sinβ1+Lcosθ(t)sin(β2+β1)
    Y (t)=d11'(t)cosβ1-Lcosθ(t)sin(β2+β1)
    Wherein, β1Represent the angle between A1A3 and A1T1', β2Represent the angle between A1T1' and T1'T2'；
    Four key feature amounts at above-mentioned all moment (t ... t+i...t+s) form the eigenmatrix of the stroke, i.e.,：
    <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mo>&amp;lsqb;</mo> <mi>&amp;theta;</mi> <mo>,</mo> <mi>&amp;Delta;</mi> <mi>h</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>T</mi> <mn>0</mn> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>&amp;Delta;</mi> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>T</mi> <mn>0</mn> <mrow> <mo>(</mo> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>&amp;Delta;</mi> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>T</mi> <mn>0</mn> <mrow> <mo>(</mo> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>&amp;Delta;</mi> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>T</mi> <mn>0</mn> <mrow> <mo>(</mo> <mi>x</mi> <mo>(</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>i</mi> </mrow> <mo>)</mo> <mo>,</mo> <mi>y</mi> <mo>(</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>i</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>&amp;Delta;</mi> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>T</mi> <mn>0</mn> <mrow> <mo>(</mo> <mi>x</mi> <mo>(</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>s</mi> </mrow> <mo>)</mo> <mo>,</mo> <mi>y</mi> <mo>(</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>s</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
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