CN107367244A - A kind of optimal striped sequence method of design twined based on time phase solution - Google Patents
A kind of optimal striped sequence method of design twined based on time phase solution Download PDFInfo
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Abstract
The invention discloses a kind of optimal striped sequence method of design twined based on time phase solution, initially set up the noise model of the three-dimensionalreconstruction under the slight out-of-focus appearance of projecting apparatus system, and the frequency response function of projecting apparatus system is measured, curve matching, according to the noise model of foundation and fit come frequency response function determine the optimal fringe frequency of coding；Then derive time phase solution twine during threshold condition, and derive that the frequency frames of different phase shift step numbers than formula, is chosen optimal phase shift step number according to the formula and combined according to the threshold condition；Threshold condition during finally being twined according to time phase solution, low frequency auxiliary fringe frequency is derived with the relational expression between high frequency fringes frequency, according to the relational expression, determine the frequency of stripeds at different levels one by one.The present invention finds the optimal fringe frequency f for encoding by being detected to the frequency response function of projecting apparatus systemopt, to improve the precision of three-dimensional measurement.
Description
Technical Field
The invention belongs to the field of fringe projection profilometry, and particularly relates to a method for designing an optimal fringe sequence based on time phase unwrapping.
Background
In the field of fringe projection profilometry, phase unwrapping is a very important process, and provides a powerful tool for the transition of wrapped phase to absolute phase, thereby providing guarantee for subsequent three-dimensional reconstruction. The phase unwrapping algorithm can be broadly divided into spatial phase unwrapping and temporal phase unwrapping. Spatial-domain phase unwrapping has been proposed to be developed to date, and has been in the past fifty years, and although it has been improved, the limitations of the algorithm itself cannot be avoided: in the treatment of complex and discontinuous objects with surfaces, it becomes difficult to adapt (Saldner HO, Huntley JM. "Temporal phase unwarping: application to surface profiling of discrete objects." Appl Opt,1997,36: 2770-. In order to break through the limitation of spatial domain phase unwrapping, a time domain phase unwrapping algorithm is gradually proposed to become a large hot spot of the phase unwrapping algorithm. In the short last two decades, there are tens of time domain phase unwrapping algorithms proposed. These algorithms can be broadly divided into three categories: one cycle, number theory and heterodyne (Chao Zuo. "Temporal phase unwrapting analysis profiler: A compatationview"). Compared with other two methods, the single period method has the strongest capacity of resisting time noise.
Although the time domain phase unwrapping algorithm can well process objects with complex or discontinuous surfaces, the time domain phase unwrapping additionally requires many auxiliary stripes (Yan Hu. "Real-time microscopic 3-D shape Measurement based on optimized pulse-width-modulation reconstruction" 2017 ") in order to obtain a high-precision three-dimensional reconstruction result, which undoubtedly seriously affects the efficiency of three-dimensional Measurement. Therefore, how to reasonably design the stripe sequence and effectively reduce the number of auxiliary stripes is a very important and deeply studied problem. In addition, in the model established in the past, the phenomenon that the projector has slight defocusing is almost ignored. And slight defocusing of the projector can obviously inhibit the modulation degree of the high-density sinusoidal stripes, thereby reducing the signal-to-noise ratio of the phase. Therefore, it is necessary and meaningful to find a coding stripe with an optimal frequency to achieve a three-dimensional measurement result with high precision. However, in the time domain phase unwrapping process, how to select the fringe frequency sequence to stabilize the time domain phase unwrapping and effectively avoid the transmission of time noise has not been proposed yet. The above three problems are three problems which are very important in fringe projection profilometry, are closely related to the accuracy, efficiency and system robustness of three-dimensional measurement, but are technically ignored and cannot be solved.
Disclosure of Invention
The invention aims to provide a method for designing an optimal fringe sequence based on fringe projection profilometry time phase unwrapping, so as to improve the precision, efficiency and robustness of three-dimensional reconstruction in fringe projection profilometry.
The technical solution for realizing the purpose of the invention is as follows: a method for designing an optimal fringe sequence based on time phase unwrapping comprises the steps of firstly establishing a three-dimensional reconstruction noise model of a projector system in a slight out-of-focus state, measuring and curve fitting a frequency response function of the projector system, and determining the encoded optimal fringe frequency according to the established noise model and the fitted frequency response function;
then deducing a threshold condition in the time phase unwrapping process, deducing a frequency-frame ratio formula of different phase shift steps according to the threshold condition, and selecting an optimal phase shift step combination according to the formula;
and finally, deducing a relational expression between the frequency of the low-frequency auxiliary stripes and the frequency of the high-frequency stripes according to a threshold condition in the time phase unwrapping process, and determining the frequency of each level of stripes one by one according to the relational expression.
Compared with the prior art, the invention has the following remarkable advantages: (1) finding the optimal fringe frequency f for encoding by detecting the frequency response function of the projector systemoptTo improve the accuracy of the three-dimensional measurement. (2) And deducing frequency-frame ratio formulas of different phase shift steps according to the threshold condition in the time phase unwrapping process, and selecting the optimal phase shift step according to the formula so as to improve the efficiency of three-dimensional measurement. (3) And deducing a mathematical relation between the frequency of the low-frequency auxiliary stripes and the frequency of the high-frequency stripes according to a threshold condition in the time phase unwrapping process, and determining the frequency of each level of stripes one by one according to the mathematical relation so as to improve the stability of three-dimensional measurement.
The present invention is described in further detail below with reference to the attached drawing figures.
Drawings
FIG. 1 is a schematic flow chart of the method for designing an optimal fringe sequence according to the present invention.
Fig. 2 is a diagram of the verification process of fig. 1 for phase shift combination 2-2-3.
FIG. 3 is a graph of phase results of the present invention measured on a standard ceramic plate.
Fig. 4 is a graph showing the results of the measurement of the standard ball according to the present invention.
Detailed Description
The invention relates to a method for designing an optimal fringe sequence based on time phase unwrapping, which comprises the steps of firstly establishing a three-dimensional reconstruction noise model of a projector system in a slight out-of-focus state, measuring and curve fitting a frequency response function of the projector system, and determining the encoded optimal fringe frequency according to the established noise model and the fitted frequency response function; then deducing a threshold condition in the time phase unwrapping process, deducing a frequency-frame ratio formula with different phase shift steps according to the threshold condition, and selecting the optimal phase shift step according to the formula; and finally, deducing a relational expression between the frequency of the low-frequency auxiliary stripes and the frequency of the high-frequency stripes according to a threshold condition in the time phase unwrapping process, and determining the frequency of each level of stripes one by one according to the relational expression.
With reference to fig. 1, the specific steps for implementing the method are as follows:
step one, establishing a three-dimensional reconstruction noise model of the projector system in a slight out-of-focus state.
Firstly, establishing a wrapped phase noise model in a projector focusing state. The noise variance of the wrapped phase (i.e. wrapped phase noise model) solved by the standard N-step phase shift algorithm and the 2+ 1-step phase shift algorithm is:
wherein,is the variance of the noise, σ, wrapping the phase2Is the noise variance of the fringe pattern captured by the camera, N is the phase shift step number, and B is the modulation of the fringe pattern captured by the camera. For the 2+1 step phase shift algorithm, N is the number of phase shift steps of the fringes that provide the average intensity.
And secondly, establishing a wrapped phase noise model of the projector in a defocusing state. In the out-of-focus state of the projector, the projected fringe modulation degree B (f) is inhibited:
wherein, B0Is the modulation degree, sigma, of the sinusoidal fringes projected by the projector in perfect focusgIs a coefficient proportional to the projector defocus level, f is the frequency of the encoded fringes, and h (f) is the frequency response function of the projector system.
Therefore, the constant B in equation (1) should be replaced by B (f) in equation (2):
and thirdly, establishing a three-dimensional reconstruction noise model of the projector in the out-of-focus state. In the process of unwrapping the wrapped phase into an absolute phase, the phase noise does not change. However, after mapping the absolute phase to the three-dimensional coordinate, the noise variance of the three-dimensional coordinate is greatly reduced, and the degree of reduction is determined by the frequency of the code stripes:
wherein,is the noise variance of the three-dimensional coordinates. In the formula (4), σ2Is the noise variance of the fringe pattern acquired by the camera, whose magnitude depends only on the measurement environment and the measurement equipment, and N is the number of phase shift steps. Although increasing the number of phase shift steps reduces noise, this reduction is very inefficient and increasing the number of phase shift steps can severely impact the efficiency of the measurement. Therefore, it is the best choice to focus on increasing the frequency of the fringes. However, the higher the fringe frequency, the better, due to the projector defocus. To reduce noise to the maximum, the maximum of the product of the fringe frequency f and the fringe modulation b (f) should be found.
And step two, measuring and curve fitting are carried out on the frequency response function of the projector system, and the optimal fringe frequency of the code is determined according to the established noise model and the fitted frequency response function.
① projecting sinusoidal stripes with different frequencies by the projector, projecting stripes with the frequency of 1 to the standard ceramic flat plate by the projector,where w is the lateral or longitudinal resolution of the projector and is acquired synchronously with the camera. The expression for these sinusoidal stripes is:
wherein, InIs the light intensity of the fringe pattern collected by the camera, a is the average light intensity, x is the horizontal or vertical pixel coordinate of the camera, n is the phase shift index, and n is 0, 1, 2.
And secondly, synchronously acquiring the sine stripes by using a camera and calculating the stripe modulation degree. From the gradation distribution of these fringe patterns, the modulation degree of the sinusoidal fringe can be calculated:
selecting the average value of the modulation degrees corresponding to the 200 pixel positions in the central area 200 × 200To eliminate errors.
③ fitting the calculated fringe modulation degrees to the 13 calculatedPerforming third-order polynomial fitting to obtainAnalysis formula (II)Where a, b, c, d are the fitted coefficients, and then plottedFinding the maximum value of the curveAnd selectValue range ofStripe frequency range [ f ] corresponding to intervalmin，fmax]Inner frequency as the optimal code stripe frequency fopt。
And step three, deducing a threshold condition in the time phase unwrapping process, deducing a frequency-frame ratio formula with different phase shift steps according to the threshold condition, and selecting the optimal phase shift step according to the formula.
Firstly, a threshold condition in the time phase unwrapping process is obtained. In the time phase unwrapping process, in order to ensure the correctness of the phase unwrapping, the fringe order error must be guaranteed to be 0, namely:
wherein, the [ alpha ], [ beta ]]Is the operator of nearest rounding, Δ khIs the fringe order error, fhIs the frequency of the high frequency fringes, flIs the frequency of the low-frequency auxiliary fringes, Δ ΦlIs the noise of the low-frequency auxiliary phase,is noise that high frequency wraps the phase. To ensure that the stripe order error is 0, then:
equation (8) can be further converted to:
to ensure the reliability of the result, we can approximate the maximum of the noise with 4.5 times the standard deviation, so we can get the threshold condition in the time phase unwrapping process:
wherein,is the standard deviation of the noise of the low frequency auxiliary phase,is the standard deviation of the noise of the high frequency phase. The purpose of the stripe sequence is to achieve the measurement system with the least stripesThe highest accuracy achieved, in short, is to be able to set the frequency foptThe wrapped phase corresponding to the stripe is accurately unwrapped using a phase shift method with a minimum number of stripe frames.
② definition of frequency-to-frame ratio for convenience of description, we will use (f, N) to denote a fringe pattern with frequency f, phase-shift step number N,shows in a striped pattern (f)lAnd N) the highest frequency of the fringe pattern corresponding to the wrapping phase that can be unwrapped when the fringe is an auxiliary fringe. To better analyze the problem, the concept of frequency-frame ratio (FFR) is proposed. When the phase shift is N and the frequency is flIf the high frequency f can be expanded when the stripes of (2) are auxiliary stripeshThe wrap phase corresponding to the stripe of which phase shift step number is 3 cannot be spread outhThe wrapped phase corresponding to +1 is the ratioIs the so-called frequency-to-frame ratio.
③ derive the frequency-frame ratio formula for different phase shift steps assuming 3 phase shift steps and f frequencylWhen the fringe of (2) is used as an auxiliary fringe, the number of phase shift steps of the fringe corresponding to the unwrapped wrapping phase is 3, and the frequency isThen, according to the formula (3) and the formula (10), it can be obtained:
similarly, when the phase shift step number is N, the frequency is flWhen the fringe of (2) is used as an auxiliary fringe, the number of phase shift steps corresponding to the unwrapped wrapping phase is 3, and the frequency isThen root ofFrom equations (3) and (10), it can be found that:
from equation (12), we can obtain:
it can thus be obtained that the auxiliary stripe frequency is flThe frequency-frame ratio when the phase shift step number is N is:
wherein, the upper mark flIndicating the frequency of the assist fringe and the subscript N indicating the number of phase shift steps of the assist fringe. Assuming that a continuous phase obtained by 2+1 step phase shift is used as the auxiliary phase, the frequency f of the auxiliary fringe can be obtained according to the above analysislThe frequency of the unwrapping phase which can be unwrapped by the two phase shifting steps and the corresponding frequency-frame ratio are as follows:
as can be seen from equations (15) and (16), for the standard N-step phase shift, N is 3, but the frequency-frame ratio of the 2+1 step phase shift is larger than that of the three step phase shift.
Selecting the optimal phase shift step number combination. From the definition of the frequency-to-frame ratio, it can be known that: the larger the frame-to-frame ratio, the greater the contribution of the corresponding number of phase shift steps to the temporal phase spread. Therefore, we should try to choose the continuous phase obtained by 2+1 step phase shift to spread out the high frequency envelope phase, but at least one set of fringe pattern with 3 phase shift steps is needed to provide the average intensity for 2+1 step phase shift. From this principle we can derive the best combination of phase shift steps: the phase shift steps of the highest frequency stripe are 3 or 4, and the phase shift steps of the other frequencies are all 2. We list the optimal phase shift step combinations for some cases, as shown in table 1.
TABLE 1 partial optimal stripe combinations
And step four, deducing a relational expression between the low-frequency auxiliary stripe frequency and the high-frequency stripe frequency according to a threshold condition in the time phase unwrapping process and determining the frequency of each level of stripes one by one according to the relational expression.
① relationship between the frequency of the low frequency assist stripes and the frequency of the high frequency stripes, when the high frequency stripes correspond to a phase shift of 2+1 steps, and the low frequency stripes correspond to a phase shift of N steps,heelWill be maximized, thus:
as can be seen from the phase shift step combinations in Table 1, N is ≦ 4 for the optimal phase shift step combination. In addition to this, the present invention is,the size of the projection system is related to the defocusing degree of the projection system, and the specific value range can be determined according to the detected frequency responseShould be determined as a function. Therefore, we can consider that,the upper limit of (d) is a constant related to the defocus level of the projector system, called PDC, which can be expressed as:
thus, equation (10) can be:
in general, for a slightly out-of-focus projector system,and in order to improve the stability of the system, twice the noise variance of the detected low-frequency auxiliary phase is used as the noise variance in the actual measurement, the formula (19) can be further converted into:
the equation (20) can be used as a relational equation that must be satisfied to determine the frequencies of the low and high frequency fringes.
② determining the optimal fringe frequency sequence step by step according to the formula (20). first, determining the optimal phase shift step combination (selecting an optimal phase shift step combination from the sequence of the frame numbers from small to large in the table 1) according to the total frame number of the fringe pattern to be used, and then, determining the fringe frequency of each stage step by step according to the selected phase shift step combination by detecting the noise standard deviation of the single period phaseWill f isl1 and detectedSubstituting into equation (20), f is calculatedh(ii) a Then detecting the frequency fhNoise standard deviation of phase ofWill f isl＝fhAnd detectedSubstituting into equation (20), f is calculatedh(ii) a Detecting and calculating one by one according to the sequence until finally calculating the frequency f corresponding to the last group of stripeshiUntil now.
③ comparison fhiAnd foptIf f is large or smallhi≥foptIf so, then the selected phase shift step combination does not meet the requirement, and if not, then the next phase shift step combination is selected according to the sequence in step ② to determine the fringe frequency sequence.
FIG. 1 is a schematic flow chart of the method for designing an optimal fringe sequence according to the present invention. Fig. 2 is a diagram of the verification process of fig. 1 for phase shift combination 2-2-3.
Fig. 3 is the phase results measured on a standard ceramic plate. FIGS. 3(a) - (c) are absolute phase diagrams of the frequencies of each stage of the fringe frequency series {1,8,180 }. As can be seen from the figure, 212 error points appear in the last absolute phase. FIGS. 3(d) - (f) are absolute phase diagrams of each stage frequency of the fringe frequency series {1,15,180}, which are designed according to the present invention, and it can be seen that no error point appears in the absolute phase of each stage frequency. FIGS. 3(g) - (i) are absolute phase diagrams of the stage frequencies of the fringe frequency series {1,25,180 }. As can be seen from the figure, 138 error points occurred in the last absolute phase. From this comparison, it can be seen that the striped sequence designed according to the present invention has a higher stability.
Fig. 4 is a measurement result for a standard ball. FIG. 4 shows a standard sphere in the fringe frequency series {1,15,90}, where,
{1,15,180} and {1,15,256}, the first row is the measurement of the standard sphere, the second row is the calculated radius profile, and the third row is the radius error map. As can be seen from the figure, the fringe sequence {1,15,180} designed according to the present invention has the highest measurement accuracy.
Claims (5)
1. A method for designing an optimal fringe sequence based on time phase unwrapping is characterized by firstly establishing a three-dimensional reconstruction noise model of a projector system in a slight out-of-focus state, measuring and curve fitting a frequency response function of the projector system, and determining the encoded optimal fringe frequency according to the established noise model and the fitted frequency response function;
then deducing a threshold condition in the time phase unwrapping process, deducing a frequency-frame ratio formula of different phase shift steps according to the threshold condition, and selecting an optimal phase shift step combination according to the formula;
and finally, deducing a relational expression between the frequency of the low-frequency auxiliary stripes and the frequency of the high-frequency stripes according to a threshold condition in the time phase unwrapping process, and determining the frequency of each level of stripes one by one according to the relational expression.
2. The method for designing optimal fringe sequence based on time-phase unwrapping as recited in claim 1, wherein the process of establishing a three-dimensional reconstructed noise model of the projector system in a slightly out-of-focus state is as follows:
establishing a wrapped phase noise model in a projector focusing state, namely solving the noise variance of a wrapped phase through a standard N-step phase shift algorithm and a 2+ 1-step phase shift algorithm, wherein the noise variance of the wrapped phase is as follows:
wherein,is the variance of the noise, σ, wrapping the phase2The noise variance of the fringe pattern acquired by the camera, N is the phase shift step number, B is the modulation degree of the fringe pattern captured by the camera, and for the 2+1 step phase shift algorithm, N is the phase shift step number of the fringe providing the average light intensity;
secondly, establishing a wrapping phase noise model of the projector in the out-of-focus state, wherein the wrapping phase noise model can inhibit the projected fringe modulation degree B (f) in the out-of-focus state:
<mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>B</mi> <mn>0</mn> </msub> <mi>H</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>B</mi> <mn>0</mn> </msub> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mn>2</mn> <msup> <mi>&pi;</mi> <mn>2</mn> </msup> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msup> <mi>f</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
wherein, B0Is the modulation degree, sigma, of the sinusoidal fringes projected by the projector in perfect focusgIs a coefficient proportional to the projector defocus level, f is the frequency of the encoded fringes, h (f) is the frequency response function of the projector system; thus, the constant B in equation (1) is replaced by B (f) in equation (2):
establishing a three-dimensional reconstruction noise model in a defocused state of the projector, wherein in the process of unwrapping the wrapped phase into an absolute phase, the phase noise cannot change, but after mapping the absolute phase into a three-dimensional coordinate, the noise variance of the three-dimensional coordinate can be greatly reduced, and the reduction degree is determined by the frequency of the coding stripes:
wherein,is the noise variance of the three-dimensional coordinates, in equation (4), σ2The noise variance of the fringe pattern acquired by the camera is only related to the measuring environment and the measuring equipment, N is the phase shift step number, and formula (4) is an established noise model of the three-dimensional reconstruction of the projector in the defocusing state.
3. The method of claim 1, wherein the determining and curve fitting are performed on a frequency response function of the projector system, and the encoded optimal fringe frequency is determined according to the established noise model and the fitted frequency response function, and the method comprises the following steps:
① sinusoidal stripes with different frequencies are projected by the projector, namely the stripes are projected to a standard ceramic flat plate by the projector with the frequency of 1,the expression of these sinusoidal stripes is:
<mrow> <msub> <mi>I</mi> <mi>n</mi> </msub> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mfrac> <mi>x</mi> <mi>w</mi> </mfrac> <mi>f</mi> <mo>*</mo> <mn>2</mn> <mi>&pi;</mi> <mo>-</mo> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> <mo>*</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
where w is the transverse or longitudinal resolution of the projector, InIs the light intensity of the fringe pattern collected by the camera, a is the average light intensity, x is the horizontal or vertical pixel coordinate of the camera, n is the phase shift index, and n is 0, 1, 2;
secondly, synchronously acquiring the sine stripes by using a camera and calculating the stripe modulation degree, and calculating the modulation degree of the sine stripes according to the gray distribution of the stripe patterns:
<mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <msub> <mi>I</mi> <mi>n</mi> </msub> <mi>sin</mi> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>&pi;</mi> </mrow> <mn>3</mn> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <msub> <mi>I</mi> <mi>n</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>&pi;</mi> </mrow> <mn>3</mn> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
selecting the average value of the modulation degrees corresponding to the 200 pixel positions in the central area 200 × 200To eliminate errors;
③ fitting the calculated fringe modulation degrees, i.e. 13 calculatedPerforming third-order polynomial fitting to obtainAnalysis formula (II)Where a, b, c, d are the fitted coefficients, and then plottedFinding the maximum value of the curveAnd selectValue range ofStripe frequency range [ f ] corresponding to intervalmin，fmax]Inner frequency as the optimal code stripe frequency fopt。
4. The method according to claim 1, wherein a threshold condition is derived during the time phase unwrapping process, and a frequency-to-frame ratio formula with different phase shift steps is derived according to the threshold condition, and an optimal phase shift step combination is selected according to the formula, wherein the specific process is as follows:
firstly, a threshold condition in the time phase unwrapping process is obtained, and in order to ensure the accuracy of the phase unwrapping process, a stripe order error is required to be ensured to be 0, namely:
wherein, the [ alpha ], [ beta ]]Is the operator of nearest rounding, Δ khIs the fringe order error, fhIs the frequency of the high frequency fringes, flIs the frequency of the low-frequency auxiliary fringes, Δ ΦlIs the noise of the low-frequency auxiliary phase,is noise of high frequency wrapped phase, in order to ensure that the fringe order error is 0, then:
equation (8) further translates to:
to ensure the reliability of the results, the maximum value of the noise is approximated with 4.5 times the standard deviation, thus obtaining the threshold condition in the time-phase unwrapping process:
wherein,is the standard deviation of the noise of the low frequency auxiliary phase,is the standard deviation of the noise of the high frequency phase;
② definition of frequency-to-frame ratio for convenience of description, a fringe pattern having a frequency f, a phase shift step number N is expressed by (f, N),shows in a striped pattern (f)lN) as fringe pattern corresponding to wrapping phase capable of being spread when auxiliary fringe is formedThe highest frequency;
when the phase shift is N and the frequency is flIf the high frequency f can be expanded when the stripes of (2) are auxiliary stripeshThe wrap phase corresponding to the stripe of which phase shift step number is 3 cannot be spread outhThe wrapped phase corresponding to +1 is the ratioIs the frequency to frame ratio;
③ derive the frequency-frame ratio formula for different phase shift steps assuming 3 phase shift steps and f frequencylWhen the fringe of (2) is used as an auxiliary fringe, the number of phase shift steps of the fringe corresponding to the unwrapped wrapping phase is 3, and the frequency isThen, according to the formula (3) and the formula (10), we obtain:
<mrow> <mfrac> <msubsup> <mi>f</mi> <mn>3</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <msub> <mi>f</mi> <mi>l</mi> </msub> </mfrac> <msqrt> <mfrac> <mrow> <mn>2</mn> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>3</mn> <msup> <mi>B</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </msqrt> <mo>=</mo> <mfrac> <mi>&pi;</mi> <mn>4.5</mn> </mfrac> <mo>-</mo> <msqrt> <mfrac> <mrow> <mn>2</mn> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>3</mn> <msup> <mi>B</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msubsup> <mi>f</mi> <mn>3</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
similarly, when the phase shift step number is N, the frequency is flWhen the fringe of (2) is used as an auxiliary fringe, the number of phase shift steps corresponding to the unwrapped wrapping phase is 3, and the frequency isThen, according to the formula (3) and the formula (10), we obtain:
<mrow> <mfrac> <msubsup> <mi>f</mi> <mi>N</mi> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <msub> <mi>f</mi> <mi>l</mi> </msub> </mfrac> <msqrt> <mfrac> <mrow> <mn>2</mn> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> </mrow> <mrow> <msup> <mi>NB</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </msqrt> <mo>=</mo> <mfrac> <mi>&pi;</mi> <mn>4.5</mn> </mfrac> <mo>-</mo> <msqrt> <mfrac> <mrow> <mn>2</mn> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>3</mn> <msup> <mi>B</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msubsup> <mi>f</mi> <mi>N</mi> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> </msqrt> <mo>&le;</mo> <mfrac> <mi>&pi;</mi> <mn>4.5</mn> </mfrac> <mo>-</mo> <msqrt> <mfrac> <mrow> <mn>2</mn> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>3</mn> <msup> <mi>B</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msubsup> <mi>f</mi> <mn>3</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> </msqrt> <mo>=</mo> <mfrac> <msubsup> <mi>f</mi> <mn>3</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <msub> <mi>f</mi> <mi>l</mi> </msub> </mfrac> <msqrt> <mfrac> <mrow> <mn>2</mn> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>3</mn> <msup> <mi>B</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
from equation (12):
<mrow> <msubsup> <mi>f</mi> <mi>N</mi> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>&le;</mo> <msqrt> <mfrac> <mi>N</mi> <mn>3</mn> </mfrac> </msqrt> <msubsup> <mi>f</mi> <mn>3</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
thereby obtaining an auxiliary fringe frequency of flThe frequency-frame ratio when the phase shift step number is N is:
<mrow> <msubsup> <mi>FFR</mi> <mi>N</mi> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>=</mo> <mfrac> <msubsup> <mi>f</mi> <mi>N</mi> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mrow> <msub> <mi>Nf</mi> <mi>l</mi> </msub> </mrow> </mfrac> <mo>&le;</mo> <mfrac> <msubsup> <mi>f</mi> <mn>3</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mrow> <msqrt> <mrow> <mn>2</mn> <mi>N</mi> </mrow> </msqrt> <msub> <mi>f</mi> <mi>l</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
wherein, the upper mark flDenotes the frequency of the auxiliary fringe, and the subscript N denotes the phase shift step number of the auxiliary fringe;
assuming that a continuous phase obtained by 2+1 step phase shift is used as an auxiliary phase, the frequency of the auxiliary fringe is flThe phase shift step number isThe frequency of the unwrapping phase and the corresponding frequency-frame ratio in the two steps are as follows:
<mrow> <msubsup> <mi>f</mi> <mn>2</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>&GreaterEqual;</mo> <mfrac> <mn>2</mn> <msqrt> <mn>8</mn> </msqrt> </mfrac> <msubsup> <mi>f</mi> <mn>3</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msubsup> <mi>FFR</mi> <mn>2</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mo>=</mo> <mfrac> <msubsup> <mi>f</mi> <mn>2</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </mrow> </mfrac> <mo>&GreaterEqual;</mo> <mfrac> <msubsup> <mi>f</mi> <mn>3</mn> <msub> <mi>f</mi> <mi>l</mi> </msub> </msubsup> <mrow> <msqrt> <mn>8</mn> </msqrt> <msub> <mi>f</mi> <mi>l</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
as can be seen from equations (15) and (16), for the standard N-step phase shift, N is 3, but the frequency-frame ratio of the 2+1 step phase shift is larger than that of the three step phase shift.
Selecting the optimal phase shift step number combination. According to the definition of the frequency-frame ratio, when the frequency-frame ratio is larger, the corresponding phase shift step number contributes more to the time phase unwrapping, so that continuous phases obtained by 2+1 phase shifts are selected as much as possible to unwrapp the high-frequency wrapped phase, but at least one group of stripe patterns with the phase shift step number of 3 is needed to provide average light intensity for the 2+1 phase shifts, so that the optimal phase shift step combination is obtained: the phase shift steps of the highest frequency stripe are 3 or 4, and the phase shift steps of the other frequencies are all 2.
5. The method for designing an optimal fringe sequence based on time phase unwrapping as claimed in claim 1, wherein a relationship between a frequency of a low frequency auxiliary fringe and a frequency of a high frequency fringe is derived according to a threshold condition during the time phase unwrapping, and the frequencies of the fringes at each level are determined one by one according to the relationship, wherein the specific process is as follows:
① relationship between the frequency of the low frequency assist stripes and the frequency of the high frequency stripes, when the high frequency stripes correspond to a 2+1 step phase shift, and the low frequency stripes correspond to an N step phase shift,heelWill be maximized, thus:
for the optimal phase shift step number combination, N is less than or equal to 4; in addition to this, the present invention is,the value of the projection system is related to the defocusing degree of the projection system, and the specific value range is determined according to the detected frequency response function; therefore, the temperature of the molten metal is controlled,is at the upper limit of the projectorA constant related to the defocus of the system, called PDC, is expressed as:
<mrow> <mi>P</mi> <mi>D</mi> <mi>C</mi> <mo>=</mo> <msqrt> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </msqrt> <mfrac> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
thus, equation (10) is:
<mrow> <mfrac> <msub> <mi>f</mi> <mi>h</mi> </msub> <msub> <mi>f</mi> <mi>l</mi> </msub> </mfrac> <mo>&le;</mo> <mfrac> <mi>&pi;</mi> <mrow> <mn>4.5</mn> <msub> <mi>&sigma;</mi> <mrow> <msub> <mi>&Delta;&Phi;</mi> <mi>l</mi> </msub> </mrow> </msub> </mrow> </mfrac> <mo>-</mo> <mi>P</mi> <mi>D</mi> <mi>C</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
in order to improve the stability of the system, twice the noise variance of the detected low-frequency auxiliary phase is used as the noise variance in the actual measurement, and the formula (19) is further converted into:
<mrow> <mfrac> <msub> <mi>f</mi> <mi>h</mi> </msub> <msub> <mi>f</mi> <mi>l</mi> </msub> </mfrac> <mo>&le;</mo> <mfrac> <mrow> <mn>2</mn> <mi>&pi;</mi> </mrow> <mrow> <mn>9</mn> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>&sigma;</mi> <mrow> <msub> <mi>&Delta;&Phi;</mi> <mi>l</mi> </msub> </mrow> </msub> </mrow> </mfrac> <mo>-</mo> <mn>2</mn> <msqrt> <mn>10</mn> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
the formula (20) is used as a relational expression for judging the satisfaction between the frequencies of the low frequency stripes and the high frequency stripes;
② determining the optimal fringe frequency sequence step by step according to the formula (20), firstly determining the optimal phase shift step combination according to the total frame number of the fringe pattern, and then determining the fringe frequency of each stage step by step according to the selected phase shift step combination by detecting the noise standard deviation of the single period phaseWill f isl1 and detectedSubstituting into equation (20), f is calculatedh(ii) a Then detecting the frequency fhNoise standard deviation of phase ofWill f isl＝fhAnd detectedSubstituting into equation (20), f is calculatedh(ii) a Detecting and calculating one by one according to the sequence until finally calculating the frequency f corresponding to the last group of stripeshiUntil the end;
③ comparison fhiAnd foptIf f is large or smallhi≥foptIf the selected combination of phase shift steps meets the requirements, otherwise if the selected combination of phase shift steps fails to meet the requirements, the total number of frames of the fringe pattern should be increased, and the combination of phase shift steps determined again in accordance with step ② to determine the fringe frequency sequence.
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