Four-step phase shift reconstruction and improved least square iterative phase unwrapping method

Abstract

The present application relates to the field of structured light three-dimensional measurement technology, and particularly relates to a method for reconstructing and improving least square iterative phase unwrapping based on four-step phase shifting. An object light field is obtained on an imaging plane after a lens in a holography process; four reference light fields are generated by changing the phase of the reference light to interfere with the object light field, four-step phase shifting holograms are obtained, and an arctangent calculation is performed to obtain a wrapped phase map; a discrete Poisson equation of the wrapped phase map is established according to the least square method, and iterative solving is performed to obtain the real phase of the wrapped phase map. The unwrapping algorithm of the present application is optimized by the least square method multiple iterations, so that the partial derivative of the real phase and the difference value of the wrapped phase difference are minimized, the unwrapping error under noise is effectively reduced, and the Poisson equation is introduced for iterative solving, so that the quality of phase unwrapping is improved, and the robustness of the algorithm is improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of structured light three-dimensional measurement technology, and particularly relates to a method for reconstructing and improving least square iterative phase unwrapping. BACKGROUND

[0002] In optical measurement, a key step for processing interference signals to extract light field information is to perform arctangent function calculation; since the arctangent function will cause the recovered phase data to be wrapped between [-π, +π], when the phase change is greater than one wavelength, the phase will be discontinuous, so the real measurement information distribution cannot be directly obtained, therefore, in order to obtain the real phase distribution, the truncated wrapped phase must be connected in actual measurement, and this process is called phase unwrapping.

[0003] At present, the classical phase unwrapping algorithms can be divided into two categories: one is the path-dependent algorithm represented by the branch-cut method and the quality map guided method, and the other is the path-independent algorithm represented by the least square method.(1) The path-dependent algorithm finds the best integral path through various strategies, and integrates the path to obtain the unwrapped real phase, and at the same time, the noise area is bypassed to reduce or avoid the error transmission accumulation effect in the phase unwrapping process;(2) The most typical representative of the path-independent algorithm is the least norm method, which converts the unwrapping into an objective optimization problem, and its principle is based on the minimum norm criterion framework, defines an equivalent function which can reflect the difference between the unwrapped phase gradient and the wrapped phase gradient, and then finds the solution that minimizes the cost function through reasonable and effective mathematical methods. From the principle, the phase unwrapping seems to be a very simple problem, but due to the problems such as noise, local shadow, stripe pattern rupture, under-sampling and error transmission caused by the rapid change of phase in space in the wrapped phase obtained in actual situation, the phase unwrapping becomes a very difficult problem.

[0004] Although many algorithms have been proposed by numerous domestic and foreign experts and scholars, at present, there is no algorithm that can completely solve all the problems in phase unwrapping, therefore, it is necessary to conduct in-depth and thorough research on the phase unwrapping algorithm, and find a digital holographic microscopic measurement phase unwrapping method suitable for various complex situations. SUMMARY

[0005] In order to solve the problems in the prior art, the application provides a four-step phase shift reconstruction and improved least square iterative phase unwrapping method. The method comprises the following steps: obtaining an object light field on an imaging surface after a light field passes through a lens in a holography process; generating four reference light fields by changing the phase of the reference light, and performing interference with the object light field to obtain four-step phase shift holograms and perform arctangent calculation to obtain a wrapped phase map; establishing a discrete Poisson equation of the wrapped phase map according to an improved least square method, and performing multiple iteration solving on the discrete Poisson equation to obtain a real phase of the wrapped phase map. The least square method multiple iteration optimization unwrapping algorithm of the application minimizes the partial derivative of the real phase and the difference value of the wrapped phase difference, effectively reduces the unwrapping error under noise, introduces the Poisson equation, and improves the robustness of the algorithm while improving the quality of phase unwrapping.

[0006] The application adopts the following technical scheme, a four-step phase shift reconstruction and improved least square iterative phase unwrapping method, comprising:

[0007] Obtaining an object light field on an imaging surface after a light field passes through a lens in a holography process;

[0008] Generating four reference light fields by changing the phase of the reference light, and performing interference with the object light field to obtain four-step phase shift holograms;

[0009] Performing arctangent calculation on the four-step phase shift holograms to obtain a wrapped phase map;

[0010] Establishing a discrete Poisson equation of the wrapped phase map according to an improved least square method, and performing multiple iteration solving on the discrete Poisson equation established by the improved least square method to obtain a real phase of the wrapped phase map.

[0011] Further, the four reference light fields are generated by changing the phase of the reference light, and interference with the object light field is performed to obtain four-step phase shift holograms, comprising:

[0012] Supposing that the object light field and the reference light field at a point (x, y) in the holography process are And R(x,y)=r(x,y)e jφ(x,y) The reference light and the object light interfere with each other, at this time, the phase of the object light is: the reference light and the object light interfere with each other to record the phase of the object light: I(x,y):

[0013]

[0014] Wherein, o(x,y) represents the amplitude of the object light wave, φ(x,y) represents the phase distribution of the object light field; r(x,y) represents the amplitude of the reference light field, and φ(x,y) represents the phase distribution of the reference light field;

[0015] The phase of the reference light is changed Four reference light fields are obtained as follows:

[0016] R i (x,y)=R(x,y)exp[j(i-1)π / 2],i=1,2,3,4

[0017] Four-step phase shift holograms are obtained as follows:

[0018]

[0019] wherein, I1 represents an image plane hologram Figure 1 ; I2 represents an image plane hologram Figure 2 ; I3 represents an image plane hologram Figure 3 ; I4 represents an image plane hologram Figure 4 ; o(x,y) represents an amplitude of an object light wave, φ(x,y) represents a phase distribution of an object light field; r(x,y) represents an amplitude of a reference light field, and φ(x,y) represents a phase distribution of a reference light field.

[0020] Further, a wrapped phase map is obtained by performing an inverse tangent calculation according to the four-step phase shift hologram, and the expression is as follows:

[0021]

[0022] wherein, φ(x,y) represents a phase distribution of an object light field; φ(x,y) represents a phase distribution of a reference light field; I1 represents an image plane hologram Figure 1 ; I2 represents an image plane hologram Figure 2 ; I3 represents an image plane hologram Figure 3 ; I4 represents an image plane hologram Figure 4 .

[0023] Further, a discrete Poisson equation of the wrapped phase map is established according to an improved least square method, and the equation comprises:

[0024] a wrapped phase is set as a real phase, and wrapped phases in x and y directions of the wrapped phase map are and then:

[0025]

[0026] wherein, W is a wrapped operator;

[0027] A wrapped phase difference expression of the wrapped phase map is established based on the improved least square algorithm:

[0028]

[0029] φ i,j Taking derivative and setting it to 0, the Poisson equation is obtained:

[0030] φ i+1,j +φ i,j +φ i,j+1 -φ i,j-1 -4φ i,j =ρ i,j

[0031] wherein, represents the value of the wrapped phase.

[0032] Further, the discrete Poisson equation is solved by multiple iterations, comprising:

[0033] The boundary condition of the discrete Poisson equation is:

[0034]

[0035]

[0036] Substituting the wrapped phase in the x direction and the y direction of the wrapped phase map into the discrete Poisson equation, the following equation is obtained:

[0037]

[0038] Let then:

[0039] k i+1,j +k i-1,j +k i,j+1 +k i,j-1 -k i,j = a i,j

[0040] The above equation is iteratively solved until the solution no longer changes, and the real phase of the wrapped phase map is obtained.

[0041] The beneficial effects of the present application are: the present application proposes an unwrapping algorithm based on multiple iterations of least square method optimization for the problem of unwrapping phase in the case of large noise or steep gradient: in order to minimize the partial derivative of the real phase and the difference of the wrapped phase difference, the Poisson equation obtained by derivation is combined with the boundary condition and the iterative least square method solving strategy to convert the unwrapping problem into an equivalent continuous minimization problem, and finally the optimal solution is obtained to obtain a better unwrapping result; through multiple iterations: (1) reduce the error to a certain extent and improve the unwrapping accuracy under noise; (2) improve the quality of phase unwrapping while minimizing the partial derivative of the real phase and the difference of the wrapped phase difference, and reduce the calculation time; (3) the boundary condition is an integer in the iteration process, which further improves the robustness of the unwrapping algorithm. BRIEF DESCRIPTION OF DRAWINGS

[0042] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0043] Figure 1 The flowchart of the four-step phase shift reconstruction and improved least square iteration phase unwrapping method of the embodiment of the present application is shown in the figure.

[0044] Figure 2 The simulation experiment result diagram of the least square algorithm unwrapping of the embodiment of the present application is shown in the figure.

[0045] Figure 3 The optical path structure diagram of the microscopic digital holographic observation of onion cells of the embodiment of the present application is shown in the figure.

[0046] Figure 4 The hologram and reconstruction phase diagram of the four-step phase shift microscopic digital holographic simulation experiment result of the embodiment of the present application is shown in the figure. DETAILED DESCRIPTION

[0047] The technical solutions in the embodiments of the present application will be described clearly and completely in the following with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only some embodiments of the present application, not all. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the scope of protection of the present application.

[0048] The flowchart of the four-step phase shift reconstruction and improved least square iteration phase unwrapping method of the embodiment of the present application is shown in the figure. Figure 1As shown, comprising:

[0049] Obtaining the object light field of the light field passing through the lens on the imaging plane in the holography process;

[0050] In the holography process, the object light field and the reference light field at a certain point (x, y) are respectively And R(x, y) = r(x, y)e jφ(x,y) The reference light and the object light interfere with each other, so that the phase of the object light is recorded as:

[0051]

[0052] If Where k = 0, ±1, ±2..., then a bright line will appear at (x, y); if Then a dark line will appear at (x, y). Therefore, the hologram is actually an interference pattern, and the phase difference between the object light and the reference light produces interference fringes with bright and dark bands, which is used to record the phase.

[0053] Four reference light fields are generated by changing the phase of the reference light, and interference is carried out with the object light field to obtain four-step phase shift holograms;

[0054] The four-step phase shift method used in the embodiment of the application is usually used in image plane holography, that is, the holographic recording surface needs to be placed on the image plane of the clear imaging system, and the phase of the reconstructed light field needs to be changed accurately by changing the phase of the reference light, and at least two digital holograms are shot. Taking a typical four-step phase shift method as an example, the phase of the reference light is changed by Then the four reference light fields are:

[0055] R i (x, y) = R(x, y)exp[j(i-1)π / 2], i = 1, 2, 3, 4

[0056] Corresponding, four different holograms are recorded in turn:

[0057]

[0058] The wrapped phase pattern is obtained by arctangent calculation according to the four-step phase shift hologram;

[0059] The difference between the object light field and the reference light field at (x, y) can be obtained by the following formula:

[0060]

[0061] The real phase of the wrapped phase pattern is obtained by solving the discrete Poisson equation of the wrapped phase pattern through multiple iterations according to the least square method;

[0062] The wrapped phase is set as It is a true phase, and The package phases in the x and y directions of the package phase diagram are respectively and but:

[0063]

[0064] Where W is the wrapper operator; its purpose is to add or subtract 2π from the wrapper phase to ensure... and It lies between [-π, π].

[0065] To minimize the difference between the partial derivative of the true phase and the wrapped phase difference, this embodiment of the invention establishes the wrapped phase difference expression of the wrapped phase map based on an improved least squares algorithm:

[0066]

[0067] For φ in the above formula i,j Taking the derivative and setting it to 0, we obtain the Poisson equation:

[0068] φ i+1,j +φ i,j +φ i,j+1 -φ i,j-1 -4φ i,j =ρ i,j

[0069] in, This represents the value of the wrapper phase.

[0070] The boundary conditions for the discrete Poisson equation are:

[0071]

[0072]

[0073] Substituting the wrapped phase in the x and y directions of the wrapped phase diagram into the discrete Poisson equation, we get:

[0074]

[0075] make Then we have:

[0076] k i+1,j +k i-1,j +k i,j+1 +k i,j-1 -k i,j =a i,j

[0077] Iterating the above formula until the solution no longer changes, the real phase of the wrapped phase map is obtained.

[0078]

[0079] The abrupt conversion caused by arctangent in the operation process becomes a smooth continuous phase map, which can approach the real phase value and improve the accuracy. Compared with the traditional iteration, the technical means proposed in the embodiment of the application solves the problem that if the phase inconsistency area is crossed in the unwrapping process, the phase error will propagate in the whole space. The application improves the least square method iteration by using a data fusion algorithm. Even when the noise is relatively large, the application can still normally solve the real wrapped phase. The wrapped phase obtained by the application is close to the result when the external noise coefficient is 0.06. The wrapped phase can be better restored, and the unwrapped phase is basically consistent with the original phase. Meanwhile, the defects of the real phase can also be smoothed, that is, if a sharp peak or steep slope occurs, corresponding smoothing can be performed, which improves the stability and accuracy of the algorithm to a certain extent.

[0080] Another specific embodiment of the application is shown in the following table. Figure 2 The simulation experiment results of unwrapping by the least square algorithm are shown in the table. Figure 2 a and 2b are the wrapped phase without noise (noise coefficient mu=0) and the unwrapped result thereof, Figure 2 c and 2d are the wrapped phase with noise (noise coefficient mu=0.3) and the unwrapped result thereof. By comparing the two, it can be seen that the least square algorithm can obtain accurate results when there is no noise. Once there is noise, the phenomenon of peak clipping and valley filling occurs. This is because the least square algorithm uses cosine transformation or Fourier transformation. As long as there is noise in one point of the wrapped phase, the noise will be propagated to other points in the two-dimensional space, thereby causing errors in the unwrapped phase of all points. Therefore, the accurate solution cannot be obtained directly by using the least square algorithm. However, since the least square method is a global algorithm, there is no discontinuity such as line drawing in the solution. The least square unwrapping algorithm obtains a smooth curve that can restore the original curve well. However, compared with the original curve, the longitudinal coordinate of the curve is smaller, so the curve is more gentle, and the value of each point changes, and the overall error is small. Although there are certain error points in the wrapped phase map, compared with the other two methods, the algorithm can approach the real phase value and improve the accuracy through multiple iterations, as shown in e and 2f. Figure 2

[0081] In another specific embodiment of the application, the related data processing of the reconstructed object light field and the reconstructed light field phase of the hologram taken by the experiment is completed, the four-step phase shift method is used for calculation, and the optical path is as shown in the following table. Figure 3 ​As shown is a micro digital holographic optical path for observing onion cells; a He-Ne laser beam (wavelength λ = 632.8 nm) is split into two beams by a beam splitter BS1, one of which is reflected by a mirror M to illuminate the sample, and the transmitted light is projected onto a charge-coupled device (CCD) by a microscope objective, and the other beam is converted into a plane beam by a beam expander BE and a pinhole filter h, and then collimated by a collimating lens L. The mirror PZT M and the beam splitter BS2 are driven by a piezoelectric ceramic tube and reflect the light onto the CCD as the reference light. In the experiment, the PZT is controlled by a computer to make accurate phase, four holograms are recorded, and the reconstructed phase is calculated by the inverse tangent function.

[0082] As shown in Figure 4 the hologram and the reconstructed phase after Matlab simulation are shown, Figure 4 (a)- Figure 4 (d) represents four-step phase-shift holograms, Figure 4 (e) represents the reconstructed phase, Figure 4 (f) is a reconstructed phase profile showing that the reconstructed phase is wrapped in .

[0083] According to the simulation results Figure 4 (f), it can be seen that the four-step phase-shift method can effectively solve the phase information of the sinusoidal fringe pattern in the presence of certain noise, thereby calculating the accurate absolute phase. By using the least squares method to iteratively unwrap the phase of the fringe pattern reflected by the mirror PZT M and the beam splitter BS2 to the CCD, only four phase-shift fringe patterns are needed to solve the phase and unwrap the phase, and finally the three-dimensional phase information of the object is obtained.

[0084] The above only describes the preferred embodiments of the present application and should not be used to limit the present application, and any modification, equivalent replacement, improvement, etc. made within the spirit and principles of the present application should be included in the protection scope of the present application.

Claims

1. A method for reconstruction and improved least square iterative phase unwrapping based on four-step phase shift, characterized in that, Comprise: acquire the object light field of the light field through the lens on the imaging plane in the holography process; generate four reference light fields by changing the phase of the reference light, and interfere with the object light field respectively to obtain four-step phase shift holograms; comprising: Let the point in the holographic process be... The object light field and the reference light field are respectively and The reference beam and the object beam interfere with each other, and the phase of the object beam is recorded as I(x,y). ； wherein, an amplitude of the representative light wave, a phase distribution of the representative light field; an amplitude of the reference light field, a phase distribution of the reference light field; The phase of the reference light is changed each time resulting in four reference light fields: ； obtain four-step phase shift holograms as follows: ； wherein I1 represents image plane hologram 1; I2 represents image plane hologram 2; I3 represents image plane hologram 3; I4 represents image plane hologram 4; represents the amplitude of the object light wave, represents the phase distribution of the object light field; represents the amplitude of the reference light field, represents the phase distribution of the reference light field; According to the four-step phase shift hologram, the wrapped phase diagram is obtained by arctangent calculation; According to the improved least square method, the discrete Poisson equation of the wrapped phase diagram is established, and the real phase of the wrapped phase diagram is obtained by solving the discrete Poisson equation established by the improved least square method multiple times.

2. The method of claim 1, wherein the four-step phase shift reconstruction and improved least squares iterative phase unwrapping is based on: According to the four-step phase shift hologram, the wrapped phase diagram is obtained by arctangent calculation, and the expression is: ； wherein a phase distribution representative of the object light field; a phase distribution representative of the reference light field; I1 represents image plane hologram 1; I2 represents image plane hologram 2; I3 represents image plane hologram 3; I4 represents image plane hologram 4.

3. The method of claim 1, wherein the four-step phase shift reconstruction and improved least squares iterative phase unwrapping is based on: According to the improved least square method, the discrete Poisson equation of the wrapped phase diagram is established, including: Set the wrapping phase as , is the real phase, and ; the wrapping phase in the x direction and the y direction in the wrapping phase map are and , respectively. ； where W is a wrapping operator; denotes an integer which forms a discontinuous plateau - a truncated plateau in two-dimensional space; Based on the improved least square algorithm, the wrapped phase difference expression of the wrapped phase diagram is established: ； For the above equation Taking the derivative and setting it to zero, the Poisson equation is obtained: ； wherein represents a value of the wrapped phase.

4. The method of claim 3, wherein the four-step phase shift reconstruction and improved least squares iterative phase unwrapping method is characterized by: Solve the discrete Poisson equation multiple times, including: The boundary condition of the discrete Poisson equation is: ； Substitute the wrapped phase in the x direction and the y direction of the wrapped phase diagram into the discrete Poisson equation to obtain: ； Let Then, there is: ； Iterative solution is performed on the above formula until the solution result no longer changes, and the real phase of the wrapped phase diagram is obtained.

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