CN109855558A - Digital hologram three-dimensional method for reconstructing - Google Patents
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Abstract
The present invention proposes a kind of digital hologram three-dimensional method for reconstructing, method includes the following steps: S0: carrying out two-dimension fourier transform to a width interference fringe image；Two dimensional inverse fourier transform is carried out after handling Fourier domain analysis；S1: judge phase diffractive plane to the distance of interference fringe image plane whether be 0, if so, will then execute S3；If not, executing S2；S2: carrying out Fresnel number Diffraction Calculation to the positive interferometric phase information of extraction, extract positive object optical phase information, and adjusting Fresnel number diffraction distance is focal position, to obtain the phase of focal position；S3: the positive information of extraction is substituted into interferometry formula and is calculated, the three-dimensional elevation information of object is obtained, to carry out the reconstruction of digital hologram three-dimensional.This method realizes high speed in real time and the measurement of the single exposure digital hologram of stable precision by carrying out frequency-domain analysis to a width interference fringe image.
Description
Technical Field
The invention belongs to the field of digital holography, and particularly relates to a digital holographic three-dimensional reconstruction method by utilizing two-dimensional Fourier interference fringe analysis.
Background
The digital holography is a method for reconstructing a digital light field by utilizing an optical interference related diffraction principle, the digital holography regards points in each three-dimensional space on the surface of an opaque object which reflects light or in the internal structure of a transparent and semitransparent object which transmits light as a point light source, the light wave phase when the point light sources emit light is calculated by a computer through a mathematical formula of digital optical interference and diffraction, and the surface of the opaque object or the shape of the transparent and semitransparent object is subjected to nano-scale ultra-precise three-dimensional measurement according to an interferometry.
When digital holographic ultra-precise three-dimensional measurement is carried out, two steps of interference fringe recording and object light reconstruction are needed. The recording of the interference fringes is completed by constructing an interference optical system and shooting the interference fringes by using a digital camera; and the object light reconstruction needs to use digital Fresnel transformation to perform digital diffraction calculation on the interference fringe image so as to obtain the phase distribution of light in an object light field, and finally, an interferometry method is used for realizing three-dimensional reconstruction.
In the prior art, the mathematical formula of the fresnel transformation used is as follows:
in the above equation (1), Γ (ξ ', η ') is the optical complex number information on the fresnel diffraction plane, including the light intensity and phase, (ξ ', η ') represents the plane coordinates on the fresnel diffraction plane, a (x, y) is the optical complex number information when the light source passes through the aperture in the aperture plane, and contains only the light intensity value, that is, the gradation value of the interference fringe image, (x, y) represents the plane coordinates on the aperture plane, d represents the vertical distance between the fresnel diffraction plane and the aperture plane, ρ ' represents the diffraction distance when a point on the aperture strikes a point on the fresnel diffraction plane, λ represents the optical wavelength value of the light source, and i represents the complex number.
As can be seen from the above, the double integration has no speed advantage in computer calculation and has the disadvantage of complicated computer programming. In addition, when the digital holography three-dimensional reconstruction is performed by using the formula (1), the interference fringes photographed by the digital camera are generally directly substituted into the formula, and since only the light intensity of the interference fringes photographed by the digital camera are in the form of sine waves, in the general three-dimensional reconstruction, as shown in the following formula (2), the positive phase, the negative phase and the direct current component of the sine waves are simultaneously reconstructed, wherein only the positive phase accurately reflects the shape information of the actual object, and the negative phase and the direct current component influence the accurate measurement as noise.
Where a (x, y) represents a gradation value of the interference fringe image, and (x, y) represents an image coordinate. A. theR(x, y) denotes the complex function of the reference light, Ao(x, y) represents a complex function of the object light,representing the conjugate, i.e. the opposite phase,representing the conjugate of the object's light complex function. B (x, y) represents DC information in the interference fringes,representing negative object light phase information in interference fringes, Ao(x, y) represents the true positive phase information of the object light needed for three-dimensional reconstruction.
In the prior art, although phase-shift digital holography can be used for extracting positive object optical phase information from one image, 2-4 phase-shift interference fringe images are needed, time is consumed for shooting a plurality of phase-shift interference fringe images, the phase-shift precision of interference fringes in the shooting process is easily influenced by environmental factors such as vibration, air convection and the like, rapid real-time digital holography three-dimensional measurement cannot be realized, the measurement environment requirement is strict, and the factors cause that digital holography measurement equipment is expensive and the application occasion is limited.
Disclosure of Invention
In order to solve the above problems, the present invention provides a digital holographic three-dimensional reconstruction method, which performs frequency domain analysis on an interference fringe image to quickly extract positive object optical phase information required for three-dimensional measurement from the interference fringe image in real time, thereby realizing high-speed, real-time, precise and stable single-exposure digital holographic measurement.
In order to achieve the purpose, the invention adopts the technical scheme that:
a digital holographic three-dimensional reconstruction method comprises the following steps:
s0: performing two-dimensional Fourier transform on one interference fringe image to convert the interference fringe image into a Fourier frequency domain; performing two-dimensional inverse Fourier transform after Fourier frequency domain analysis processing to form an image domain; extracting positive interference phase information;
s1: judging whether the distance from the phase diffraction plane to the interference fringe image plane is 0 or not, if so, executing S3; if not, executing S2;
s2: performing Fresnel digital diffraction calculation on the extracted positive interference phase information, extracting positive object light phase information, and adjusting the Fresnel digital diffraction distance to be a focusing position to obtain the phase of the focusing position;
s3: and substituting the extracted positive information into an interferometry formula for calculation to obtain the three-dimensional height information of the object so as to carry out digital holographic three-dimensional reconstruction.
As a further optimization of the present invention, in step S0, the method specifically includes the following steps: firstly, performing two-dimensional Fourier transform on an interference fringe image, wherein in a Fourier frequency domain space after the two-dimensional Fourier transform, positive interference phase components and negative interference phase components are symmetrically distributed on direct current components positioned in the center of the frequency domain space; then extracting frequency domain information of a positive interference phase through a frequency domain space band-pass filter, and frequency shifting the frequency domain information of the positive interference phase to the center of a frequency domain space; and finally, carrying out inverse Fourier transform on the positive interference phase information translated to the center of the frequency domain space, and extracting the positive interference phase information.
As a further optimization of the present invention, in step S1, the method specifically includes the following steps: firstly, setting a Gemini Fresnel diffraction plane, namely respectively setting a positive phase diffraction plane and a negative phase diffraction plane in positive and negative directions symmetrically to an interference fringe image plane; secondly, substituting the positive interference phase into an Euler formula to obtain complex amplitude of the interference fringes, substituting the complex amplitude of the interference fringes into a two-dimensional Fresnel digital diffraction transformation formula, and adjusting the Fresnel digital diffraction distance until a focusing position is obtained to obtain object light complex amplitude of the focusing position; and finally, obtaining the object light phase and amplitude distribution from the object light complex amplitude according to an Euler formula.
As a further optimization of the present invention, in step S0, the mathematical expression of the interference fringe image is formulated as
C(x,y)＝2AO(x,y)AR(x,y)
Wherein, A (x, y) is the gray value of the interference fringe image, and (x, y) is the coordinate of the interference fringe image; a. theR(x, y) andthe reference light complex amplitude and its conjugate, respectively; a. theO(x, y) andrespectively object light complex amplitude and conjugate thereof; b (x, y) is direct current information in the interference fringes;negative interference light phase information in the interference fringes;the phase information of the interference light which is positive in the interference fringe image; the mathematical expression formula of the two-dimensional Fourier transform of the interference fringe image is as follows: a (f)x，fy)＝B(fx，fy)+C(fx-f0，fy-f0)+C*(fx+fo，fy+fo)，
Wherein, A (f)z，fy) Is a two-dimensional Fourier transform of the interference fringe image, (f)x，fy) Is a two-dimensional Fourier frequency domain space center coordinate of the interference fringe image, B (f)x，fy) For two-dimensional Fourier transformation of direct current information in the interference fringe image, C (f)z-fo，fy-fo) Two-dimensional Fourier transform of the phase information of the positive interference light in the interference fringes, C*(fx+f0，fy+f0) Two-dimensional Fourier transform of negative interference light phase information in interference fringesoThe phase information of the interference light is positive or negative, and the frequency domain coordinate is deviated from the central coordinate in the middle of the two-dimensional Fourier frequency domain after the two-dimensional Fourier transform.
As a further optimization of the present invention, in step S2, the two-dimensional Fresnel digital diffraction transformation formula isWherein Γ (ξ) is the object complex amplitude on the positive phase diffraction plane,E0(x, y) represents the complex amplitude of the interference fringes;
as a further optimization of the present invention, in step S1, the two-dimensional Fresnel digital diffraction transformation formula isWherein Γ (ξ ', η') is the complex amplitude of the object light on the negative phase diffraction plane,wherein, B0(x, y) represents the complex amplitude of the interference fringes;
as a further optimization of the present invention, in step S1, the euler formula is:wherein,for phase, i represents a complex number.
As a further optimization of the present invention, in step S0, the fresnel transformation formula according to which the fresnel digital diffraction is based is:wherein Γ (ξ ', η ') is the optical complex number information on the Fresnel diffraction plane, which is the light intensity and phase, respectively, (ξ ', η ') is the plane coordinate on the Fresnel diffraction plane, (A (x, y) is the optical complex number information when the light source passes through the aperture in the aperture plane, (x, y) is the plane coordinate on the aperture plane, (d) is the vertical distance between the Fresnel diffraction plane and the aperture plane, (ρ ') is the diffraction distance from a point on the aperture to a point on the Fresnel diffraction plane, and λ is the optical wavelength value of the light source, i represents the complex number.
Compared with the prior art, the invention has the advantages and positive effects that: according to the digital holographic three-dimensional reconstruction method, through carrying out frequency domain analysis on one interference fringe image, positive object optical phase information required by three-dimensional measurement is extracted from the one interference fringe image in real time, so that high-speed real-time precise and stable single-exposure digital holographic measurement is realized.
Drawings
FIG. 1 is a flow chart of a digital holographic three-dimensional reconstruction method of the present invention;
FIG. 2 is a frequency domain distribution diagram of an interference fringe image and a two-dimensional Fourier transform thereof in the method of the present invention;
FIG. 3 is a plan view of a Gemini setup Fresnel diffraction in the method of the present invention;
FIG. 4 is a schematic diagram of moving band-pass filtered frequency domain information to the center of the entire frequency domain in the method of the present invention;
FIG. 5 is a diagram of a two-dimensional Fourier transformed phase profile in the method of the present invention;
FIG. 6 is a graph of the amplitude distribution after two-dimensional Fourier transform in the method of the present invention;
FIG. 7 is a schematic diagram of adjusting the diffraction distance to obtain focused diffraction imaging in the method of the present invention;
FIG. 8 is a diagram illustrating the results of digital holographic three-dimensional reconstruction in the method of the present invention.
Detailed Description
The invention is described in detail below by way of exemplary embodiments. It should be understood, however, that elements, structures and features of one embodiment may be beneficially incorporated in other embodiments without further recitation.
As shown in fig. 1, the present invention provides a digital holographic three-dimensional reconstruction method, which includes the following steps:
s0: performing two-dimensional Fourier transform on one interference fringe image to convert the interference fringe image into a Fourier frequency domain; performing two-dimensional inverse Fourier transform after Fourier frequency domain analysis processing to form an image domain; extracting positive interference phase information;
s1: judging whether the distance from the phase diffraction plane to the interference fringe image plane is 0 or not, if so, executing S3; if not, executing S2;
s2: performing Fresnel digital diffraction calculation on the extracted positive interference phase information, extracting positive object light phase information, and adjusting the Fresnel digital diffraction distance to be a focusing position to obtain the phase of the focusing position; as shown in fig. 7, the lowest image in the graph is the phase information of the focusing position;
s3: and substituting the extracted positive information into an interferometry formula for calculation to obtain the three-dimensional height information of the object so as to carry out digital holographic three-dimensional reconstruction.
Further, in step S0, the method specifically includes the following steps: firstly, performing two-dimensional Fourier transform on an interference fringe image, wherein in a Fourier frequency domain space after the two-dimensional Fourier transform, positive interference phase components and negative interference phase components are symmetrically distributed on direct current components positioned in the center of the frequency domain space; then extracting frequency domain information of a positive interference phase through a frequency domain space band-pass filter, and frequency shifting the frequency domain information of the positive interference phase to the center of a frequency domain space; and finally, carrying out inverse Fourier transform on the positive interference phase information translated to the center of the frequency domain space, and extracting the positive interference phase information.
As shown in fig. 3, in the fourier frequency domain space after the two-dimensional fourier transform, the frequency domain value at the center is the dc component of the interference fringe, the value range at the lower left corner is the positive object optical phase component in the interference fringe, and the value range at the upper right corner is the negative object optical phase component in the interference fringe. The positive phase component at the lower left corner can be extracted through a two-dimensional frequency domain band-pass filter, and the direct current component and the negative phase component are directly filtered. Specifically, as shown in fig. 4, the object light complex-change function in the time domain can be obtained only by performing two-dimensional inverse fourier transform on the band-pass filtered frequency domain information, but the direct current and negative phase information is lost in the band-pass filtered frequency domain information, and the band-pass filtered frequency domain component needs to be moved to the center of the frequency domain to compensate for the lost direct current and negative phase component.
As shown in fig. 5 and 6, the frequency domain shown on the right side in fig. 4 is subjected to two-dimensional inverse fourier transform, and a complex function of the interference fringe, i.e., a (x, y) in expression 1, can be obtained, and the phase and amplitude distribution can be obtained from the euler's expression.
In addition, in the above, in step S0, the mathematical expression of the interference fringe image is expressed as
C(x,y)＝2AO(x,y)AR(x,y)
Wherein, A (x, y) is the gray value of the interference fringe image, and (x, y) is the coordinate of the interference fringe image; a. theR(x, y) andthe reference light complex amplitude and its conjugate, respectively; a. theO(x, y) andrespectively object light complex amplitude and conjugate thereof; b (x, y) is direct current information in the interference fringes;negative interference light phase information in the interference fringes;the phase information of the interference light which is positive in the interference fringe image; the mathematical expression formula of the two-dimensional Fourier transform of the interference fringe image is as follows: a (f)x，fy)＝B(fx，fy)+C(fx-f0，fy-f0)+C*(fx+f0，fy+f0)，
Wherein, A (f)x，fy) Is a two-dimensional Fourier transform of the interference fringe image, (f)x，fy) Is a two-dimensional Fourier frequency domain space center coordinate of the interference fringe image, B (f)x，fy) For two-dimensional Fourier transformation of direct current information in the interference fringe image, C (f)x-fo，fy-fo) Two-dimensional Fourier transform of the phase information of the positive interference light in the interference fringes, C*(fx+f0，fy+f0) Two-dimensional Fourier transform of negative interference light phase information in interference fringes0The phase information of the interference light is positive or negative, and the frequency domain coordinate is deviated from the central coordinate in the middle of the two-dimensional Fourier frequency domain after the two-dimensional Fourier transform.
In step S0, the fresnel transformation formula according to which the fresnel digital diffraction is based is:wherein Γ (ξ ', η') is the optical complex number information on the Fresnel diffraction plane, which is the light intensity and phase, respectively, (ξ ', η') is the plane coordinate on the Fresnel diffraction plane, (A (x, y) is the optical complex number information when the light source passes through the aperture in the aperture plane, (x, y) is the plane coordinate on the aperture plane, (d) is the vertical distance between the Fresnel diffraction plane and the aperture plane, (A) is the diffraction distance from a point on the aperture to a point on the Fresnel diffraction plane, and λ is the optical wavelength value of the light source, i represents the complex number.
Continuing with fig. 2, in step S1, the method specifically includes the following steps: firstly, a gemini fresnel diffraction plane is set, namely a positive phase diffraction plane 2 and a negative phase diffraction plane 3 are respectively set in the positive direction and the negative direction symmetrically to an interference fringe image plane 1, the distance between the positive phase diffraction plane 2 and the interference fringe image plane 1 is a positive diffraction distance 4, and the distance between the negative phase diffraction plane 3 and the interference fringe image plane 1 is a negative diffraction distance 5; secondly, substituting the positive interference phase into an Euler formula to obtain complex amplitude of the interference fringes, substituting the complex amplitude of the interference fringes into a two-dimensional Fresnel digital diffraction transformation formula, and adjusting the Fresnel digital diffraction distance until a focusing position is obtained to obtain object light complex amplitude of the focusing position; and finally, obtaining the object light phase and amplitude distribution from the object light complex amplitude according to an Euler formula.
In the above, in step S1, the two-dimensional fresnel digital diffraction reconstruction formula isWherein Γ (ξ ', η') is the complex amplitude of the object light on the negative phase diffraction plane,wherein，E0(x, y) represents the complex amplitude of the interference fringes;meanwhile, in step S1, the euler equation is:wherein,for phase, i represents a complex number.
In step S2, the two-dimensional Fresnel digital diffraction transformation formula isWherein Γ (ξ) is the object complex amplitude on the positive phase diffraction plane,E0(x, y) represents the complex amplitude of the interference fringes;
by the digital holographic three-dimensional reconstruction method, the reconstruction result is shown in fig. 8, and the method performs frequency domain analysis on one interference fringe image by a two-dimensional Fourier transform method, and extracts positive object optical phase information required by three-dimensional measurement from the one interference fringe image in real time, so that high-speed, real-time, precise and stable single-exposure digital holographic measurement is realized. Meanwhile, the digital holographic three-dimensional reconstruction method does not need to shoot a plurality of phase-shift interference fringe images as in the prior art, reduces the shooting time, and is not influenced by environmental factors such as vibration, air convection and the like, so that the rapid real-time digital holographic three-dimensional measurement is realized, the application occasions are wider, and the cost is reduced.
The above description is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily conceive of the changes or substitutions within the technical scope of the present invention, and all the changes or substitutions should be covered within the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the appended claims.
Claims (8)
1. A digital holographic three-dimensional reconstruction method is characterized in that: the method comprises the following steps:
s0: performing two-dimensional Fourier transform on one interference fringe image to convert the interference fringe image into a Fourier frequency domain; performing two-dimensional inverse Fourier transform after Fourier frequency domain analysis processing to form an image domain; extracting positive interference phase information;
s1: judging whether the distance from the phase diffraction plane to the interference fringe image plane is 0 or not, if so, executing S3; if not, executing S2;
s2: performing Fresnel digital diffraction calculation on the extracted positive interference phase information, extracting positive object light phase information, and adjusting the Fresnel digital diffraction distance to be a focusing position to obtain the phase of the focusing position;
s3: and substituting the extracted positive information into an interferometry formula for calculation to obtain the three-dimensional height information of the object so as to carry out digital holographic three-dimensional reconstruction.
2. The digital holographic three-dimensional reconstruction method of claim 1, wherein: in step S0, the method specifically includes the following steps: firstly, performing two-dimensional Fourier transform on an interference fringe image, wherein in a Fourier frequency domain space after the two-dimensional Fourier transform, positive interference phase components and negative interference phase components are symmetrically distributed on direct current components positioned in the center of the frequency domain space; then extracting frequency domain information of a positive interference phase through a frequency domain space band-pass filter, and frequency shifting the frequency domain information of the positive interference phase to the center of a frequency domain space; and finally, carrying out inverse Fourier transform on the positive interference phase information translated to the center of the frequency domain space, and extracting the positive interference phase information.
3. The digital holographic three-dimensional reconstruction method according to claim 1 or 2, characterized in that: in step S1, the method specifically includes the following steps: firstly, setting a Gemini Fresnel diffraction plane, namely respectively setting a positive phase diffraction plane and a negative phase diffraction plane in positive and negative directions symmetrically to an interference fringe image plane; secondly, substituting the positive interference phase into an Euler formula to obtain complex amplitude of the interference fringes, substituting the complex amplitude of the interference fringes into a two-dimensional Fresnel digital diffraction transformation formula, and adjusting the Fresnel digital diffraction distance until a focusing position is obtained to obtain object light complex amplitude of the focusing position; and finally, obtaining the object light phase and amplitude distribution from the object light complex amplitude according to an Euler formula.
4. The digital holographic three-dimensional reconstruction method of claim 2, wherein: in step S0, the interference fringe image is mathematically expressed as
Wherein, A (x, y) is the gray value of the interference fringe image, and (x, y) is the coordinate of the interference fringe image; a. theR(x, y) andthe reference light complex amplitude and its conjugate, respectively; a. theO(x, y) andrespectively object light complex amplitude and conjugate thereof; b (x, y) is direct current information in the interference fringes;negative interference light phase information in the interference fringes;the phase information of the interference light which is positive in the interference fringe image; the mathematical expression formula of the two-dimensional Fourier transform of the interference fringe image is as follows:
A(fx，fy)＝B(fx，fy)+C(fx-fo，fy-fo)+C*(fx+fo，fy+f0)，
wherein, A (f)x，fy) Is a two-dimensional Fourier transform of the interference fringe image, (f)x，fy) Is a two-dimensional Fourier frequency domain space center coordinate of the interference fringe image, B (f)x，fy) For two-dimensional Fourier transformation of direct current information in the interference fringe image, C (f)x·f0，fy·f0) Is a two-dimensional fourier transform of the positive interfering light phase information in the interference fringes,
C*(fx+f0，fy+f0) Two-dimensional Fourier transform of negative interference light phase information in interference fringesoThe phase information of the interference light is positive or negative, and the frequency domain coordinate is deviated from the central coordinate in the middle of the two-dimensional Fourier frequency domain after the two-dimensional Fourier transform.
5. The digital holographic three-dimensional reconstruction method of claim 3, wherein: in step S2, the two-dimensional Fresnel digital diffraction reconstruction formula isWherein Γ (ξ) is the object complex amplitude on the positive phase diffraction plane,E0(x, y) represents the complex amplitude of the interference fringes;
6. the digital holographic three-dimensional reconstruction method of claim 3, wherein: in step S1, the two-dimensional Fresnel digital diffraction reconstruction formula isWherein Γ (ξ ', η') is the complex amplitude of the object light on the negative phase diffraction plane,wherein E is0(x, y) represents the complex amplitude of the interference fringes;
7. the digital holographic three-dimensional reconstruction method of claim 3, wherein: in step S1, the euler formula is:wherein,for phase, i represents a complex number.
8. The digital holographic three-dimensional reconstruction method of claim 1, wherein: in step S0, the fresnel transformation formula according to which the fresnel digital diffraction is based is:wherein Γ (ξ ', η ') is the optical complex number information on the Fresnel diffraction plane, which is the light intensity and phase, respectively, (ξ ', η ') is the plane coordinate on the Fresnel diffraction plane, (A (x, y) is the optical complex number information when the light source passes through the aperture in the aperture plane, (x, y) is the plane coordinate on the aperture plane, (d) is the vertical distance between the Fresnel diffraction plane and the aperture plane, (ρ ') is the diffraction distance from a point on the aperture to a point on the Fresnel diffraction plane, and λ is the optical wavelength value of the light source, i represents the complex number.
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