CN109060723B - Simple and quick dual-wavelength digital holographic phase imaging method - Google Patents

Simple and quick dual-wavelength digital holographic phase imaging method Download PDF

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CN109060723B
CN109060723B CN201810936403.5A CN201810936403A CN109060723B CN 109060723 B CN109060723 B CN 109060723B CN 201810936403 A CN201810936403 A CN 201810936403A CN 109060723 B CN109060723 B CN 109060723B
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王凡
郭荣礼
张维光
刘王云
胡小英
张玉虹
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Xian Technological University
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Abstract

The invention provides a simple and quick dual-wavelength digital holographic phase imaging method, which solves the problems that the prior art can not work normally when the noise is large and is not suitable for occasions with large synthetic wavelength. The method has the advantages of high imaging speed: because discrete cosine transform or other complex operations are not involved, the time required by the operations is shorter than that of other phase unwrapping methods; it is not sensitive to noise: when the two wavelengths are separated by a small distance, the method can be used for quickly imaging. When the interval between the two wavelengths is small and the synthesized wavelength is large, the method can still correctly unpack to obtain the optical path distribution of the measured sample, because the method respectively carries out the averaging operation on the noise in each region and carries out the rounding operation after dividing the optical path subjected to the averaging by the single wavelength, the method is insensitive to the noise, and when the noise is large, the method can still normally work; meanwhile, the method has wide application range and simple method.

Description

Simple and quick dual-wavelength digital holographic phase imaging method
Technical Field
The invention belongs to the technical field of digital holography, and particularly relates to a simple and rapid dual-wavelength digital holographic phase imaging method.
Background
Digital holography typically uses a CCD (or CMOS) camera to record holograms, and subsequent data processing is done in a computer. The digital holographic technology has the advantages of real-time measurement, full-field imaging, high axial measurement precision and the like, and is widely applied to the fields of biomedical imaging, surface topography analysis, microscopic imaging and the like. In single-wavelength digital holographic phase imaging, holograms of only one wavelength are recorded. When reconstructing, firstly obtaining wrapping phase from hologram
Figure BDA0001767942890000011
Then, the unwrapped phase phi (x, y) is obtained through a phase unwrapping algorithm. Both are generally related as follows:
Figure BDA0001767942890000012
is an integer at point (x, y). The core idea of the unwrapping algorithm is therefore to determine the integer m (x, y). However, the unwrapping algorithm has two major drawbacks: 1) the optical path difference between two adjacent pixel points must be smaller than a wavelength lambda, otherwise, the unwrapping algorithm will make a mistake; 2) the algorithm takes longer, especially when there are more pixels in the image (e.g. 2000 × 2000 pixels). These problems can be solved by dual-wavelength digital holography in which two wavelengths λ are recorded separately1、λ2A corresponding hologram. According to the principle, in a two-wavelength digital hologram, the synthetic wavelength is Λ ═ λ1λ2/|λ1-λ2The smaller the separation of two wavelengths, the larger the synthesized wavelength. By selecting two suitable wavelengths, unwrapped phase imaging can be achieved. However, in the two-wavelength digital hologram, the phase noise amplitude in a single wavelength is amplified by M ═ 2 Λ/λi(i ═ 1, 2) times (Optics Letters, volume 28, issue 13,2003, j. gass), wavelength interval | λ1-λ2The smaller the | the greater the resulting phase noise. The amplified phase noise can mask detail variations of the measured sample and seriously affect the accuracy of the measurement.
Gass (Optics Letters, volume 28, issue 13,2003) et al propose a dual wavelength phase imaging (phase unwrapping) method that removes noise, but this method requires that the noise amplitude of the single wavelength wrapped phase be less than λi2M, where M2 Λ/λiIs the amplification of the noise, otherwise the method is not effective. To meet this condition, the synthetic wavelength Λ is typically relatively small (meaning that the magnification M is also small), for example only 3.33 μ M in their experiments. Khmaladze (Optics Letters, volume 36, issue 6,2011) et al propose another two-wavelength phase formationLike the method, the method utilizes the linear regression principle to obtain the unwrapped phase, but is still sensitive to noise and is not suitable for occasions with large synthetic wavelengths lambda.
Disclosure of Invention
The invention provides a simple and quick dual-wavelength digital holographic phase imaging method, which aims to solve the problems that the prior art can not work normally when the noise is large and is not suitable for occasions with large synthetic wavelength.
A simple and quick dual-wavelength digital holographic phase imaging method sequentially comprises the following steps:
(1) from the hologram, two wavelengths λ at the image plane are obtained1、λ2The corresponding object-light wave complex amplitude O1、Ο2；
(2) Obtaining two wavelengths lambda by using an arctangent function1、λ2Respectively corresponding wrapped phase and correcting linear phase distortion for (-pi, 0)]Is added with 2 pi to obtain a main value interval of (0,2 pi)]And is noted as
Figure BDA0001767942890000021
(3) Wrapped phase
Figure BDA0001767942890000022
Conversion into corresponding optical path distribution
Figure BDA0001767942890000023
Or 2);
(4) subtracting the two wrapped phase maps to obtain the synthetic wavelength lambda ═ lambda1λ2/|λ1-λ2Phase distribution corresponding to |
Figure BDA0001767942890000031
The phase distribution phi (x, y) is then converted into a corresponding optical path distribution
Figure BDA0001767942890000032
In consideration of amplified noise, the actually obtained optical path distribution is represented by l' (x, y) l (y)x, y) + Δ (x, y), where Δ (x, y) is the noise term;
(5) according to
Figure BDA0001767942890000033
The characteristics of phase distribution
Figure BDA0001767942890000034
Dividing the interference area into n different areas which are respectively marked as region (n), wherein each area belongs to the same interference level;
(6) optical path distribution l' (x, y) minus optical path distribution l corresponding to one of the wrapped phasesi(x, y) (i ═ 1 or 2), yielding: lsub(x,y)＝l′(x,y)-li(x,y)；
(7) Define a
Figure BDA0001767942890000035
And (3) assigning the new matrixes with the same size to the masks according to the regions region (n) obtained in the step (5) in sequence according to the regions:
mask(x,y)＝round[mean(lsub(x,y))/λi](x, y) e region (n), (i ═ 1 or 2)
Wherein mean () represents averaging the regions, round () represents rounding, (x, y) is e region (n) represents the coordinates within region (n);
(8) the optical path distribution h (x, y) after final noise removal is:
h(x,y)＝mask(x,y)·λi+li(x, y), (i ═ 1 or 2).
In the step (5), for the continuous object, the jump point of the phase value in the wrapping phase needs to be found, and then
Figure BDA0001767942890000036
Dividing the image into different areas, and comprising the following steps:
firstly, pair
Figure BDA0001767942890000037
The derivatives are taken in the x and y directions, respectively, and are recorded as
Figure BDA0001767942890000038
Based on
Figure BDA0001767942890000041
The noise level of (d), setting two thresholds + -th,
Figure BDA0001767942890000042
in
Figure BDA0001767942890000043
The value of the pixel of (a) is 1,
Figure BDA0001767942890000044
the value of the pixel is-1, and the value of other pixel points is 0;
③ define an
Figure BDA0001767942890000045
The matrix m with the same size assumes that m (i, j) of the pixel point in the ith row and the jth column in m takes a value of 0, and the pixel values of other points (l, k) are determined by the following formula:
Figure BDA0001767942890000046
where C (i, j) → (l, k) represents an integration path from the point (i, j) to the point (l, k).
And dividing the matrix m into n different areas according to different pixel values in the matrix m, and marking the areas as regions (n).
In the step (5), for the step-shaped object, the phase value of different areas in the wrapping phase is directly calculated according to the difference of the phase values
Figure BDA0001767942890000047
The method comprises the following steps:
according to
Figure BDA0001767942890000048
The difference of phase values of different areas in the middleMatrix array
Figure BDA0001767942890000049
Segmentation into distinct regions, labeled region (n);
compared with the prior art, the invention has the following advantages:
(1) the imaging speed is high: because discrete cosine transform or other complex operations are not involved, the time required by the operations is shorter than that of other phase unwrapping methods;
(2) it is not sensitive to noise: when the two wavelengths are separated by a small distance, the method can be used for quickly imaging. When the interval between the two wavelengths is small and the synthesized wavelength is large, the method can still correctly unpack to obtain the optical path distribution of the measured sample, because the method respectively carries out the averaging operation on the noise in each region and carries out the rounding operation after dividing the optical path subjected to the averaging by the single wavelength, the method is insensitive to the noise, and when the noise is large, the method can still normally work;
(3) the application range is wide: the method is not only suitable for continuous samples, but also suitable for step-shaped samples with the optical path difference of adjacent pixels exceeding one wavelength;
(4) the method is simple: the noise is effectively suppressed by using averaging and rounding according to different regions.
Drawings
FIG. 1 is a schematic diagram of continuous object dual wavelength unwrapping.
Wherein: fig. 1(a), true phase distribution; FIG. 1(b), wrapped phase distribution
Figure BDA0001767942890000051
FIG. 1(c), wrapped phase distribution
Figure BDA0001767942890000052
FIG. 1(d), wrapped phase distribution
Figure BDA0001767942890000053
Corresponding optical path distribution; FIG. 1(e) shows an optical path distribution l 'obtained by direct subtraction of wrapped phases'(x, y); FIG. 1(f), segmented regions, different gray levels representing different regions; fig. 1(g), optical path distribution h (x, y) after removing noise; FIG. 1(h), and height sectional views in the white line direction in FIGS. 1(a), 1(e) and 1 (g).
Fig. 2 is a schematic diagram of dual wavelength unwrapping of a stepped object.
Wherein: fig. 2(a), true phase distribution; FIG. 2(b), wrapped phase distribution
Figure BDA0001767942890000054
FIG. 2(c), wrapped phase distribution
Figure BDA0001767942890000055
FIG. 2(d), wrapped phase distribution
Figure BDA0001767942890000056
Corresponding optical path distribution; fig. 2(e), optical path distribution l' (x, y) by direct subtraction of wrapped phases; FIG. 2(f), segmented regions, different gray levels representing different regions; fig. 2(g), optical path distribution h (x, y) after removing noise; fig. 2(h), and height sectional views in the white line direction in fig. 2(a), 1(e), and 1 (g).
Detailed Description
The present invention is further illustrated by the following examples, which are presented for the purpose of providing a better understanding of the present invention and are presented for purposes of simplicity and clarity of illustration.
Example 1:
referring to fig. 1, the sample to be measured is a continuous variation type sample, and the maximum optical path length variation is 5 μm, wherein random noise is added to the sample, and the maximum amplitude of the noise is 15nm, as shown in fig. 1 (a). Two wavelengths lambda1＝633nm、λ2650nm, synthetic wavelength lambda ═ lambda1λ2/|λ1-λ2I 24.2 μm, which corresponds to the wavelength λ1The noise in the wrapped phase corresponding to 633nm is amplified by 2 Λ/λ176 times.
The invention provides a simple and quick dual-wavelength digital holographic phase imaging method, which sequentially comprises the following steps:
(1) starting from hologramsObtaining two wavelengths lambda at the image plane1、λ2The corresponding object-light wave complex amplitude O1、Ο2；
(2) Obtaining two wavelengths lambda by using an arctangent function1、λ2Respectively corresponding wrapped phase and correcting linear phase distortion for (-pi, 0)]Is added with 2 pi to obtain a main value interval of (0,2 pi)]And is noted as
Figure BDA0001767942890000061
As shown in FIGS. 1(b) and 1 (c);
(3) wrapped phase
Figure BDA0001767942890000062
Conversion into corresponding optical path distribution
Figure BDA0001767942890000063
As shown in FIG. 1 (d);
(4) subtracting the two wrapped phase maps to obtain the synthetic wavelength lambda ═ lambda1λ2/|λ1-λ2Phase distribution corresponding to |
Figure BDA0001767942890000064
The phase distribution phi (x, y) is then converted into a corresponding optical path distribution
Figure BDA0001767942890000065
In consideration of amplified noise, the actually obtained optical path distribution is represented by l' (x, y) ═ l (x, y) + Δ (x, y), as shown in fig. 1(e), where Δ (x, y) is a noise term;
(5) according to
Figure BDA00017679428900000711
The characteristics of phase distribution
Figure BDA00017679428900000712
The method is divided into n different regions which are respectively marked as regions (n), wherein each region belongs to the same interference level, and the method specifically comprises the following steps:
firstly, pair
Figure BDA0001767942890000071
The derivatives are taken in the x and y directions, respectively, and are recorded as
Figure BDA0001767942890000072
Based on
Figure BDA0001767942890000073
Two thresholds are set to be + -6,
Figure BDA0001767942890000074
in
Figure BDA0001767942890000075
The value of the pixel of (a) is 1,
Figure BDA0001767942890000076
the value of the pixel is-1, and the value of other pixel points is 0;
③ define an
Figure BDA0001767942890000077
The matrix m with the same size assumes that a pixel m (1,1) of a 1 st row and a 1 st column in m takes a value of 0, and pixel values of other points (l, k) are determined by the following formula:
Figure BDA0001767942890000078
where C (i, j) → (l, k) represents an integration path from the point (i, j) to the point (l, k).
Here, in the specific operation, starting from m (1,1), the value of each element m (1, k) in the first row in the matrix m is obtained:
Figure BDA0001767942890000079
then, the value m (l, k) of any element in the matrix m is determined by:
Figure BDA00017679428900000710
dividing the matrix m into 8 different regions marked as regions (n) according to different pixel values in the matrix m, for example, all pixels with a pixel value of 0 constitute the region (0), and all pixels with a pixel value of 1 constitute the region (1), as shown in fig. 1 (f);
(6) optical path distribution l' (x, y) (shown in FIG. 1 (e)) minus optical path distribution l corresponding to the wrapped phase1(x, y) (shown in FIG. 1 (d)) to obtain: lsub(x,y)＝l′(x,y)-l1(x,y)；
(7) Define a
Figure BDA0001767942890000081
And assigning the new matrix mask with the same size to the mask according to the region (n) obtained by the previous steps (as shown in fig. 1 (f)), and sequentially assigning the new matrix mask according to the regions:
mask(x,y)＝round[mean(lsub(x,y))/λ1],(x,y)∈region(n)，
wherein mean () represents the averaging, round () represents the rounding operation, (x, y) is e.g. region (n) represents the coordinates within the region (n);
(8) the optical path distribution h (x, y) after final noise removal is: h (x, y) ═ mask (x, y) · λ1+l1(x, y) as shown in FIG. 1 (g).
The height profile distribution along the white line direction in fig. 1(a), 1(e) and 1(g) is shown in fig. 1(h), it can be seen that the noise of the result is very large when the two wrapped phases are directly subtracted, and after the method provided by the present invention is adopted, it can be seen that the noise can be effectively reduced while the correct height distribution is obtained.
Example 2:
referring to fig. 2, the sample to be measured is a step-shaped sample, the optical path changes corresponding to the step are 2 μm and 4 μm, respectively, random noise is added to the sample, and the maximum amplitude of the noise is 15nm, as shown in fig. 2 (a). Two wavelengths lambda1＝633nm、λ2＝650nmThe synthetic wavelength lambda ═ lambda1λ2/|λ1-λ2I 24.2 μm, which corresponds to the wavelength λ1The noise in the wrapped phase corresponding to 633nm is amplified by 2 Λ/λ176 times.
The invention provides a simple and quick dual-wavelength digital holographic phase imaging method, which sequentially comprises the following steps:
(1) from the hologram, two wavelengths λ at the image plane are obtained1、λ2The corresponding object-light wave complex amplitude O1、Ο2；
(2) Obtaining two wavelengths lambda by using an arctangent function1、λ2Respectively corresponding wrapped phase and correcting linear phase distortion for (-pi, 0)]Is added with 2 pi to obtain a main value interval of (0,2 pi)]And is noted as
Figure BDA0001767942890000091
As shown in FIGS. 2(b) and 2 (c);
(3) wrapped phase
Figure BDA0001767942890000092
The corresponding optical path distribution l is obtained1(x,y)，
Figure BDA0001767942890000093
As shown in FIG. 2 (d);
(4) subtracting the two wrapped phase maps to obtain the synthetic wavelength lambda ═ lambda1λ2/|λ1-λ2Phase distribution corresponding to |
Figure BDA0001767942890000094
The phase distribution phi (x, y) is then converted into a corresponding optical path distribution
Figure BDA0001767942890000095
In consideration of the amplified noise, the actually obtained optical path distribution is represented as l' (x, y) ═ l (x, y) + Δ (x, y), where Δ (x, y) is a noise term, as shown in fig. 2 (e);
(5) for step-like objects, straight rootsAccording to the difference of phase values of different areas in the wrapped phase
Figure BDA0001767942890000096
The method is divided into 3 different areas, and comprises the following specific steps:
due to the fact that
Figure BDA0001767942890000097
The distribution of (2) is still step-shaped, according to the overall phase distribution of 3 different regions, the threshold values are selected to be 0.5rad and 1.5rad, and the matrix is processed
Figure BDA0001767942890000098
Dividing into 3 different regions, namely, the region with phase value less than 0.5rad is marked as region (0), the region with phase value greater than 0.5rad and less than 1.5rad is marked as region (1), the region with phase value greater than 1.5rad is marked as region (2), and the final division result is shown in FIG. 2(f), and the different regions are marked as region (n);
(6) optical path profile l' (x, y) (as shown in FIG. 2 (e)) minus the optical path profile l corresponding to the wrapped phase1(x, y) (as shown in FIG. 2 (d)) to give: lsub(x,y)＝l′(x,y)-l1(x,y)；
(7) Define a
Figure BDA0001767942890000099
New matrix mask of the same size and according to the data
Figure BDA00017679428900000910
And (n) assigning values to the mask according to the regions in sequence:
mask(x,y)＝round[mean(lsub(x,y))/λ1],(x,y)∈region(n)，
wherein mean () represents the averaging, round () represents the rounding operation, (x, y) is e.g. region (n) represents the coordinates within the region (n);
(8) the optical path distribution h (x, y) after final noise removal is: h (x, y) ═ mask (x, y) · λ1+l1(x, y) as shown in FIG. 2 (g);
the height profile in the white line direction in fig. 2(a), 2(e) and 2(g) is shown in fig. 2(h), and it can be seen that the noise is very large when the two wrapped phases are directly subtracted, and the noise can be effectively reduced while obtaining the correct height profile by the proposed method.
The invention is not limited to the examples, and any equivalent changes to the technical solution of the invention by a person skilled in the art after reading the description of the invention are covered by the claims of the invention.

Claims (3)

1. A simple and quick dual-wavelength digital holographic phase imaging method sequentially comprises the following steps:
(1) from the hologram, two wavelengths λ at the image plane are obtained1、λ2The corresponding object-light wave complex amplitude O1、Ο2；
(2) Obtaining two wavelengths lambda by using an arctangent function1、λ2Respectively corresponding wrapped phase and correcting linear phase distortion for (-pi, 0)]Is added with 2 pi to obtain a main value interval of (0,2 pi)]And is noted as
Figure FDA0002745279820000011
(3) Wrapped phase
Figure FDA0002745279820000012
(i ═ 1 or 2) to the corresponding optical path length distribution li(x,y)，
Figure FDA0002745279820000013
(i ═ 1 or 2);
(4) subtracting the two wrapped phase maps to obtain the synthetic wavelength lambda ═ lambda1λ2/|λ1-λ2Phase distribution corresponding to |
Figure FDA0002745279820000014
The phase distribution phi (x, y) is then converted into a corresponding optical path distribution
Figure FDA0002745279820000015
In consideration of amplified noise, the actually obtained optical path distribution is represented as l' (x, y) ═ l (x, y) + Δ (x, y), where Δ (x, y) is a noise term;
(5) according to
Figure FDA0002745279820000016
(i ═ 1 or 2) by phase distribution
Figure FDA0002745279820000017
(i ═ 1 or 2) into n distinct regions, labeled region (n), respectively, each region belonging to the same interference order;
(6) optical path distribution l' (x, y) minus optical path distribution l corresponding to one of the wrapped phasesi(x, y) (i ═ 1 or 2), yielding: lsub(x,y)＝l′(x,y)-li(x,y)；
(7) Define a
Figure FDA0002745279820000018
And (3) assigning the new matrixes with the same size to the masks according to the regions region (n) obtained in the step (5) in sequence according to the regions:
mask(x,y)＝round[mean(lsub(x,y))/λi](x, y) e region (n), (i ═ 1 or 2)
Wherein mean () represents averaging the regions, round () represents rounding, (x, y) is e region (n) represents the coordinates within region (n);
(8) the optical path distribution h (x, y) after final noise removal is:
h(x,y)＝mask(x,y)·λi+li(x, y), (i ═ 1 or 2).
2. The method of claim 1, further comprising:
in the step (5), for the continuous object, the jump point of the phase value in the wrapping phase needs to be found, and then
Figure FDA0002745279820000021
Dividing the image into different areas, and comprising the following steps:
firstly, pair
Figure FDA0002745279820000022
(i ═ 1 or 2) derivatives in the x and y directions, respectively, are reported
Figure FDA0002745279820000023
Based on
Figure FDA0002745279820000024
The noise level of (d), setting two thresholds + -th,
Figure FDA0002745279820000025
in
Figure FDA0002745279820000026
The value of the pixel of (a) is 1,
Figure FDA0002745279820000027
the value of the pixel is-1, and the value of other pixel points is 0;
③ define an
Figure FDA0002745279820000028
The matrix m with the same size assumes that m (i, j) of the pixel point in the ith row and the jth column in m takes a value of 0, and the pixel values of other points (l, k) are determined by the following formula:
Figure FDA0002745279820000029
where C (i, j) → (l, k) represents an integration path from the point (i, j) to the point (l, k);
and dividing the matrix m into n different areas according to different pixel values in the matrix m, and marking the areas as regions (n).
3. The method of claim 1, further comprising:
in the step (5), aiming at the step-shaped object, the phase value of different areas in the wrapping phase is directly calculated according to the difference of the phase values
Figure FDA0002745279820000031
The method comprises the following steps:
according to
Figure FDA0002745279820000032
The difference of phase values of different areas in (i 1 or 2) wraps the phase
Figure FDA0002745279820000033
The segmentation into distinct regions is labeled region (n).
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