CN104407507A - Hilbert transform based high-precision digital holography phase reconstruction method - Google Patents

Abstract

An embodiment of the present invention provides a Hilbert transform based high-precision digital holography phase reconstruction method, relating to the field of digital holography microscopy. By using the method, the precision of phase reconstruction can be increased. The method includes: recording a hologram; performing Fourier transform on the digital hologram to obtain a hologram frequency spectrum; filtering the hologram frequency spectrum to remove a zero-order spectrum; performing inverse Fourier transform on the filtered frequency spectrogram to obtain a hologram without the zero order item; performing two-dimensional Hilbert transform on the hologram; constructing an analytic function with the hologram without the zero order item and the Hilbert transform thereof; carrying out argument calculation for the analytic function to obtain a wrapping phase of the phase difference between an object light field and a reference light field; unwrapping the wrapping phase to obtain an unwrapped phase of the phase difference between the object light field and the reference light field; and extracting phase data from a region where the phase is a constant in the object light field, and performing linear fitting such that linear phase distortion correction is achieved and phase distribution of the object light field is obtained.

Description

A kind of digital hologram high precision position phase method for reconstructing based on Hilbert conversion

Technical field

The present invention relates to digital holographic microscopy field, particularly relate to a kind of digital hologram high precision position phase method for reconstructing based on Hilbert conversion.

Background technology

Digital holographic microscopy is a kind of novel quantitative imaging and visualization measurement technology, and it achieving the height fusion of optics, photoelectric conversion technique, computer technology, is a kind of whole audience, noncontact, quasi real time 3 Dimension Image Technique.Utilize this technology can obtain the quantitative phase information of object, and can carry out neatly storing, transmit, filtering, automatic focus, phase aberrated beams compensate and all-round display etc., above-mentioned plurality of advantages makes digital holographic microscopy obtain a wide range of applications.

Digital holography is based on interference imaging technology, and thus relative to other measuring technique, the precision comparison of its phase measurement is high.But, owing to being subject to the impact of speckle and Zero-order diffractive noise, making to be subject to noise based on the multiple links in the conventional reconstruction algorithm reconstruction mutually in place of Fourier transform, reduce the precision that position rebuilds mutually on the one hand, have impact on the speed that position rebuilds mutually simultaneously.In order to reduce the impact of speckle noise, multiple noise-reduction method and technology are carried out, and wherein representative method has the noise reduction technology that superposes based on several holograms and selects short-coherence light source to reduce the technology of speckle noise.But first method can not be used for real-time phase imaging at all, second method makes the contrast of reproduction image decline, what is more important, and what these two kinds of methods did not all fundamentally solve noise affects problem.For the elimination of Zero-order diffractive item, frequency domain filtering is the most frequently used method, but this method is only applicable to off-axis digital holography situation, also there is the halfway problem of noise filtering simultaneously.

Summary of the invention

Embodiments of the invention provide a kind of digital hologram high precision position phase method for reconstructing based on Hilbert conversion, can improve a phase reconstruction precision.

For achieving the above object, embodiments of the invention adopt following technical scheme:

Based on a digital hologram high precision position phase method for reconstructing for Hilbert conversion, comprising:

Utilize image planes digital hologram light path system to carry out holographic recording to object light field, obtain digital hologram;

Fourier transform is carried out to recorded digital hologram, obtains hologram frequency spectrum;

Carry out filtering to described hologram frequency spectrum, filtering zero level is composed;

Inverse Fourier transform is carried out to filtered spectrogram, obtains the hologram of filtering zero-order term;

To the hologram two dimension Hilbert conversion after filtering zero-order term, filtering residual DC item, and obtain shape and intensity is constant, the stripe pattern of pi/2 phase shift only occurs;

Converted by the hologram after filtering zero-order term and Hilbert thereof, structure is containing object light field and the analytical function with reference to light field phasic difference information;

Argument computing is got to the analytical function built, obtains object light field and the parcel position phase with reference to light field phasic difference value;

Position is carried out mutually to above-mentioned parcel position and separates parcel mutually, obtain object light field and the expansion position phase with reference to light field phasic difference;

In the extracted region position phase data that object light field meta is constant mutually, line linearity matching of going forward side by side, thus linear phase aberrated beams rectification is carried out mutually to above-mentioned linear position, obtain the phase mehtod of object light field.

The digital hologram high precision position phase method for reconstructing based on Hilbert conversion that the embodiment of the present invention provides, because Hilbert conversion itself has the effect eliminating DC terms, the impact that Hilbert conversion can eliminate Zero-order diffractive item is further carried out on the hologram after frequency domain filtering, thus can obtain more high-precision position phase images.With in prior art based on compared with the conventional reconstruction algorithm of Fourier transform, in the experiment of haemocyte in the present invention, the standard deviation based on the position phase reconstructed results of Hilbert conversion reduces 14.0%.

Accompanying drawing explanation

The schematic flow sheet of a kind of digital hologram high precision position phase method for reconstructing based on Hilbert conversion that Fig. 1 provides for the embodiment of the present invention;

The structural representation of a kind of iso-curvature thing ginseng light image face digital holography recording system that Fig. 2 provides for the embodiment of the present invention;

Fig. 3 is the experimental result of human body erythrocyte, and 3 (a) is hologram; 3 (b) is hologram frequency spectrum;

Fig. 4 is the position phase obtained based on conventional reconstruction algorithm, and 4 (a) is the parcel position phase before distortion correction; Parcel position phase after 4 (b) distortion correction; 4 (c) is for launching position phase; 4 (d) is the 3-D display of 4 (c);

Fig. 5 is the position phase based on Hilbert transform reconstruction, the hologram after 5 (a) filtering, 0 grade of item; The imaginary part that 5 (b) is analytical function; 5 (c) is the parcel position phase before distortion correction; 5 (d) is the parcel position phase after distortion correction; The expansion position phase of 5 (e); 5 (f) is the 3-D display of 5 (e);

Fig. 6 is based on conventional reconstruction algorithm with based on the phase mehtod figure on the center line of a cell of Hilbert mapping algorithm;

Fig. 7 is hologram and the frequency spectrum of Lygodium japonicum cell, and 7 (a) is hologram; 7 (b) is hologram frequency spectrum;

Fig. 8 is the Lygodium japonicum cell experiment result that conventional reconstruction obtains, and 8 (a) is the Two dimensional Distribution launching position phase; The 3-D display that Fig. 8 (b) is Fig. 8 (a); Fig. 8 (c) is the Two-dimensional Position phase after removing cell peripheral liquid; The 3-D display that Fig. 8 (d) is Fig. 8 (c);

Fig. 9 converts based on Hilbert the Lygodium japonicum cell experiment result obtained, and Fig. 9 (a) launches the Two dimensional Distribution of position phase; The 3-D display that 9 (b) is Fig. 9 (a); Fig. 9 (c) is the Two-dimensional Position phase after removing cell peripheral liquid; Fig. 9 (d) is the 3-D display of figure (c);

Figure 10 is the one dimension distribution curve schematic diagram of cell position phase on white lines in Fig. 8 (c) and Fig. 9 (c).

Embodiment

Below in conjunction with the accompanying drawing in the embodiment of the present invention, be clearly and completely described the technical scheme in the embodiment of the present invention, obviously, described embodiment is only the present invention's part embodiment, instead of whole embodiments.Based on the embodiment in the present invention, those of ordinary skill in the art, not making the every other embodiment obtained under creative work prerequisite, belong to the scope of protection of the invention.

Embodiments provide a kind of digital hologram high precision position phase method for reconstructing based on Hilbert conversion, as shown in Figure 1, said method comprising the steps of:

101, utilize image planes digital hologram light path system to carry out holographic recording to object light field, obtain digital hologram.

Figure 2 shows that iso-curvature thing, ginseng light Digital holographic microscopy register system schematic diagram.The light pencil sent by the He-Ne laser instrument of wavelength X=623.8nm, power P=60mW is divided into two bundles by polarization beam apparatus PBS after half-wave plate, and respectively through beam-expanding collimation system BE ₁and BE ₂after be transformed into two bundle directional lights, the light beam wherein transmitted from PBS is as object light, and the light beam of reflection is as with reference to light.Two identical microcobjective MO ₁and MO ₂be placed in object light road and reference path respectively, one for amplifying object, another is then for the formation of spherical reference wave.Two-beam carries out conjunction by prism BS and restraints and produce interference, forms continuous print interference light wave field, is positioned to carry out sampling as the photoelectric coupled device CCD of plane and discrete, forms digital hologram, then sends into computing machine and carry out numerical reconstruction.Meticulously MO need be adjusted during application ₂position, make MO ₁and MO ₂equal to beam splitter BS centre distance, thoroughly to eliminate MO ₁to the secondary phase aberrated beams that object is introduced.By adjust the orientation of BS can realize hologram from axle record.The microcobjective MO parameter specification that can adopt in the embodiment of the present invention is: enlargement factor is 40, and numerical aperture is 0.65, and focal length is the theoretical limit resolution of 4.65mm, this MO is 0.59 μm; The parameter specification of CCD is: Pixel Dimensions is 4.65 μm × 4.65 μm, pixel count is 1392 × 1032,16bit level gray scale exports.System as shown in Figure 2, computing machine just can realize the record of digital hologram.

Represent the coordinate of CCD plane with x-y, its initial point is positioned at CCD photosurface center.If the light field complex amplitude that object light and reference light are formed in this plane is respectively O (x, y)=O ₀(x, y) exp [j φ _o(x, y)] and R (x, y)=R ₀(x, y) exp [j φ _r(x, y)], then the continuous interference field distribution that CCD plane is formed is:

I(x，y)＝|O(x，y)+R(x，y)| ²

＝|O(x，y)| ²+|R(x，y)| ²+O(x，y)R ^*(x，y)+O ^*(x，y)R(x，y)

＝a(x，y)+b(x，y)cos[φ _o(x，y)-φ _r(x，y)] (1)

O in formula ^*(x, y), R ^*(x, y) represents the complex conjugate light wave of Object light wave and reference light wave respectively, a (x, y)=O ₀ ²(x, y)+R ₀ ²(x, y) is zero-order term, i.e. DC terms; B (x, y)=2O ₀(x, y) R ₀(x, y) is the distribution of amplitudes of+1 striped formed with-1 grade of item sum.Visible, the wavefront information of object light field encoded with the shape of interference fringe, density and intensity distributions in hologram.The hologram of CCD physical record is discrete and the function be limited in scope, and formula (1) is the optical interference intensity distributions of hologram plane, is continuous function.

Corner dimension between object light and reference light needs to adjust according to the recording parameters of element each in light path, to make in hologram frequency spectrum 0 grade to be just separated with+1 ,-1 grade.In order to ensure MO in light path ₁and MO ₂the spherical light wave produced has equal radius-of-curvature relative to CCD, can under the condition of not setting-out product, accurate adjustment MO ₂position make record holographic fringes be straight line.

According to the feature of image planes digital holography, the resolution of reproduction image and digital hologram size have nothing to do, and therefore, in order to improve reconstruction speed, preferably, recorded digital hologram can be cut to 1024 × 1024 or 512 × 512 and rebuild.

Computing machine can carry out Fourier transform to recorded digital hologram after obtaining digital hologram, obtains hologram frequency spectrum.

102, Fourier transform is carried out to recorded digital hologram, obtain hologram frequency spectrum.

103, carry out filtering to described hologram frequency spectrum, filtering zero level is composed.

104, inverse Fourier transform is carried out to filtered spectrogram, obtain the hologram after filtering zero-order term.

Carry out Fourier transform to formula (1), then carry out spectral filtering by zero-order term filtering, finally carry out inverse Fourier transform to filtered spectrogram, then the hologram that can obtain after filtering zero-order term (i.e. DC terms) is as follows:

I ₁(x，y)＝b(x，y)cos[φ _o(x，y)-φ _r(x，y)] (2)

105, to the hologram two dimension Hilbert conversion after filtering zero-order term, filtering residual DC item, and obtain shape and intensity is constant, the stripe pattern of pi/2 phase shift only occurs.

For simplicity, the one dimension form of each amount is only write out in below analyzing.To spatial domain signal u (x), its Hilbert conversion can be expressed as:

$w\left(x\right)=\frac{1}{\mathrm{\πx}}*u\left(x\right)---\left(3a\right)$

In formula, symbol * represents convolution algorithm.In frequency domain, Hilbert transformation relation can be expressed as:

$\left(f\right)\left\{\begin{array}{cc}-\mathrm{jU}\left(f\right)& f>0\\ \mathrm{jU}\left(f\right)& f<0\end{array}\right.---\left(3b\right)$

Wherein, W (f) and U (f) is the Fourier transform of w (x) and u (x) respectively, and j is imaginary unit.From (3b) formula, asking the Hilbert of a real function u (x) to convert, is exactly by all positive frequency composition phase rotating-pi/2s of u (x), to all negative frequency composition phase rotating+pi/2s.Therefore, the Hilbert of cos (2 π fx) is sin (2 π fx).

In addition, Hilbert conversion also has following character: if b (x) frequency is relatively low in function b (x) cos (2 π fx), then its Hilbert is transformed to b (x) sin (2 π fx).

Because b (x, y) in formula (2) is relative to φ _o(x, y)-φ _r(x, y) component frequency is much lower, therefore, according to the character of Hilbert conversion, does Hilbert conversion, obtain formula (2) both sides:

I ₂(x，y)＝H{I ₁(x，y)}＝b(x，y)sin[φ _o(x，y)-φ _r(x，y)] (3)

In formula, H represents that two-dimentional Hilbert converts.

106, converted by the hologram after filtering zero-order term and Hilbert thereof, structure is containing object light field and the analytical function with reference to light field phasic difference information.

Analytical function is constructed as follows by (2) formula and (3) formula:

Z(x，y)＝I ₁(x，y)+jI ₂(x，y)

＝b(x，y){cos[φ _o(x，y)-φ _r(x，y)]+jsin[φ _o(x，y)-φ _r(x，y)]} (4)

107, argument computing is got to the analytical function built, obtain object light field and the parcel position phase with reference to light field phasic difference value.

Undertaken getting argument computing by formula (4), object light field and the parcel phase mehtod with reference to light field phasic difference can be obtained:

${\mathrm{\&Delta;\&phi;}}_{\mathrm{wrap}}(x,y)=[{\mathrm{\&phi;}}_o(x,y)-{\mathrm{\&phi;}}_r(x,y){]}_{\mathrm{wrap}}=\arctan \{\frac{\mathrm{Im}\left[Z(x,y)\right]}{\mathrm{Re}\left[Z(x,y)\right]}\}---\left(5\right)$

108, position is carried out mutually to described parcel position and separate parcel mutually, obtain object light field and the expansion position phase with reference to light field phasic difference.

The formula launching position phase is: Δ φ _unwrap(x, y)=[φ _o(x, y)-φ _r(x, y)] _unwrap(6)

109, in object light field, a certain flat site (position is the region of constant mutually) extracts position phase data, line linearity matching of going forward side by side, thus carries out linear phase aberrated beams rectification mutually to above-mentioned linear position, obtains the phase mehtod φ of object light field _o(x, y).

Compared with the existing conventional reconstruction method realized based on Fourier transform, two kinds of method for reconstructing all need to carry out filtering process to hologram frequency spectrum, all need filtering Zero-order diffractive item, and this is their something in common.Object due to reality is not band limit, and thus the zero level of hologram can not be composed thorough filtering by frequency domain filtering, and this will cause direct impact to the reconstruction of digital hologram one-level picture.In addition; no matter take which kind of method for reconstructing, what first obtain during digital hologram position rebuilds mutually is the parcel position phase reproducing light field, and there occurs the position phase of distortion often; therefore, rear usually also needs carries out solution parcel and phase aberrated beams compensation to it mutually to obtain parcel position.Zero-order diffractive noise remaining in existing method not only can the contraposition precision of separating parcel and distortion compensation mutually impact, and these two kinds of calculating processes also can be made slack-off.But, because Hilbert conversion has the effect of elimination DC terms (i.e. Zero-order diffractive item), although the hologram after frequency domain filtering includes a small amount of remaining DC terms, after Hilbert conversion, the flip-flop of these remnants is just completely eliminated.Thus, compared with the conventional reconstruction method based on Fourier transform, in literary composition carry based on Hilbert conversion digital hologram method for reconstructing there is higher position phase reconstruction precision.

Apply the existing conventional reconstruction method based on Fourier transform below and provided by the inventionly carry out Experimental Comparison based on Hilbert transform reconstruction method.

2.1 based on the result of the conventional reconstruction method of Fourier transform

Light path shown in Fig. 1 is utilized to carry out image planes digital hologram record to human body erythrocyte.Fig. 3 (a) is 512 × 512 size holograms obtained from recorded hologram core cutting, and Fig. 3 (b) is the spectrogram of 3 (a).Utilize transparency window function+1 grade of spectrum in rectangle frame in Fig. 3 (b) to be leached, then carry out position and rebuild mutually.

Fig. 4 (a) is the parcel position phase before distortion correction, and from enlarged drawing, position there occurs linear distortion mutually.Distortionless parcel position phase is obtained, as shown in Fig. 4 (b) after distortion compensation.Fig. 4 (c) is the expansion position phase utilizing the least-squares algorithm based on lateral shear to obtain, and Fig. 4 (d) is the 3-D display of Fig. 4 (c).

2.2 based on the result of Hilbert transform reconstruction

The position phase method for reconstructing based on Hilbert conversion is utilized to rebuild the hologram shown in Fig. 3 (a), first+1 in Fig. 3 (b) rectangle frame and-1 grade of spectrum are leached in reconstruction, obtain the hologram after filtering 0 grade, as shown in Fig. 5 (a).Then the imaginary part that Hilbert conversion obtains analytical function is done, as shown in Fig. 5 (b) to Fig. 5 (a).Comparison diagram 5 (a) and 5 (b), find no obvious difference, but pass through the enlarged drawing in region, the lower left corner in comparison diagram, can find out nuance therebetween.Fig. 5 (c) is the parcel phase mehtod that the reproduction image light field obtained contains distortion, equally, utilize linear phase aberrated beams compensation method to carry out distortion compensation with the least square method based on lateral shear and separate parcel mutually with position, obtain distortionless parcel position phase and launch phase mehtod as shown in Fig. 5 (d), (e).Fig. 5 (f) gives the 3-D display result of Fig. 5 (e).Comparison diagram 4 (d) and Fig. 5 (f), the microslide in visible Fig. 4 (d) slightly tilts, and this is that the phase aberrated beams error caused due to noise causes.

In order to the difference that relatively above two kinds of algorithm contrapositions more are intuitively rebuild mutually, made the phase mehtod figure on the center line of same cell in Fig. 4 (c) and 5 (e) respectively, result as shown in Figure 6.Visible, maximum larger with minimum bit phase difference in the position phasor obtained by Hilbert mapping algorithm.

In order to the precision that quantitative comparison two kinds of algorithm contrapositions are rebuild mutually, we choose three rectangular areas not containing cellular informatics respectively in Two-dimensional Position phase Fig. 4 (c) and 5 (e) of reconstruct, calculate its standard deviation.Following formula is the computing formula of standard deviation:

$S=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{({\mathrm{\&phi;}}_{\mathrm{oi}}-\stackrel{\&OverBar;}{{\mathrm{\&phi;}}_o})}^2}---\left(7\right)$

In formula, for the draw value of selected position, region phase data.According to formula (7), respectively these rectangular areas are calculated, shown in result table 1.

Table 1 standard deviation correlation data

Can obviously be found out by upper table, no matter for which region, the standard deviation obtained by the reconstructing method converted based on Hilbert is all little than the standard deviation obtained by conventional Fourier transformation method.This shows that the position phase precision obtained than conventional Fourier transformation method based on the method for Hilbert conversion is high.Data in table be averaged, can obtain following result: compared with conventional reconstruction result, the standard deviation based on the position phase reconstructed results of Hilbert conversion reduces 14.0%.

In order to compare the quality of two kinds of method for reconstructing further, identical optical path pair medicine medicine materical crude slice Lygodium japonicum cell is utilized to make experimental study.Fig. 7 (a), 7 (b) are recorded hologram and spectrum distribution thereof, and wherein hologram size is 1024 × 1024.Fig. 8 and Fig. 9 is the result utilizing conventional method respectively and obtain based on Hilbert transform method.From (a), (b) figure in Fig. 8 and Fig. 9, owing to needing, to the process of doing of Lygodium japonicum cell, to cause cell peripheral to there is the encirclement of liquid, have impact on the display effect of the form of cell own before hologram record.Eliminated the impact of cell peripheral liquid by limb recognition technology, improve display effect.

Figure 10 is the one dimension distribution curve of cell position phase on white lines in Fig. 8 (c) and Fig. 9 (c), obvious, and the phase mehtod curve that two kinds of methods obtain is closely similar, but highly slightly high mutually in the position that cell centre is obtained by Hilbert algorithm.This is consistent with the conclusion that haemocyte situation obtains.In addition, in Fig. 8 (a) and 9 (a), choose several flat site respectively calculate its standard deviation, obtain the conclusion that position phase precision that the algorithm based on Hilbert conversion obtains than the conventional method based on Fourier transform is high equally.

As from the foregoing, because Hilbert conversion itself has the effect eliminating DC terms, the impact that Hilbert conversion can eliminate Zero-order diffractive item is further carried out on the hologram after frequency domain filtering, thus can obtain more high-precision position phase images.Compared with the conventional reconstruction algorithm based on Fourier transform, in the experiment of haemocyte in literary composition, the standard deviation based on the position phase reconstructed results of Hilbert conversion reduces 14.0%.

The above; be only the specific embodiment of the present invention, but protection scope of the present invention is not limited thereto, is anyly familiar with those skilled in the art in the technical scope that the present invention discloses; change can be expected easily or replace, all should be encompassed within protection scope of the present invention.Therefore, protection scope of the present invention should described be as the criterion with the protection domain of claim.

Claims (2)

1., based on a digital hologram high precision position phase method for reconstructing for Hilbert conversion, it is characterized in that, comprising:

Utilize image planes digital hologram light path system to carry out holographic recording to object light field, obtain digital hologram;

Fourier transform is carried out to recorded digital hologram, obtains hologram frequency spectrum;

Carry out filtering to described hologram frequency spectrum, filtering zero level is composed;

Inverse Fourier transform is carried out to filtered spectrogram, obtains the hologram of filtering zero-order term;

To the hologram two dimension Hilbert conversion after filtering zero-order term, filtering residual DC item, and obtain shape and intensity is constant, the stripe pattern of pi/2 phase shift only occurs;

Converted by the hologram after filtering zero-order term and Hilbert thereof, structure is containing object light field and the analytical function with reference to light field phasic difference information;

Argument computing is got to the analytical function built, obtains object light field and the parcel position phase with reference to light field phasic difference value;

Position is carried out mutually to above-mentioned parcel position and separates parcel mutually, obtain object light field and the expansion position phase with reference to light field phasic difference;

In the extracted region position phase data that object light field meta is constant mutually, line linearity matching of going forward side by side, thus linear phase aberrated beams rectification is carried out mutually to above-mentioned linear position, obtain the phase mehtod of object light field.

2. method according to claim 1, is characterized in that,

Represent the coordinate of CCD plane with x-y, its initial point is positioned at CCD photosurface center, if the light field complex amplitude that object light and reference light are formed in this plane is respectively O (x, y)=O ₀(x, y) exP [j φ _o(x, y)] and R (x, y)=R ₀(x, y) exp [j φ _r(x, y)], then the hologram that interference plane is formed is distributed as:

$\begin{array}{c}I(x,y)={|O(x,y)+R(x,y)|}^2\\ ={\left|O(x,y)\right|}^2+{\left|R(x,y)\right|}^2+O(x,y)R^{*}(x,y)+O^{*}(x,y)R(x,y)---\left(1\right)\\ =a(x,y)+b(x,y)\cos [{\mathrm{\&phi;}}_0(x,y)-{\mathrm{\&phi;}}_r(x,y)]\end{array}$

O in formula ^*(x, y), R ^*(x, y) represents the complex conjugate light wave of Object light wave and reference light wave respectively, a (x, y)=O ₀ ²(x, y)+R ₀ ²(x, y) is zero-order term; B (x, y)=2O ₀(x, y) R ₀(x, y) is the distribution of amplitudes of+1 striped formed with-1 grade of item sum;

Carry out Fourier transform to formula (1), then carry out spectral filtering by zero-order term filtering, finally carry out inverse Fourier transform to filtered frequency spectrum, then the hologram that can obtain after filtering DC terms is as follows:

I ₁(x，y)＝b(x，y)cos[φ _o(x，y)-φ _r(x，y)] (2)

Because b (x, y) in formula (2) is relative to φ _o(x, y)-φ _r(x, y) component frequency is much lower, therefore, according to the character of Hilbert conversion, does Hilbert conversion, obtain formula (2) both sides:

I ₂(x，y)＝H{I ₁(x，y)}＝b(x，y)sin[φ _o(x，y)-φ _r(x,y)] (3)

In formula, H represents that two-dimentional Hilbert converts;

Analytical function is constructed as follows by (2) formula and (3) formula:

Z(x，y)＝I ₁(x，y)+jI ₂(x，y)

＝b(x，y){cos[φ _o(x,y)-φ _r(x,y)]+jsin[φ _o(x,y)-φ _r(x,y)]} (4)

Argument computing is got to the analytical function built, obtains formula (5):

$\mathrm{\&Delta;}{\mathrm{\&phi;}}_{\mathrm{wrap}}(x,y)={[{\mathrm{\&phi;}}_o(x,y)-{\mathrm{\&phi;}}_r(x,y)]}_{\mathrm{wrap}}=\arctan \left\{\frac{\mathrm{Im}\left[Z(x,y)\right]}{\mathrm{Re}\left[Z(x,y)\right]}\right\}---\left(5\right).$