Wideband snapshot-type polarization interferometry imaging system and method for detecting breast tumors

Abstract

This invention relates to the field of optical imaging and detection technology, specifically providing a broadband snapshot-type polarization interferometry imaging detection system and method for breast tumors. A broadband light source illuminates a breast tumor tissue sample. The reflected signal light is collected and collimated by a pre-optical system, then sequentially passes through a first and a second ray shaping unit for two shearing and two polarization state transformations, resulting in two beams of linearly polarized light and two beams of circularly polarized light. These beams are imaged by an imaging lens to an area array detector to obtain a polarization interferogram, which is then demodulated to obtain all Stokes parameters of the breast tumor tissue sample. This invention achieves broadband snapshot imaging of breast tumors with all Stokes parameters without using an analyzer, doubling the light throughput compared to traditional polarization imaging instruments using analyzers. Furthermore, this invention has advantages such as compact structure, no moving parts, easy assembly and adjustment, and no need for image registration.

Description

Technical Field

[0001] This invention belongs to the field of optical imaging and detection technology, specifically providing a wideband snapshot-type polarization interferometric imaging detection system and method for breast tumors. Background Technology

[0002] In breast cancer surgery, determining whether any tumor tissue remains at the edges is crucial. If tumor cells are not completely removed, a second surgery is required, increasing medical costs and harming the patient's health. Traditional methods of breast tumor pathological diagnosis are limited in sample collection and time-consuming, and their assessment of tumor tissue edges is inaccurate. Polarization imaging technology can provide high-contrast images of normal and tumor tissue, quickly and directly determining the extent of tumor tissue. It offers fast detection and analysis speed and high accuracy, representing an important direction for tumor diagnosis.

[0003] Currently, most polarization imaging detection techniques used in medicine are based on the rotating polarizer method. This method obtains images of samples in different linear polarization states by changing the polarization direction of the polarizer, and then further calculates some Stokes parameters and other polarization information of the sample. However, this method requires scanning time, has low diagnostic efficiency, and involves moving parts, making it difficult to guarantee instrument performance stability. Furthermore, existing methods for acquiring interferometric images all use analyzers, but analyzers are typically placed at a 45° angle, resulting in a 50% energy loss of polarized light after passing through the analyzer. Summary of the Invention

[0004] In view of this, the present invention aims to provide a broadband snapshot-type polarization interferometry imaging detection system and method for breast tumors. The system shears the broadband light used to detect breast tumor tissue into two sets of polarized light with different polarization states, and then images and demodulates the two sets of polarized light. Without using an analyzer, all Stokes parameters of the polarization interferometric image containing information about the breast tumor tissue are obtained, thereby obtaining information about the breast tumor tissue. Furthermore, by avoiding the use of an analyzer, the light throughput is doubled, improving the accuracy of obtaining information about the breast tumor tissue.

[0005] To achieve the above objectives, the technical solution created by this invention is implemented as follows:

[0006] A broadband snapshot-type polarization interferometry imaging detection system for breast tumors includes a broadband light source, a front optical unit, a first ray shaping unit, a second ray shaping unit, an imaging lens, an area array detector, and a demodulation processor. The broadband light source generates broadband light and illuminates the breast tumor tissue. The breast tumor tissue reflects the broadband light into the front optical unit for collimation, generating collimated light. The first ray shaping unit shears the collimated light to obtain two beams of first circularly polarized light and converts their polarization states to obtain two beams of first linearly polarized light. The second ray shaping unit shears and converts the polarization states of the two first linearly polarized beams to obtain four beams of second circularly polarized light, and then converts the polarization states of two of these beams to obtain two beams of second linearly polarized light. The two beams of second linearly polarized light and the two beams of second circularly polarized light are imaged onto the area array detector via the imaging lens to obtain a polarization interferometry image containing information about the breast tumor tissue. The area array detector transmits the polarization interferometry image to the demodulation processor for demodulation processing to obtain all Stokes parameters of the polarization interferometry image.

[0007] Furthermore, the first ray shaping unit includes a first polarizing grating, a second polarizing grating, and a first achromatic quarter-wave plate; wherein the first polarizing grating and the second polarizing grating are placed perpendicular to the optical axis, and the grating periods of the first polarizing grating and the second polarizing grating are the same and they are placed parallel to each other; the first polarizing grating and the second polarizing grating shear the collimated light to obtain two beams of first circularly polarized light, and the two beams of first circularly polarized light enter the first achromatic quarter-wave plate for polarization state conversion to obtain two beams of first linearly polarized light.

[0008] Furthermore, the second ray shaping unit includes a third polarization grating, a fourth polarization grating, and a second achromatic quarter-wave plate; wherein, the third polarization grating and the fourth polarization grating are placed perpendicular to the optical axis, and the grating periods of the third polarization grating and the fourth polarization grating are the same and they are placed parallel to each other; the third polarization grating and the fourth polarization grating shear and convert the polarization state of the two first linearly polarized beams to obtain four second circularly polarized beams, two of the four second circularly polarized beams enter the second achromatic quarter-wave plate for polarization state conversion to obtain two second linearly polarized beams, which together with the remaining two second circularly polarized beams enter the imaging lens.

[0009] Furthermore, the size of the second achromatic quarter-wave plate is half that of the fourth polarization grating.

[0010] Furthermore, the front optical unit includes an objective lens, an aperture stop, and a collimating lens; wherein, the objective lens receives the broadband light reflected from the breast tumor tissue and images it on the aperture stop, the aperture stop restricts the broadband light, and after collimation by the collimating lens, collimated light is obtained and incident on the first ray shaping unit.

[0011] A broadband snapshot-type polarization interferometry imaging method for detecting breast tumors, applicable to the broadband snapshot-type polarization interferometry imaging system for detecting breast tumors provided by this invention, specifically includes the following steps:

[0012] S1: Broadband light source with Stokes parameters , , and The reference light illuminates the front optical unit, and the frequency domain channel of the reference light is obtained at the array detector. , and That is, the frequency domain channel of the reference light. , and ;

[0013] S2: A broadband light source illuminates breast tumor tissue with broadband light of different polarization states. The frequency domain channel of the polarization interference image of the breast tumor tissue under the current broadband light is obtained at the area array detector. , and This refers to the frequency domain channels of breast tumor tissue. , and ;

[0014] S3: Based on the two frequency domain channels of the reference light , Two frequency domain channels of breast tumor tissue , The first and second decoupling factors were obtained, and then the Stokes parameters of the breast tumor tissue were obtained. , , and .

[0015] Furthermore, in step S3, the two frequency domain channels of the reference light are... , Inverse Fourier transform calculations were performed, and the results were compared with two frequency domain channels of breast tumor tissue. , Divide the results of the inverse Fourier transform calculations to obtain the first and second decoupling factors.

[0016] Furthermore, in step S3, the Stokes parameters of the breast tumor tissue are obtained. , , and They are respectively:

[0017] ;

[0018] ;

[0019] ;

[0020] ;

[0021] in, This represents the inverse Fourier transform, where j represents the imaginary number. This indicates the first elimination of coupling factor. This indicates the second coupling elimination factor. This indicates the operation of taking the real part. This indicates the operation of taking the imaginary part.

[0022] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0023] (1) The wideband snapshot-type breast tumor polarization interferometric imaging detection system created by the present invention utilizes two light shaping units to perform two shearing and polarization state conversions on the wideband light reflected from the breast tumor tissue, resulting in two sets of polarized light with different polarization states. The two sets of polarized light are then interfered to obtain a clear and high-contrast polarization interferometric image. The polarization interferometric image is then subjected to Fourier transform calculation and Stokes parameter demodulation to obtain all Stokes parameters and polarization image containing information about the breast tumor tissue, thereby realizing wideband snapshot full Stokes parameter measurement.

[0024] (2) In the band snapshot type breast tumor polarization interferometric imaging detection system created by the present invention, two sets of polarized light with different polarization states are obtained by performing two shearing and polarization state conversion on the wide band light. This avoids the use of a polarizer in the subsequent process of obtaining polarization interferometric images, thereby doubling the light transmission and further improving the quality of polarization interferometric images, and thus improving the accuracy of obtaining information on breast tumor tissue. Attached Figure Description

[0025] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments and descriptions of the invention are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0026] Figure 1 A schematic diagram of the structure of the wideband snapshot-type breast tumor polarization interferometry imaging detection system described in the embodiment of the present invention;

[0027] Figure 2 A schematic diagram of the Fourier frequency domain channel of the polarization interference image as described in the embodiment of the present invention;

[0028] Figure 3The flowchart illustrates the broadband snapshot-type polarization interferometry imaging detection method for breast tumors as described in the embodiments of this invention.

[0029] Explanation of reference numerals in the attached figures:

[0030] 1. Wideband light source; 2. Front optical unit; 201. Objective lens; 202. Aperture stop; 203. Collimating lens; 3. First ray shaping unit; 301. First polarization grating; 302. Second polarization grating; 303. First achromatic quarter-wave plate; 4. Second ray shaping unit; 401. Third polarization grating; 402. Fourth polarization grating; 403. Second achromatic quarter-wave plate; 5. Imaging lens; 6. Area array detector; 7. Demodulation processor; 8. Breast tumor tissue. Detailed Implementation

[0031] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof.

[0032] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0033] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on this invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, features defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0034] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art will understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0035] The invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0036] like Figure 1 As shown, the wideband snapshot-type breast tumor polarization interferometry imaging detection system of this invention includes a wideband light source 1, a front optical unit 2, a first light shaping unit 3, a second light shaping unit 4, an imaging lens 5, an area array detector 6, and a demodulation processor 7. The wideband light source 1 generates wideband light and illuminates the breast tumor tissue 8. The breast tumor tissue 8 reflects the wideband light into the front optical unit 2 for collimation, generating collimated light.

[0037] The front optical unit 2 includes an objective lens 201, an aperture 202, and a collimating lens 203. The objective lens 201 receives the broadband light reflected from the breast tumor tissue 8 and images it onto the aperture 202. After the aperture 202 restricts the broadband light, it is collimated by the collimating lens 203 to obtain collimated light, which is then incident on the first light shaping unit 3.

[0038] The first ray shaping unit 3 includes a first polarizing grating 301, a second polarizing grating 302, and a first achromatic quarter-wave plate 303. The first polarizing grating 301 and the second polarizing grating 302 are placed perpendicular to the optical axis, and the grating periods of the first polarizing grating 301 and the second polarizing grating 302 are the same and they are placed parallel to each other.

[0039] The collimation process of the first beam shaping unit 3 is as follows: When the collimated light is incident on the first polarization grating 301, two beams of first circularly polarized light with different diffraction angles and perpendicular polarization directions are generated. The diffraction effect of the second polarization grating 302 on these two beams of first circularly polarized light is opposite to that of the first polarization grating 301, making the final exit angle of the two beams of first circularly polarized light the same as the incident angle. However, since there is a gap between the first polarization grating 301 and the second polarization grating 302, the two beams of light exit from different positions of the second polarization grating 302, and the two beams of first circularly polarized light are parallel to each other. The two beams of first circularly polarized light enter the first achromatic quarter-wave plate 303 for polarization state conversion, resulting in two beams of first linearly polarized light, which then enter the second beam shaping unit 4.

[0040] The distance between the two incident positions of the first circularly polarized beams obtained by shearing on the polarization grating 302 is the shearing distance. shear distance Spacing between the first polarization grating 301 and the second polarization grating 302 Positively correlated with the grating period of the first polarization grating 301 and the second polarization grating 302. They are negatively correlated, specifically:

[0041] ;

[0042] ;

[0043] Where m represents the diffraction order of the first polarization grating 301 and the second polarization grating 302. Indicates the wavelength of broadband light. This represents the +1st order diffraction angle and the -1st order diffraction angle of the first polarization grating 301. This indicates the angle of incidence of the collimated light entering the first polarization grating 301.

[0044] The second ray shaping unit 4 includes a third polarizing grating 401, a fourth polarizing grating 402, and a second achromatic quarter-wave plate 403. The third and fourth polarizing gratings 401 and 402 are placed perpendicular to the optical axis, and their grating periods are the same and they are placed parallel to each other. The third and fourth polarizing gratings 401 and 402 shear and convert the polarization state of two beams of first linearly polarized light to obtain four beams of second circularly polarized light. Two of these second circularly polarized beams are left-handed circularly polarized light, and the other two are right-handed circularly polarized light. The shearing process of the two beams of first linearly polarized light by the second ray shaping unit 4 is the same as the shearing process of the first ray shaping unit 3 for aligned direct light.

[0045] Two of the four second circularly polarized beams enter the second achromatic quarter-wave plate 403 for polarization state conversion, resulting in two second linearly polarized beams, which, together with the remaining two second circularly polarized beams, enter the imaging lens 5.

[0046] In this embodiment of the invention, the size of the second achromatic quarter-wave plate 403 is half that of the fourth polarization grating 402, and the fast axis of the second achromatic quarter-wave plate 403 is perpendicular to the optical axis. One beam of second circularly polarized light and one beam of second linearly polarized light are defined as one set of light, and another beam of second circularly polarized light and another beam of second linearly polarized light are defined as two sets of light. Based on the above formula, by reducing the grating period of the third polarization grating 401 and the fourth polarization grating 402, or by increasing the spacing between the third polarization grating 401 and the fourth polarization grating 402, the first set of light and the two sets of light output by the fourth polarization grating 402 are completely separated in space, and the spatial distance between the first set of light and the two sets of light is greater than the diameter of the collimated light.

[0047] In this embodiment of the invention, the imaging lens 5 is designed using a freeform surface for aberration correction, based on the paper "Research on Freeform Surfaces in Imaging Optical Systems". First, the characteristics of the incident and outgoing rays from the first polarization grating 301, the second polarization grating 302, the third polarization grating 401, and the fourth polarization grating 403 are analyzed to establish a ray propagation model. The characteristic parameters of the lens surface are then solved to establish the initial structure of the imaging lens. The freeform surface is characterized using Zenike polynomials, and an evaluation function and related constraint methods are established according to requirements. Optical simulation software is used to optimize the initial structure and obtain the optimal surface parameters, thus yielding the imaging lens 5.

[0048] Imaging lens 5 images two beams of second linearly polarized light and two beams of second circularly polarized light onto area array detector 6 to obtain a polarization interference image containing information about breast tumor tissue 8. Area array detector 6 then transmits the polarization interference image to demodulation processor 7 for demodulation processing to obtain all Stokes parameters of the polarization interference image. All Stokes parameters contain all the optical information of breast tumor tissue 8, which is convenient for subsequent processing and analysis.

[0049] Demodulation processor 7 receives polarization interference images and acquires them as shown in the image. Figure 2 The polarization interferometric image shown has three channels in the Fourier frequency domain. , and This refers to the three frequency channels of breast tumor tissue in the Fourier frequency domain. (By...) Figure 2 It can be seen that the frequency domain channels , and The tissues are separated and independently distributed, with minimal interference between channels. Therefore, the Stokes parameters of the reconstructed breast tumor tissue are highly accurate.

[0050] By analyzing the three frequency domain channels , and Inverse Fourier transform was performed to obtain the four Stokes parameters of breast tumor tissue 8. , , and ,Right now:

[0051] ;

[0052] ;

[0053] ;

[0054] in, denoted by inverse Fourier transform; π represents pi; j represents the imaginary number; x and y are the coordinates of the detector; U1 and U2 represent the carrier frequencies, where , ε1 and ε2 are the shearing distances of the first ray shaping unit 3 and the second ray shaping unit 4, respectively; λ is the wavelength; and f is the focal length of the imaging lens 5.

[0055] From the above formula, we can see that and This invention addresses the systemic influences present in the wideband snapshot-type breast tumor polarization interferometry imaging detection system provided in this embodiment. To eliminate these influences and obtain high-precision Stokes parameters of breast tumor tissue, this invention provides a wideband snapshot-type breast tumor polarization interferometry imaging detection method based on the proposed system. This method utilizes reference light demodulation technology to reconstruct the polarization parameters of breast tumor tissue, eliminating the detection influences caused by the system itself, thereby obtaining high-precision Stokes parameters of breast tumor tissue.

[0056] like Figure 3 As shown in the embodiments of the present invention, the broadband snapshot-type polarization interferometry imaging detection method for breast tumors specifically includes the following steps:

[0057] S1: Broadband light source 1 with Stokes parameters , , and The reference light illuminates the front optical unit 2, and the frequency domain channel of the reference light is obtained at the array detector 6. , and .

[0058] At this time, the frequency domain channel of the reference light and Stokes parameters of the reference light are , and The relationship is:

[0059] ;

[0060] .

[0061] S2: Broadband light source 1 illuminates breast tumor tissue 8 with broadband light of different polarization states, and the frequency domain channel of the polarization interference image of breast tumor tissue 8 under the current broadband light is obtained at the area array detector 6. , and This refers to the frequency domain channel of breast tumor tissue 8. , and .

[0062] At this point, the frequency domain channels of breast tumor tissue 8 are obtained. , and Stokes parameters of breast tumor tissue 8 , , and The relationships are as follows:

[0063] ;

[0064] ;

[0065] .

[0066] S3: Based on the two frequency domain channels of the reference light , Two frequency domain channels of breast tumor tissue 8 , The first and second decoupling factors were obtained, and then the Stokes parameters of breast tumor tissue 8 were obtained. , , and .

[0067] The frequency domain channel of the reference light is obtained from the above formula. The results of inverse Fourier transform calculations and the frequency domain channels of breast tumor tissue 8 The first decoupling factor is obtained by dividing the result of the inverse Fourier transform calculation. First, eliminate coupling factors Eliminate system effects Frequency domain channels The results of inverse Fourier transform calculations and the frequency domain channels of breast tumor tissue 8 The second decoupling factor is obtained by dividing the result of the inverse Fourier transform calculation. Second elimination of coupling factor Eliminate system effects .

[0068] At this point, the Stokes parameters of breast tumor tissue 8 were further obtained. , , and They are respectively:

[0069] ;

[0070] ;

[0071] ;

[0072] ;

[0073] in, This indicates the operation of taking the real part. This indicates the operation of taking the imaginary part.

[0074] It should be understood that the various forms of processes shown above can be used to reorder, add, or delete steps. For example, the steps described in this invention disclosure can be executed in parallel, sequentially, or in different orders, as long as the desired result of the technical solution disclosed in this invention can be achieved, and this is not limited herein.

[0075] The specific embodiments described above do not constitute a limitation on the scope of protection of this invention. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this invention should be included within the scope of protection of this invention.

Claims

1. A broadband snapshot-type polarization interferometry imaging detection system for breast tumors, characterized in that: The system includes a broadband light source, a front optical unit, a first ray shaping unit, a second ray shaping unit, an imaging lens, an area array detector, and a demodulation processor. The broadband light source generates broadband light and illuminates breast tumor tissue. The breast tumor tissue reflects the broadband light into the front optical unit for collimation, generating collimated light. The first ray shaping unit shears the collimated light to obtain two first circularly polarized beams and converts their polarization states to obtain two first linearly polarized beams. The second ray shaping unit shears and converts the polarization states of the two first linearly polarized beams to obtain four second circularly polarized beams, and then converts the polarization states of two of these second circularly polarized beams to obtain two second linearly polarized beams. The two second linearly polarized beams and the two second circularly polarized beams are imaged onto the area array detector by the imaging lens to obtain a polarization interference image containing information about the breast tumor tissue. The area array detector transmits the polarization interference image to the demodulation processor for demodulation processing to obtain all Stokes parameters of the polarization interference image.

2. The broadband snapshot-type polarization interferometry imaging detection system for breast tumors according to claim 1, characterized in that: The first light shaping unit includes a first polarizing grating, a second polarizing grating, and a first achromatic quarter-wave plate; wherein the first polarizing grating and the second polarizing grating are placed perpendicular to the optical axis, and the grating periods of the first polarizing grating and the second polarizing grating are the same and they are placed parallel to each other; the first polarizing grating and the second polarizing grating shear the collimated light to obtain two beams of first circularly polarized light, and the two beams of first circularly polarized light enter the first achromatic quarter-wave plate for polarization state conversion to obtain two beams of first linearly polarized light.

3. The broadband snapshot-type polarization interferometry imaging detection system for breast tumors according to claim 1, characterized in that: The second light shaping unit includes a third polarizing grating, a fourth polarizing grating, and a second achromatic quarter-wave plate. The third and fourth polarizing gratings are placed perpendicular to the optical axis, have the same grating period, and are placed parallel to each other. The third and fourth polarizing gratings shear and convert the polarization state of two first linearly polarized beams to obtain four second circularly polarized beams. Two of these second circularly polarized beams enter the second achromatic quarter-wave plate for polarization state conversion, resulting in two second linearly polarized beams, which, along with the remaining two second circularly polarized beams, enter the imaging lens.

4. The broadband snapshot-type polarization interferometry imaging detection system for breast tumors according to claim 3, characterized in that: The size of the second achromatic quarter-wave plate is half that of the fourth polarization grating.

5. The broadband snapshot-type polarization interferometry imaging detection system for breast tumors according to claim 1, characterized in that: The front optical unit includes an objective lens, an aperture stop, and a collimating lens; wherein, the objective lens receives the broadband light reflected by the breast tumor tissue and images it on the aperture stop, the aperture stop restricts the broadband light, and after collimation by the collimating lens, the collimated light is obtained and incident on the first light shaping unit.

6. A broadband snapshot-type polarization interferometry imaging method for detecting breast tumors, applicable to the broadband snapshot-type polarization interferometry imaging system for detecting breast tumors as described in any one of claims 1 to 5, characterized in that: Specifically, the following steps are included: S1: The broadband light source uses Stokes parameters as... , , and The reference light illuminates the front optical unit, and the frequency domain channel of the reference light is obtained at the array detector. , and ; S2: The broadband light source illuminates the breast tumor tissue with broadband light of different polarization states, and the frequency domain channel of the polarization interference image of the breast tumor tissue under the current broadband light is obtained at the area array detector. , and That is, the frequency domain channel of the breast tumor tissue. , and ; S3: Based on the two frequency domain channels of the reference light , and the two frequency domain channels of the breast tumor tissue , The first and second decoupling factors are obtained, and then the Stokes parameters of the breast tumor tissue are obtained. , , and .

7. The broadband snapshot-type polarization interferometry imaging method for detecting breast tumors according to claim 6, characterized in that: In step S3, the two frequency domain channels of the reference light are... , Perform inverse Fourier transform calculations, and compare the results with the two frequency domain channels of the breast tumor tissue. , Divide the results of the inverse Fourier transform calculations to obtain the first decoupling factor and the second decoupling factor.

8. The broadband snapshot-type polarization interferometry imaging method for detecting breast tumors according to claim 7, characterized in that: In step S3, the Stokes parameters of the breast tumor tissue are obtained. , , and They are respectively: ； ； ； ； in, This represents the inverse Fourier transform, where j represents the imaginary number. This indicates the first elimination coupling factor. This represents the second decoupling factor. This indicates the operation of taking the real part. This indicates the operation of taking the imaginary part.

Images