A method for modeling three-dimensional blurring characteristics of a conventional optical microscope

Abstract

The application provides a modeling method for three-dimensional blur characteristics of a traditional optical microscope, and belongs to the technical field of optical microscopic observation. The method comprises the following steps: establishing an optical amplitude expression of an arbitrary point on an imaging plane in an optical system based on Fresnel diffraction theory; constructing a relationship between object distance variation and image distance variation without ideal object distance by using a geometric relationship between a diffraction spot diameter and an observed object; deducing an optical amplitude expression of the arbitrary point on the imaging plane containing object distance variation from the optical amplitude expression of the arbitrary point on the imaging plane in the optical system and the relationship between the object distance variation and the image distance variation without the ideal object distance; deducing an optical intensity distribution expression of the arbitrary point on the imaging plane from the optical amplitude expression of the arbitrary point on the imaging plane containing the object distance variation; and establishing a relationship between blur degree and defocus depth variation according to the optical intensity distribution expression of the arbitrary point on the imaging plane, so as to obtain a variation law of the blur degree of the traditional optical microscope system.

Description

Technical Field

[0001] This invention belongs to the field of optical microscopy observation technology, specifically relating to a modeling method for the three-dimensional blurring characteristics of a traditional optical microscope. Background Technology

[0002] Traditional optical microscopes offer advantages such as low cost, no need for special observation environments, non-destructive treatment of samples, and fast imaging speed. Therefore, optical microscopy techniques based on traditional optical microscopes have irreplaceable advantages in various fields.

[0003] While microscopic observation methods based on traditional optical microscopes have unique advantages, they face the problem of image blurring in practical applications. There are two main reasons affecting image clarity: First, due to the small depth of field of traditional optical microscopes, accurate focusing of the traditional optical system is difficult, resulting in defocusing and significantly impacting optical image quality; second, diffraction, an unavoidable phenomenon in optical systems, also causes image blurring.

[0004] To address the blurring problem in conventional optical microscopes, the blurring characteristics of the microscopic optical system are typically analyzed to determine the variation law of its blurring, thereby further resolving the blurring issue. Currently, the point spread function (PSF) is generally used for quantitative analysis of the blurring characteristics of conventional optical microscopes. Obtaining the PSF in an optical system usually employs two methods: modeling and experimentation. For modeling, a mathematical model of the optical diffraction mechanism in free space is typically performed to determine the PSF, which is then used to further analyze the blurring characteristics of the optical system. For experimental methods, this is generally applied to fluorescence imaging systems, where small particles are fixed in an optical gel and their morphological images are measured to obtain the PSF, thus revealing the blurring law of the imaging system. However, due to the complex structure of the microscopic system itself, accurately modeling the blurring characteristics of a specific microscopic system is very difficult, and it is challenging to simultaneously quantitatively analyze both optical diffraction and defocus, the two main causes affecting image sharpness. However, for obtaining the point spread function of an optical system through experimental methods, thereby indirectly obtaining the system's fuzziness characteristics, the inherent defects of the material itself, such as the change in refractive index and the inability to measure continuously, make it impossible to obtain an accurate mathematical model of the fuzziness variation law of the optical system through experimental methods. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides a modeling method for the three-dimensional blur characteristics of a traditional optical microscope. Based on the optical parameters of the traditional optical microscope, and taking into account the influence of optical diffraction and defocus on image sharpness, a mathematical model of the optical imaging blur variation law is established. Furthermore, by changing the defocus depth of the system, the imaging blur degree of the optical system can be quantitatively analyzed.

[0006] The technical solution of this invention is as follows:

[0007] A method for modeling the three-dimensional blur characteristics of a traditional optical microscope, comprising the following steps:

[0008] Step 1: Establish the expression for the light amplitude at any point on the imaging plane in the optical system based on Fresnel diffraction theory;

[0009] Step 2: Construct the relationship between object distance variation and image distance variation without ideal object distance using the geometric relationship between the diameter of the blur spot and the observed object;

[0010] Step 3: Based on the expression for the light amplitude at any point on the imaging plane in the optical system established in Step 1 and the relationship between the change in object distance and the change in image distance without the ideal object distance constructed in Step 2, derive the expression for the light amplitude at any point on the imaging plane that includes the change in object distance.

[0011] Step 4: Based on the expression for the light amplitude at any point on the imaging surface, which includes the change in object distance, derive the expression for the light intensity distribution at any point on the imaging surface.

[0012] Step 5: Establish the relationship between blur and defocus depth variation based on the light intensity distribution expression at any point on the imaging plane, thereby obtaining the blur variation law of the traditional optical microscopy system.

[0013] Furthermore, according to the modeling method for the three-dimensional blur characteristics of a traditional optical microscope, step 2 includes the following steps:

[0014] Step 2.1: Establish the proportional relationship between the change in image distance and the ideal image distance using the ratio of the diameter of the diffuse spot to the object height when the observed object moves:

[0015]

[0016] In the above formula, p1 is the diameter of the diffuse spot when the observed target moves; D is the object height; Δb is the change in image distance before and after the object moves; b0 is the ideal image distance of the microscope.

[0017] Step 2.2: Establish the proportional relationship between the ideal object distance and the ideal image distance using the ratio of the diameter of the blur spot after the imaging surface moves to the change in distance along the light path:

[0018]

[0019] In the above formula, p2 is the diameter of the blur spot when the image distance moves; h is the change in radial distance of the optical path after the object distance changes; d0 is the ideal object distance of the microscope.

[0020] Step 2.3: Using geometric relationships, establish an expression for the radial change in the light path before and after the object moves:

[0021]

[0022] In the above formula, Δd represents the change in distance between the object and the object before and after the object moves.

[0023] Step 2.4: Keeping the diameter of the blur spot after the observed object moves and the diameter of the blur spot after the imaging plane moves constant, establish the relationship between the change in object distance and the change in image distance, including the ideal object distance:

[0024]

[0025] In the above formula, Δb is the change in image distance;

[0026] Step 2.5: Based on the imaging formula of geometric optics, obtain the relationship between the change in object distance and the change in image distance without an ideal object distance:

[0027]

[0028] ξ=b0-f (11)

[0029] In the above formula, f is the focal length of the microscope; ξ is the difference between the ideal image distance and the focal length.

[0030] Furthermore, based on the modeling method for the three-dimensional blur characteristics of a traditional optical microscope, the expression for the light amplitude E at any point on the imaging plane, which includes the change in object distance, is... P as follows:

[0031]

[0032]

[0033]

[0034]

[0035]

[0036] In the above formula, a is the effective radius of the microscope lens.

[0037] Furthermore, based on the aforementioned modeling method for the three-dimensional blurring characteristics of traditional optical microscopes, the expression for the light intensity distribution at any point on the imaging surface is as follows:

[0038]

[0039] Furthermore, according to the modeling method for the three-dimensional blur characteristics of a traditional optical microscope, step 5 includes the following steps:

[0040] Step 5.1: Based on the properties of orthogonal polynomials, change the light intensity distribution expression at any point on the imaging surface to obtain the light intensity distribution expression based on orthogonal polynomials;

[0041] Step 5.2: Add parameter K to the light intensity distribution expression based on the orthogonal polynomial to construct a more accurate light intensity distribution expression; the parameter is a set of constants between 1 and 7;

[0042] Step 5.3: Based on the more accurate expression for light intensity distribution, perform Gaussian fitting on the Bessel function and the exponential function to obtain the simplified expression for light intensity distribution;

[0043] Step 5.4: Based on the simplified light intensity distribution expression, obtain the relationship expression between blur and defocus depth change.

[0044] Furthermore, based on the aforementioned modeling method for the three-dimensional blurring characteristics of traditional optical microscopes, the expression for the light intensity distribution based on orthogonal polynomials is:

[0045]

[0046]

[0047] In the above formula, e is the natural constant; J0 is the zeroth-order Bessel function; and n is the order of the Bessel function.

[0048] Furthermore, based on the aforementioned modeling method for the three-dimensional blurring characteristics of traditional optical microscopes, the more accurate expression for the light intensity distribution is:

[0049]

[0050]

[0051] Furthermore, based on the aforementioned modeling method for the three-dimensional blurring characteristics of traditional optical microscopes, the simplified expression for the light intensity distribution is:

[0052]

[0053] Furthermore, based on the aforementioned modeling method for the three-dimensional blur characteristics of traditional optical microscopes, the relationship between the blur degree and the change in defocus depth is expressed as follows:

[0054]

[0055] In the formula, the parameter σ represents the blurriness of the microscope.

[0056] Compared with the prior art, the technical solution proposed in this invention has the following beneficial effects:

[0057] Based on Fresnel diffraction theory, this invention provides a modeling method for traditional optical microscopes that simultaneously considers the impact of optical diffraction and defocus on image sharpness. It also incorporates actual optical system parameters, cleverly simplifying various complex point spread function models based on different microscopic systems into a universal light intensity distribution model. By fitting some complex functions in the expression, the complex light intensity expression is simplified, further yielding the expression for the blurriness of the optical system. This invention can be directly designed based on actual optical systems. For microscopic systems with different parameters, it directly obtains the variation law of the blurriness of the microscopic system without relying on complex calculations, and the construction process is simple. Attached Figure Description

[0058] To more clearly illustrate the specific methods in the embodiments of the present invention, the relevant drawings involved in the embodiments will be briefly described below. The drawings below are only preferred embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative changes.

[0059] Figure 1 This is a flowchart illustrating the modeling method for the three-dimensional blur characteristics of a conventional optical microscope in this embodiment.

[0060] Figure 2 This is a schematic diagram of the spherical wave propagation model.

[0061] Figure 3 This is a schematic diagram of the spherical wave propagation model in the optical system of this embodiment;

[0062] Figure 4 This is a diagram showing the relationship between the change in object distance and the change in image distance in this embodiment;

[0063] Figure 5 This is a distribution diagram of parameter K in this embodiment;

[0064] Figure 6 This is a diagram showing the results of the microscope's blurring experiment in this embodiment. Detailed Implementation

[0065] To facilitate understanding of this application, a more complete description will be provided below with reference to the accompanying drawings. Preferred embodiments of this application are shown in the drawings. However, this application can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a more thorough and complete understanding of the disclosure of this application.

[0066] The key technical point of this invention is:

[0067] 1. Based on Fresnel diffraction theory, an optical propagation model under the dual influence of optical diffraction and defocus was established by introducing parameters of an actual optical system;

[0068] 2. By introducing a model with a constant spot diameter, the relationship between changes in object distance and image distance is introduced, thereby establishing a model relating light intensity distribution to defocus depth.

[0069] 3. Based on the light intensity distribution model, by simplifying higher-order Bessel functions and other complex functions, the relationship between the blurriness of the microscopic system and the change in defocus depth was further obtained.

[0070] Figure 1 This is a flowchart illustrating the modeling method for the three-dimensional blurring characteristics of a conventional optical microscope according to this embodiment. The modeling method for the three-dimensional blurring characteristics of a conventional optical microscope includes the following steps:

[0071] Step 1: Establish the expression for the light amplitude at any point on the imaging plane in the optical system based on Fresnel diffraction theory;

[0072] Step 1.1: Based on Figure 2 The spherical wave propagation model shown yields the amplitude E at any point on the imaging surface. p The expression is:

[0073]

[0074]

[0075]

[0076] In the above formula, i is the imaginary unit; E0 is the optical amplitude at the intersection of the imaging plane and the optical axis; λ is the wavelength of the light source; d is the object distance; b is the image distance; k is the wave number; ρ is the distance from a point on the imaging plane to the optical axis; q is the distance from a point on the spherical wave to the optical axis; g is the distance from a point on the spherical wave to the line parallel to the optical axis from a point on the imaging plane; Φ is the wavefront of the spherical wave; dΦ is the surface element of the spherical wave; α is the angle formed by the line connecting a point on the wavefront to the optical axis and the perpendicular line to the optical axis; and r is the distance from a point on the wavefront to a point on the imaging plane.

[0077] exist Figure 2In the diagram, P0 is the source of the spherical wave; Φ is the wavefront of the spherical wave; Q0 is the center of the spherical wave; R is the radius of the spherical wave; Q is any point on the spherical wave; θ is the angle between R and r; and P is any point on the propagation path of the spherical wave.

[0078] Step 1.2: Based on the system parameters of the microscope, and on... Figure 3 The propagation model of spherical waves in the optical system shown is used to obtain the amplitude E at a point on the imaging plane of the optical system shown in equation (4). p The system parameters of the microscope include the wavelength λ of the light source, the wave number k, the ideal image distance b0 of the system, and the focal length f of the system.

[0079]

[0080]

[0081] In the above formula, n is the order of the Bessel function; J n is the nth-order Bessel function; a is the effective radius of the objective lens.

[0082] Step 2: Construct the relationship between object distance variation and image distance variation without ideal object distance using the geometric relationship between the diameter of the blur spot and the observed object;

[0083] Step 2.1: Establish the proportional relationship between the change in image distance and the ideal image distance using the ratio of the diameter of the diffuse spot to the height of the object as it moves.

[0084]

[0085] In the above formula, p1 is the diameter of the diffuse spot when the observed target moves; D is the object height; Δb is the change in image distance before and after the object moves; and b0 is the ideal image distance of the microscope.

[0086] Step 2.2: Establish the proportional relationship between the ideal object distance and the ideal image distance using the ratio of the diameter of the blur spot after the imaging surface moves to the change in the distance along the light path, as follows:

[0087]

[0088] In the above formula, p2 is the diameter of the blur spot when the image distance moves; h is the change in radial distance of the optical path after the object distance changes; and d0 is the ideal object distance of the microscope.

[0089] Step 2.3: Using geometric relationships, establish an expression for the radial change in the light path before and after the object moves. The expression is as follows:

[0090]

[0091] In the above formula, Δd represents the change in distance between the object and the object before and after the object moves.

[0092] Step 2.4: After the observed object and the imaging plane are moved twice, the diameter of the blur spot remains unchanged, such as... Figure 4 As shown, the relationship between the change in object distance and the change in image distance, including the ideal object distance, is established, and its expression is:

[0093]

[0094] In the above formula, Δb is the change in image distance.

[0095] Step 2.5: Based on the imaging formula of geometric optics, the relationship between the change in object distance and the change in image distance without an ideal object distance is obtained, and its expression is:

[0096]

[0097] ξ=b0-f (11)

[0098] In the above formula, f is the focal length of the microscope; ξ is the difference between the ideal image distance and the focal length.

[0099] Step 3: Based on the optical amplitude expression for any point on the imaging plane established in Step 1 and the relationship between the change in object distance and the change in image distance without the ideal object distance constructed in Step 2, obtain the optical amplitude expression E for any point on the imaging plane that includes the change in object distance. P :

[0100]

[0101]

[0102]

[0103]

[0104]

[0105] In the above formula, a is the effective radius of the microscope lens.

[0106] Step 4: Based on the expression for the light amplitude at any point on the imaging plane, which includes the change in object distance, the expression for the light intensity distribution at any point on the imaging plane is obtained as follows:

[0107]

[0108] Step 5: Establish the relationship between blur and defocus depth change based on the light intensity distribution expression at any point on the imaging plane;

[0109] Step 5.1: Based on the properties of orthogonal polynomials, the expression for the light intensity distribution at any point on the imaging plane is transformed to obtain the light intensity distribution expression based on orthogonal polynomials:

[0110]

[0111]

[0112] In the formula, e is the natural constant, J0 is the zeroth-order Bessel function, and n is the order of the Bessel function.

[0113] Step 5.2: Based on the light intensity distribution expression using orthogonal polynomials, add parameter K to construct a more accurate light intensity distribution expression as shown in equation (20). The distribution of parameter K is described in [reference needed]. Figure 5 The parameter K is a set of constants between 1 and 7.

[0114]

[0115]

[0116] Step 5.3: Based on the more accurate expression for light intensity distribution, Gaussian fitting is performed on the Bessel function and the exponential function to obtain the simplified expression for light intensity distribution:

[0117]

[0118]

[0119]

[0120] In the formula, parameters A1 and A2 are the fitted amplitudes, and ε1 and ε2 are the fitted standard deviations.

[0121] Step 5.4: Based on the simplified light intensity distribution expression, i.e., Equation 24, the relationship between blur and defocus depth change is obtained as follows:

[0122]

[0123] In the formula, parameter σ represents the blurriness of the microscope, and the experimental results are as follows: Figure 6 As shown. By Figure 6 It can be seen that the imaging blur σ of an optical microscope is related to the change in object distance Δd, and σ follows a symmetrical curve distribution with respect to Δd.

[0124] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; therefore, these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the claims of the present invention.

Claims

1. A method of modeling the three-dimensional blurring characteristics of a conventional light microscope, characterized in that, The method comprises the following steps: Step 1: establishing an optical amplitude expression of any point on an imaging plane in an optical system based on Fresnel diffraction theory; Step 2: constructing a relationship between object distance change and image distance change without ideal object distance based on a geometric relationship between a diffraction spot diameter and an observed object; Step 3: deducing an optical amplitude expression of any point on the imaging plane containing object distance change based on the optical amplitude expression of any point on the imaging plane in the optical system established in step 1 and the relationship between object distance change and image distance change without ideal object distance constructed in step 2; Step 4: deducing an optical intensity distribution expression of any point on the imaging plane based on the optical amplitude expression of any point on the imaging plane containing object distance change; Step 5: establishing a relationship between defocus depth change and blur degree based on the optical intensity distribution expression of any point on the imaging plane, so as to obtain a variation law of the blur degree of a traditional optical microscopic system.

2. The method of modeling the three-dimensional blurring characteristics of a conventional light optical microscope according to claim 1, wherein, The step 2 comprises the following steps: Step 2.1: establishing a proportional relationship between image distance change and ideal image distance by using a ratio of a diffraction spot diameter when the observed object moves to an object height; （6） In the above formula, is the diameter of the diffraction spot when the observed object is moving; is the object height; is the change in image distance before and after the object moves; is the ideal image distance of the microscope; Step 2.2: establishing a proportional relationship between ideal object distance and ideal image distance by using a ratio of a diffraction spot diameter after the imaging plane moves to a distance change of an optical path; （7） In the above formula, is the diameter of the diffraction spot when the image distance is moved; is the change in the radial distance of the optical path after the object distance is changed; is the ideal object distance of the microscope; Step 2.3: establishing an expression of radial distance change of the optical path before and after the object moves by using geometric relationship; （8） In the above formula, is the change in object distance before and after the object moves; Step 2.4: establishing a relationship between object distance change and image distance change containing ideal object distance by keeping the diffraction spot diameter after the observed object moves and the diffraction spot diameter after the imaging plane moves unchanged; In the above formula, Δb is an image distance change; （9） Step 2.5: obtaining a relationship between object distance change and image distance change without ideal object distance based on an imaging formula of geometric optics; The optical intensity distribution expression of any point on the imaging plane is as follows: （10） （11） In the above formula, f is the focal length of the microscope; ξ is the difference between the ideal image distance and the focal length.

3. The method of modeling the three-dimensional blurring characteristics of a conventional light optical microscope according to claim 2, wherein, An optical amplitude expression of a change amount of the inclusion distance at any point on the imaging surface As follows: （12） （13） （14） （15） （16） In the above formula, i is the imaginary unit; E0 is the light amplitude at the intersection of the imaging plane and the optical axis; is the distance from a certain point on the imaging plane to the optical axis; n is the order of the Bessel function; is the n-order Bessel function; is the wave number; is the wavelength of the light source; is the effective radius of the microscope lens.

4. The method of modeling the three-dimensional blurring characteristics of a conventional light microscope according to claim 3, wherein, The step 5 comprises the following steps: （17）。 5. The method of modeling the three-dimensional blurring characteristics of a conventional light microscope according to claim 4, wherein, Step 5.1: changing the optical intensity distribution expression of any point on the imaging plane based on the property of orthogonal polynomials to obtain an optical intensity distribution expression based on orthogonal polynomials; Step 5.2: adding a parameter K to the optical intensity distribution expression based on orthogonal polynomials to construct a more accurate optical intensity distribution expression; the parameter is a group of constants between 1 and 7; Step 5.3: performing Gaussian fitting on Bessel functions and exponential functions based on the more accurate optical intensity distribution expression to obtain a simplified optical intensity distribution expression; Step 5.4: obtaining a relationship expression between defocus depth change and blur degree based on the simplified optical intensity distribution expression; The optical intensity distribution expression based on orthogonal polynomials is as follows: In the above formula, e is a natural constant; J0 is a zero-order Bessel function; n is an order of the Bessel function; （18） （19） The more accurate optical intensity distribution expression is as follows: The simplified optical intensity distribution expression is as follows: （20） （21） The relationship expression between defocus depth change and blur degree is as follows: （24）。 6. The method of modeling the three-dimensional blurring characteristics of a conventional light microscope according to claim 5, characterized in that In the formula, the parameter σ is the blur degree of the microscope. （25） ​

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