Spectral transmittance correction method suitable for DMD coded aperture imaging system

Abstract

The invention provides a spectral transmittance correction method suitable for a DMD (Digital Micromirror Device) coded aperture imaging system. The method comprises the following steps of: 1, performing spectral calibration on a coded aperture spectral imaging system by adopting a plurality of groups of uniform monochromatic light sources; 2, loading two encoding templates to a DMD encoder of the system, respectively carrying out imaging acquisition, and calculating a transformation matrix from the DMD encoder to a detector and a system point spread function; and 3, acquiring spectral transmittance distribution on the detector based on the grating dispersion model and the transformation matrix, and correcting the spectral transmittance distribution from a binary matrix to a gray matrix through a point spread function for reconstructing a spectral image of a detection target. According to the invention, the point spread function of the optical system between the DMD encoder and the detector is utilized to carry out gray scale correction on the spectrum transmittance distribution on the detector modulated by the encoding template, so that the problems of low calibration data accuracy and low reconstructed spectrum image quality caused by the problems of diffraction effect, mismatching of pixel units and the like of the system are effectively solved.

Description

Technical Field

[0001] The present invention relates to a precise calibration method for a coded aperture spectral imaging system, which is aimed at a spectral coded aperture spectral imaging system based on a double grating dispersion type. According to the spectral calibration result, the coded aperture matrix from the coding template to the detector plane in each spectral band is corrected from a binary matrix to a grayscale matrix, thereby reducing the coded aperture calibration error, the spectral data reconstruction error, and improving the quality of the reconstructed spectral image. Background Art

[0002] Coded Aperture Snapshot Spectral Imaging (CASSI) usually uses binary coded compressed sampling to reconstruct the input hyperspectral image. The binary random sampling on the spectrum is achieved by coded aperture and spectroscopic elements (such as gratings and dispersion prisms), that is, the coded aperture randomly samples the scene image in the spatial domain, and the spectroscopic element offsets the coded template according to the wavelength of each spectral channel.

[0003] In a dual-dispersion coded aperture system, the target's optical signal is dispersed and converged on the DMD encoder plane to obtain a spatial and spectral aliased image staggered along the dispersion direction. After coding gating, the light is combined and converged to obtain a light intensity image on the image plane. The DMD encoder controls the transmittance distribution of different spectral channels at each point after light combination by gating or blocking different spatial positions of the aliased image, thereby realizing coding gating of the spectral channel on the image plane. The coded aperture spectral imaging system can change the loaded coding template through the DMD encoder to flexibly switch between a variety of different detection parameter combinations, and the coding template loaded on the DMD encoder plane determines the transmittance of each spectral channel at each point on the image plane. Therefore, the reconstruction of spectral data requires determining the transformation relationship between the DMD encoder plane and the detector image plane under each spectral channel. At the same time, according to the corresponding relationship, the central wavelength and spectral resolution of the reconstructed spectral image can be determined, that is, the spectral calibration and template transformation matrix calibration of the coded aperture spectral imaging system are completed.

[0004] However, for the actual CASSI system, the spectral spatial modulation formed by its binary coded aperture will be affected by the actual optical structure, resulting in a certain degree of ambiguity in the information received by the detector. At this time, the coding template used to reconstruct the data is not completely binary. Most of the existing reconstruction algorithms are based on the imaging model of binary modulation of the target information by the coding template, which can be described as a simple imaging model. This simple imaging model cannot fully describe the information modulation performed by the actual optical system, and the hyperspectral data reconstruction method is highly dependent on the imaging model to obtain prior information: the traditional regularization iterative optimization algorithm relies on the system measurement matrix determined by the coding template to reconstruct the collected multi-dimensional information, and the data training of the deep learning model will also be affected by the coding template. Therefore, the error between the simple imaging model and the actual optical modulation will reduce the quality of the reconstructed hyperspectral data.

[0005] In summary, it is necessary to effectively correct the coded transmittance distribution used for reconstruction according to the various errors of the system optical elements corresponding to the coding template, such as optical distortion, diffraction effect, aberration and mismatch between the coding elements and the detector elements, so as to improve the quality of the reconstructed spectral image. Summary of the invention

[0006] The purpose of the present invention is to provide a spectral transmittance correction method suitable for a DMD coded aperture imaging system. The method calculates the point spread function of the optical system between the DMD encoder plane and the detector plane, and calculates the grayscale coded aperture modulation actually introduced by the system in combination with the transformation relationship between the DMD encoder and the detector. The method is used to describe the optical distortion, diffraction effect, aberration and mismatch between the coding element and the detector element caused by various errors of the optical elements of the actual system, thereby reducing the coded aperture calibration error and improving the quality of the reconstructed spectral image.

[0007] The technical solution of the present invention:

[0008] Step 1: First, load the checkerboard template T used to calibrate the transformation relationship between the encoder plane and the detector plane on the DMD encoder. D1 , the calibration template features corner point position features to match the DMD encoder coordinates with the detector coordinates. Use a monochromator or laser to obtain monochromatic light with a wavelength of λ1, which is homogenized by an integrating sphere and then illuminates the DMD encoder plane, forming an image on the detector plane, and then collects the corresponding single-wavelength checkerboard calibration image f λ1 .

[0009] Calibrate the checkerboard template T D1 and the checkerboard calibration image f collected by the detector λ1 Perform sub-pixel corner detection to obtain the calibration template and the position coordinate set of the calibration image feature corner points (xD ,y D ) and (x i ,y i ). According to the calibration code checkerboard template T D1 The corner features on (x D ,y D ) and the calibration checkerboard image f λ1 The corner feature (x i ,y i ) between (x i ,y i ,1)=H1·(x D ,y D ,1), use affine transformation to solve the transformation matrix H λ1 =[a,c,t x (λ1); b, d, t y ; 0,0,1], where (a,b,c,d) determines the rotation, scaling and shearing operations, t x ,t y is the translation parameter at the current wavelength. According to the properties of grating linear dispersion (along the horizontal direction), we can know that the transformation matrix H λ1 Medium x It changes linearly with the wavelength, that is, t x (λ) = k λ λ+t0.

[0010] Change the calibration wavelength to λ2, λ3, and use the same checkerboard calibration template T D1 , collect calibration images at different wavelengths f λ2 , f λ3 . Calculate the transformation matrix H at wavelengths λ2 and λ3 respectively λ2 , H λ3 According to H λ1 , H λ2 , H λ3 The translation parameter t in the horizontal dispersion direction of the species x (λ1), t x (λ2), t x (λ3), the least squares method is used to fit k λ and t0, thereby outputting the transformation matrix H of the DMD encoder and detector plane λ =[a,c,k λ λ+t0; b, d, t y ; 0,0,1]. The parameters a, b, c, d and t y The value is the average value of each transformation matrix parameter. Theoretically, these parameters should remain unchanged as the wavelength changes, but there will be certain random errors in actual processing. Therefore, using the average value can reduce the random error of the transformation matrix.

[0011] According to the translation parameter t in the transformation matrix at different wavelengths x (λ1), t x (λ2), t x (λ3) and the spectral resolution calculation formula Δ λ =(λ1-λ2)Δ D ×cosθcosθ d / (t x (λ1)-t x (λ2))Δ i Get the highest theoretical spectral resolution of the system, where Δ D , Δ i where θ and θ are the DMD encoder symbol size and detector pixel size, respectively. d They are the relative roll angle of the DMD encoder and the detector along the optical axis and the flip angle of the DMD encoder micromirror when the system is installed. At least two groups of wavelength light sources are used in the spectrum and template calibration process. When three or more groups of light sources are used, the linear fitting wavelength λ and the transformation matrix translation parameter t x The relationship between Δ λ Generally, when the number of wavelengths is increased until the root mean square error of the fit no longer decreases, it can be considered that the highest calibration accuracy has been achieved.

[0012] At this point, the template calibration and spectrum calibration are completed, and the transformation matrix H of the encoding plane and the detector plane is obtained. λ and the system spectral resolution Δ λ .

[0013] Step 2: Select any wavelength λ1, λ2, λ3 calibrated in step 1 as the reference wavelength. For example, select λ1 as the reference wavelength and uniformly irradiate it into the imaging system. At this time, the DMD encoder loads the Gaussian binary random coding template T D2 , collect the binary random coding template image g under the current single wavelength λ1 In the effective wavelength range of the system (λ min ,λ max ), the spectral resolution obtained in step 1 is used to increase or decrease the wavelength of monochromatic light, that is, the wavelengths of different spectral bands satisfy the inequality: λ min ≤λ1-m1×Δ λ and λ1+m2×Δ λ ≤λ max ; m1, m2 are positive integers and m1+m2=n2, then collect binary random coding template image sequences G(λ1+m i ×Δ λ ), where -m1≤m i ≤m2, collect images with a total of n2 spectral bands.

[0014] Step 3: H obtained in step 1 λ , bring the wavelength values ​​of different spectrum bands in step 2 into H λ In the corresponding spectrum, the transformation matrix set H(λ1+m i ×Δ λ ). Combine it with the binary random encoding template T used in step 2 D2 The binary random transmittance distribution sequence T(λ1+m i ×Δ λ ), which is consistent with the image sequence G(λ1+m i ×Δ λ ) correspond one to one.

[0015] The binary random encoding template T in step 2 D2 When loaded onto the DMD encoder, it is completely binary, and the transformation matrix set H(λ1+m i ×Δ λ ) and then the binary random transmittance distribution sequence T(λ1+m i ×Δ λ ) is also completely binary, but in the actual system, after the optical system and space transformation between the DMD encoder and the detector, due to the transmittance error caused by optical diffraction and the mismatch between the DMD code element and the detection pixel, the actual collected binary random coding template image sequence G(λ1+m i ×Δ λ ) presents grayscale and fuzzy characteristics. To describe the influence of the optical system on the coded aperture transmittance distribution, the point spread function h(x, y, λ) can be used to characterize these errors. The binary transmittance distribution sequence T(λ1+m i ×Δ λ ) and the same randomly coded acquired image sequence G(λ1+m i ×Δ λ ), the point spread function of each spectral band can be calculated, thereby describing the influence of the system on the coded aperture transmittance distribution as accurately as possible.

[0016] The matrix least squares method is used to estimate and solve the point spread function. For example, for uniform monochromatic light λ1 in a specific spectral band, the relationship between the collected image and the binary transmittance distribution is:

[0017]

[0018] Where f(i,j,λ) is the original image of the target, Describes the dispersion effect caused by the dispersion element, and expresses the above formula in matrix form g v =TM h v , where g v 、h v are the vector forms of g(i,j,λ1) and h(i,j,λ1), T M is the convolution matrix form of T(i,j,λ1). Matrix least squares estimation solves the point spread function: h v =(T M T T M ) -1 T M T g v . Matrix least squares estimation needs to satisfy the formula T M T T M The full rank requirement is met, so the coding template used for point spread function calculation is a binary random coding template.

[0019] Due to the optical distortion and aberration in the imaging system, the point spread function under each spectral channel generally changes with the spatial position (i, j), and due to projection mapping, there is also a mismatch between the coded aperture code element size and the detector pixel. Therefore, it is assumed that the point spread function within a specific image size area is displacement invariant, and the corresponding randomly coded binary transmittance distribution and collected image under the same spectral band are processed by image block processing, and the point spread function at different spatial positions is solved to solve the spatial variation problem.

[0020] Step 4: Load the coding template used in actual detection and acquisition on the DMD encoder. Different coding templates can be used according to different reconstruction principles, such as Hadamard template or Gaussian random template. Traditional regular term optimization iterative reconstruction methods or deep learning reconstruction networks have high requirements for the orthogonality of coding templates, so Gaussian random coding is usually used. Taking random coding as an example, during actual detection, a random coding template T is loaded on the DMD encoder. D3 , the random template can be different from the template loaded when calculating the point spread function, and then the measurement target is imaged and collected on the detector to obtain the target hyperspectral compressed measurement image g(i,j). The random template used for measurement acquisition is first transformed by the transformation matrix H obtained in step 1 λ Get the binary transmittance distribution T(λ) of the random template, and then convolve it with the point spread function model h(x, y, λ) obtained in step 2 to get the grayscale transmittance distribution T gray (λ).

[0021] According to the principle of dual-dispersion coded aperture compression imaging, we can get: It can be written in matrix vector form g = Φf + η, where η is the system detection noise, Φ is the measurement matrix, which is uniquely determined by the binary transmittance distribution T(λ) and the point spread function h(x, y, λ). According to the principle of compressed sensing, the measurement matrix Φ is composed of the grayscale transmittance distribution T in each spectral band. gray (λ) is vectorized and diagonalized, that is, Φ = [diag(vec(T gray (λ1-m1×Δ λ )))|……|diag(vec(T gray (λ1+m2×Δ λ )))], vec() means vectorizing the matrix, and diag() means arranging the vectors diagonally into a matrix.

[0022] The target hyperspectral data cube can be reconstructed using the traditional iterative optimization model: f = argmin(1 / 2)||g-Φf||2+kP(f), where f is the reconstructed data cube, k is the prior model coefficient, Φ is the measurement matrix, which is uniquely determined by the binary transmittance distribution T(λ) and the point spread function h(x, y, λ), and P(f) is a manually designed prior representation model to narrow the solution space and solve the problem of underdetermination. Different prior models can be selected according to different image characteristics, such as sparsity, low rank, etc. The deep learning reconstruction network also requires the measurement matrix Φ to construct training data. Whether it is a traditional iterative algorithm based on an optimization model or a deep learning network, the measurement matrix Φ of the reconstruction problem has high-precision description requirements. The point spread function h(x, y, λ) calculated during the calibration process can more accurately describe the measurement matrix affected by the optical system.

[0023] Compared with the existing reconstruction and correction methods, the present invention has the following advantages:

[0024] In the process of spectral reconstruction, the transmittance of each spectral band on the detector plane and the point spread function of the optical system between the DMD encoder and the detector are obtained according to the dispersion projection transformation of the coding template. The value of the coded aperture modulation template is corrected from the binary value of 0 / 1 to the grayscale value, which improves the calibration accuracy of the measurement matrix used for data reconstruction and effectively solves the problems of distortion of reconstructed spectral data and low quality of spectral images caused by transmittance differences due to optical diffraction and mismatch between DMD code elements and detection pixels in actual imaging. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 A comparison diagram between the simple spectrum compression imaging model of the DMD encoder plane assumed by the traditional optical imaging principle and the actual point spread function spectrum compression imaging model;

[0026] Among them: 1. Simple spectral compression imaging model; 2. Point spread function spectral compression imaging model.

[0027] Figure 2 The system calibration and grayscale correction flow chart of the present invention;

[0028] Figure 3 The checkerboard coding template loaded at a wavelength of 515nm and the checkerboard calibration image projected onto the detector plane;

[0029] Among them: 1. chessboard coding template; 2. chessboard coding template system imaging diagram;

[0030] Figure 4 Random coding template loaded for calculating point spread function;

[0031] Figure 5 It is the binary transmittance distribution diagram of the random coding template loaded at a wavelength of 515nm, the random coding calibration image projected onto the detector plane, the point spread function distribution diagram at different positions and the grayscale random coding transmittance distribution diagram after grayscale correction;

[0032] Among them: 1. Binary random coding transmittance distribution map; 2. Random coding calibration image on the detector plane; 3. Point spread function distribution map at different positions; 4. Grayscale random coding transmittance distribution map.

[0033] Figure 6 Comparison diagram of the reference image, compressed coded image, image reconstructed based on the calibration data of the traditional simple model, and image reconstructed based on the calibration data of the precise calibration model in the 515nm spectrum band;

[0034] Among them: 1. Reference image; 2. Compressed coded image; 3. Reconstructed image based on traditional simple model; 4. Reconstructed image based on accurate calibration model. DETAILED DESCRIPTION

[0035] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0036] The flowchart of system spectrum calibration, template calibration, optical system point spread function calculation and correction and spectral image reconstruction is as follows: Figure 2 The spectrum and template calibration uses a monochromator connected to an integrating sphere to obtain uniform monochromatic light after homogenization. Laser incident on the integrating sphere can also be used as a uniform monochromatic light source.

[0037] In one implementation case, first, within the spectral response range of the coded aperture spectral imaging system to be calibrated, the wavelength λ1=515nm and the bandwidth 2nm of the calibration light source are selected, and the coded aperture spectral imaging system to be calibrated is illuminated with the full field of view after being homogenized by an integrating sphere, and the DMD encoder is controlled to load the checkerboard coding template used for calibration. Figure 3 1 in the calibration template. D,y D ) as the detection feature, and collect the corresponding imaging system image (x i ,y i )like Figure 3 2. Sub-pixel detection corner point detection obtains multiple sets of corresponding corner point coordinate sets C1(x D1 ,y D1 ;x i1 ,y i1 ), use affine transformation to obtain the transformation matrix H λ1 =[2.6154,2.6089,-886.7826;2.8866,-2.7792,366.0804;0,0,1].

[0038] The calibration wavelength λ2 is changed to 532nm, the bandwidth is 2nm, the checkerboard coding template loaded by the DMD encoder remains unchanged, and the image acquisition and sub-pixel corner point detection of the imaging system are performed in the same way to obtain the corresponding corner point coordinate set C2(x D2 ,y D3 ;x i2 ,y i2 ), and use affine transformation to get the transformation matrix H at the current wavelength λ2 =[2.6129,2.6185,-868.6425;2.8830,-2.7806,367.6976;0,0,1]. For the theoretical invariants a,b,c,d,t y To find the arithmetic mean, use t x (λ1) and t x (λ2) fits the linear relationship of grating dispersion, and finally obtains the transformation matrix H related to wavelength λ =[2.6142,2.6137,1.067λ-1436.320;2.8848,-2.7799,366.889;0,0,1].

[0039] Using the spectral resolution calculation formula Δ λ =(λ1-λ2)Δ D ×cosθcosθ d / (t x (λ1)-t x (λ2))Δ i Get the highest theoretical spectral resolution of the system, where Δ D =13.68μm, Δ i =3.45μm,θ=45°,θ d =24° are the relative rolling angle of the DMD encoder and the detector along the optical axis and the flip angle of the DMD encoder micromirror (-12° to +12°) when the system is installed. λ =2.40nm.

[0040] According to the calculated spectral resolution, λ1=515nm is used as the reference wavelength to illuminate the coded aperture spectral imaging system. At this time, the DMD encoder loads a binary random coding template as follows: Figure 4 . Collect the corresponding system imaging diagram under the current spectrum band, the target field of view is as follows Figure 5 As shown in 2 in the figure, it can be observed that the edge blur of the binary code in the system imaging image is observed. λ1 The random coding template of the DMD encoder plane is transformed to the detector plane to obtain the randomly coded binary transmittance distribution, where the target field of view is as follows: Figure 5 As shown in 1, it can be seen that compared with the system imaging diagram, the transmittance distribution calculated by transformation is binary. For the 600×600 pixel area of ​​the target field of view in the system imaging diagram under the current band, each 120×120 pixels is regarded as an area where the point spread function displacement remains unchanged, and the point spread function in this area is estimated by least squares, and finally the spatially varying point spread function distribution under the wavelength of 515nm is obtained, as shown in Figure 5 3 in . It is convolved with the randomly encoded binary transmittance distribution to obtain the randomly encoded grayscale transmittance distribution, as Figure 5 4. The grayscale transmittance distribution can describe the actual spectral spatial modulation introduced by the system with high calibration accuracy. After completing the above steps, the calculation of the point spread function at 515nm is completed.

[0041] In the effective wavelength range of the system (510nm, 590nm), the spectral resolution Δ λ =2.40nm increases or decreases the wavelength of monochromatic light, that is, the wavelengths of different spectral bands satisfy the inequality: 510nm≤515-m1×2.40 and 515+m2×2.40≤590nm; it can be calculated that m1=2, m2=31 are positive integers and m1+m2=n2=33 spectral bands, and then the binary random coding template image sequence G(515+m i ×2.40), where -2≤m i ≤31, and collect images of a total of n2=33 spectral bands. i ×2.40, -2≤m i ≤31 into H λ In the corresponding spectrum, the transformation matrix set H(515+m i ×2.40). It is calculated with the binary random coding template used to obtain the binary random transmittance distribution sequence T(515+m i ×2.40), which is consistent with the image sequence G(515+m i×2.40) one by one. The method for calculating the point spread function is the same as that for the 515nm spectrum. So far, the point spread function h(i,j,515+m i ×2.40).

[0042] The coding template used in actual detection and acquisition is loaded on the DMD encoder, such as random coding, and then the measurement target is imaged and acquired on the detector to obtain the target hyperspectral compressed measurement image g(i,j). The random template used for measurement acquisition is first transformed through the transformation matrix H(515+m i ×2.40) to obtain the binary transmittance distribution T(515+m i ×2.40), and then compare it with the point spread function model h(i,j,515+m i ×2.40) phase convolution to obtain the grayscale transmittance distribution T gray (λ).

[0043] According to the principle of dual-dispersion coded aperture compression imaging, we can get: It can be written in matrix vector form g = Φf + η, where η is the system detection noise, Φ is the measurement matrix, which is composed of the binary transmittance distribution T(515+m i ×2.40) and the point spread function h(i,j,515+m i ×2.40) is uniquely determined. According to the principle of compressed sensing, the measurement matrix Φ is composed of the grayscale transmittance distribution T in each spectral segment. gray (λ) is vectorized and diagonalized, that is, Φ = [diag(vec(T gray (515-m1×2.40)))|……|diag(vec(T gray (515+m2×2.40)))], m1=2, m2=31, vec() means vectorizing the matrix, and diag() means arranging the vectors diagonally into a matrix.

[0044] The target hyperspectral data cube is reconstructed using the traditional iterative optimization model: f = argmin(1 / 2)||g-Φf||2+kP(f), where f is the reconstructed data cube, k is the prior model coefficient, and P(f) is a manually designed prior representation model to reduce the solution space and solve the problem of underdetermination. Common models include sparsity and low rank. After selecting the prior model, iterative solution can be performed to reconstruct the original spectral image data cube. Figure 6The results of the 515nm spectrum reconstructed based on the low-rank prior model are shown in Figure 1. 1 is the reference image, 2 is the compressed measurement image, 3 is the reconstruction result of the simple imaging model, and 4 is the reconstruction result of the point spread function imaging model. It can be clearly seen from the reconstruction results that, compared with the simple imaging model, the imaging model that takes into account the point spread function of the system accurately describes the actual spectral spatial modulation of the system and significantly improves the quality of the reconstructed spectral image.

[0045] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention. The contents not described in detail in the specification of the present invention belong to the prior art known to professional and technical personnel in the field.

Claims

1. A spectral transmittance correction method suitable for a DMD coded aperture imaging system, characterized in that: The following steps are involved: Step 1: Use n1 (n1 ≥ 2) groups of uniform monochromatic light to be incident on the imaging system respectively. The system's DMD (Digital Mirror Device, DMD) encoder first loads the checkerboard coding template required for calibrating the projection transformation matrix, and uses the corner point feature matching method to calibrate the transformation relationship between the DMD encoder plane and the detector plane, as well as the linear transformation relationship between different wavelengths, to complete the coding template calibration and spectrum calibration, and obtain the transformation matrix and spectral resolution; Step 2, using n2 (determined according to the spectral resolution) groups of uniform monochromatic light to be incident on the imaging system respectively, the DMD encoder loads the binary random coding template required for calculating the point spread function, and performs imaging detection on the uniform light field of view, and collects the calibration image of the binary random coding template; Step 3, obtaining the binary transmittance distribution of the binary random coding template loaded in step 2 according to the transformation matrix calculated in step 1, and calculating the point spread function model between the detector plane and the DMD encoder plane under n2 wavelengths in blocks; Step 4, load the binary random template used for actual detection on the DMD encoder. The binary random template at this time may be different from the template loaded when calculating the point spread function. Perform imaging acquisition on the detector to obtain a measurement image. The binary random template used for measurement acquisition is combined with the transformation matrix obtained in step 1 and the point spread function model obtained in step 2 to calculate the grayscale coded aperture modulation introduced by the actual system, which is used to reconstruct the spectral data of the measurement image.

2. The spectral transmittance correction method for a DMD coded aperture imaging system according to claim 1, characterized in that: In step 4, the grayscale coded aperture transmittance distribution for spectral reconstruction is calculated based on the system point spread function and the DMD encoder and detector plane transformation matrix: Among them, g(i,j) is the grayscale coded measurement image of the detector, A is the image input to the DMD encoder plane under a single wavelength λ0, is the binary spectral transmittance distribution at this wavelength, h(i,j,λ0) is the point spread function from the DMD encoder to the detector focal plane at this wavelength, is the convolution operation.

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