METHOD FOR ESTIMATE THE GEOMETRY OF A REFLECTIVE SURFACE OF AN OBJECT

Description

[0001] The present invention relates to a method for estimating the geometry of a reflective surface of an object, in particular that of a mirror. BACKGROUND OF THE INVENTION

[0002] The quality control of a reflective surface on an object can be carried out by a deflectometric measurement, the principle of which is illustrated in the figure 1 .

[0003] Phase-shift deflectometry is a technique for estimating the geometry of an object's surface from the deformation of a target image reflected by the object's surface.

[0004] For this purpose, a screen displays a target consisting of a network of horizontal fringes with a sinusoidal profile, and a camera is arranged to capture an image of the fringe network reflected by the surface of the object.

[0005] The screen acts as a graduated ruler, allowing the ordinate of the screen's light source points to be determined. This ordinate is proportional to the phase of the sinusoid that constitutes the fringe profile. The phase shift of a regular fringe step allows the phase of the corresponding light source point, and therefore its vertical position, to be evaluated for each pixel of the camera.

[0006] Similarly, displaying a test pattern consisting of a network of vertical fringes allows us to know the abscissa of the light source points on the screen.

[0007] Knowing the relative positions of the screen, the object, and the camera, it is then possible to model the path of the light rays emitted by the screen and thus have access, at any point on the object's surface, to the direction of the normal to the surface. This allows us to measure the slope field of the object's surface, which, once integrated, allows us to reconstruct the geometry of said object surface. Recall that the slope field comprises a gradient component and a rotational component, and can be modeled as follows: P → = ∇ → ϕ + rot → A → Or ϕ And A are respectively the gradient component and the rotational component of the slope field P.

[0008] There are two integration methods: the so-called "modal" method, which consists of estimating the surface geometry as the sum of predefined polynomials such as Legendre or Zernike polynomials; and the so-called "zonal" method, which consists of estimating the surface geometry for each pixel of the camera. Modal and zonal methods are described in the following documents: XIAO YONG-LIANG ET AL: "Optical fringe-reflection deflectometry with sparse representation", OPTICS AND LASERS IN ENGINEERING, vol. 104, 1 mai 2018 (2018-05-01), pages 62-70 LEI HUANG ET AL: "Review of phase measuring deflectometry",OPTICS AND LASERS IN ENGINEERING, vol. 107, 1 août 2018 (2018-08-01), pages 247-257 POUYA FARD ALI ET AL: "Characterization of the interpolation bias in the analysis of deflectometry measurement data",PROCEEDINGS OF SPIE; vol. 10749, 18 août 2018 (2018-08-18), pages 107490G-1 à 6 HUANG LEI ET AL: "Three-dimensional shape measurement with modal phase measuring deflectometry",PROCEEDINGS OF SPIE; vol. 10449, 13 juin 2017 (2017-06-13), pages 1044909-1 à 10 JONQUIÈRE HUGO ET AL: "Study of linear phase shift algorithms and application to deflectometry",OPTICS AND LASERS IN ENGINEERING, ELSEVIER, AMSTERDAM, NL, vol. 143, 10 avril 2021 (2021-04-10), pages 1-11.

[0009] The modal integration method amounts to projecting the measured slope field onto a polynomial basis and can therefore prove to be inaccurate for the reconstruction of defects at high spatial frequencies, in other words, deformations varying over a short distance on the order of a millimeter.

[0010] On the contrary, a zonal integration method based for example on a Southwell or Fried type data model, which in its simplest version consists of solving the inverse problem by the least squares method, is preferable for the reconstruction of defects at high spatial frequencies.

[0011] However, such an integration method does not take into account the rotational components of the measured slope field, which can lead to a significant reconstruction error of high-frequency defects, especially when the rotational components of the measured slope field dominate the gradient components of said field.

[0012] Moreover, in the context of a deflectometry carried out with an uncalibrated device, errors in alignment of the device are the cause of a significant rotational component of the measured slope field, which can be limiting for the geometric reconstruction of an optical surface, particularly one from Freeform technology, or one that is highly aspheric. SUBJECT OF THE INVENTION

[0013] The invention therefore aims to improve the estimation of the geometry of a reflective surface of an object, by limiting the influence of the aforementioned disadvantages. SUMMARY OF THE INVENTION

[0014] To this end, the invention proposes a method for estimating the geometry of a reflective surface. The method comprises the steps of measuring a slope field of the surface using a deflectometric device connected to a data processing computer, then integrating the slope field by modeling the presence of a rotational component in the field, and simultaneously searching for a gradient component of said field and rotational components of alignment errors of the deflectometric device. This constitutes a self-calibration of the alignment errors.

[0015] According to a particular embodiment, the integration step is carried out using the following equation (E): ϕ ^ = arg min D → ϕ ∓ P a → − P → 2 + C P a → Or : φ̂ is the estimated geometry of the reflective surface; ϕ is the gradient component of the slope field P; D is a derivation matrix modeling the slope field measurement P ; C( P a ) is a penalty term relating to alignment errors P a of the deflectometric device; and with C(P a ) = µ|| div P a ∥ 2 , Or µ is a regularization coefficient, in order to ensure that divP a is close to zero.

[0016] Such an equation is insensitive to the presence of a rotational component in the slope field, which allows for accurate estimation of the geometry of the reflective surface, especially when it is highly aspheric.

[0017] According to another particular embodiment, the integration step is carried out using the following equation (E): ϕ ^ = arg min D → ϕ ∓ P a → − P → 2 + C P a → + R ϕ Or : φ̂ is the estimated geometry of the reflective surface; ϕ is the gradient component of the slope field P; Dis a derivation matrix modeling the slope field measurement P ; C ( P a ) is a penalty term relating to alignment errors P a of the deflectometric device; R ( ϕ ) is a regularization term; and with C ( P a ) = µ || div P a ∥ 2 Or µ is a first regularization coefficient.

[0018] In particular, the term regularization R ( ϕ ) relates to a power spectral density of the estimated geometry of the reflective surface.

[0019] In particular, R ( ϕ ) = λ ∥ D'ϕ ∥ n< with n≥2 and where D' is a differentiation matrix and l is a second regularization coefficient.

[0020] In particular, R ( ϕ ) = l || ϕ || 2< .

[0021] In particular, the reflective surface is a mirror.

[0022] Specifically, the mirror is an aspherical mirror. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] The invention will be better understood in light of the following description, which is purely illustrative and not limiting, and should be read in conjunction with the accompanying drawings, among which: There figure 1 is a schematic view of a deflectometric device; The figure 2 is a schematic view of the process of the invention; The figure 3A is a map representing the geometry of an aspheric concave mirror estimated according to a particular embodiment of the method of the invention; The figure 3B is a view analogous to the figure 3Arepresenting the geometry of the mirror estimated according to a deflectometric measurement using the Southwell integration method; The figure 3C is a view analogous to the figure 3A , representing the geometry of the mirror estimated according to a phase-shift interferometric measurement; and La 3D figure is a view analogous to the figure 3A representing a difference between the interferometric measurement illustrated in the figure 3C and the measure illustrated in the figure 3A . DETAILED DESCRIPTION OF THE INVENTION

[0024] With reference to the figure 1, a mirror M includes a reflective surface whose quality we wish to control by a deflectometric device 1. The deflectometric device 1 includes a screen S arranged to display a target and a camera D arranged to capture an image of the target reflected by the surface of the mirror M. The target consists of a vertical periodic pattern comprising an alternation of dark and light fringes with a sinusoidal profile extending along an axis Oy.

[0025] A technique known in itself consisting of shifting the fringes by a regular step makes it possible to evaluate, for each pixel of the camera D, the phase of the corresponding light source point of the screen S and therefore its horizontal position along an Ox axis.

[0026] Similarly, the display of a test pattern consisting of a horizontal periodic pattern with a sinusoidal profile allows the vertical position of the light source points of the screen S to be evaluated along the Oy axis.

[0027] Knowing the relative positions of the screen S, the mirror M and the camera D, it is then possible to model the trajectory of the light rays emitted by the screen S and thus to have access at any point on the surface of the mirror M to the direction of the normal.

[0028] For this purpose, the deflectometric device 1 is connected to a computer comprising a processor and memory containing a program executable by the processor. The program contains instructions arranged to implement the method of the invention. This method comprises step 10 of measuring a slope field P of the surface of the mirror M and then step 20 of integrating the slope field P in order to reconstruct the geometry of said surface of the mirror M.

[0029] The slope field P is defined as follows: P → = ∇ → ϕ + rot → A → Or ϕ And A are respectively the gradient component and the rotational component of the slope field P .

[0030] Integration 20 of the slope field P is achieved by the following equation E: ϕ ^ = arg min D → ϕ ∓ P a → − P → 2 + C P a → + R ϕ Or : φ̂ is the estimated geometry of the reflective surface of mirror M; D is a differentiation matrix; C ( P a ) is a penalty term applied to alignment errors P a of the deflectometric device 1; and R ( ϕ ) is a so-called regularization term imposing a regularity on the solution sought and here relating to a power spectral density of the estimated geometry of the surface of the mirror M; and with C(P a ) = µ ∥ div P a ∥ 2 and for example R ( ϕ ) = λ ∥ D ' ϕ ∥ 2 Or µ And λ are regularization coefficients and D' is a differentiation matrix applied to the gradient component ϕ of the slope field P, as is classic in regularized inverse problems.

[0031] We understand that the operator sign “∓” used in the first part of equation E, arg min(∥ Dϕ ∓ P a - P ∥ 2< ), means that equation E can be written: ϕ ^ = arg min D → ϕ − P a → − P → 2 + C P a → + R ϕ Or ϕ ^ = arg min D → ϕ + P a → − P → 2 + C P a → + R ϕ

[0032] This stems from a modeling choice: if we consider that the alignment errors belong to the measured slopes, we model P ≈ Dϕ +P a ; If we consider that the alignment errors belong to the modeled slopes, we model P ≈ Dϕ-P a .

[0033] We also understand that the first part of equation E, arg min(∥ Dϕ ∓ P a , - P∥ 2< ), is a "raw" estimate of the shape and rotational components based on the calibration method.

[0034] In order to implement the process of the invention, it is necessary to have previously modeled the components. P a errors in the alignment of the deflectometer setup, for example when measuring the slope field of a surface of known shape. The components P a Alignment errors are physically characterized by a limited number of analytically calculated polynomials. The alignment terms are then described by a set ca of coefficients and a matrix of polynomials M a such as P a = Maca. Alignment errors have a gradient component and a rotational component. The penalty term C(P a ) introduced as an additional term to the least squares criterion aims to eliminate from the estimated geometry φ̂the rotational component due to alignment errors P a of the deflectometric setup 1.

[0035] The term regularization R ( ϕ The aim is to stabilize the inversion (zonal, therefore with a large number of unknowns) by introducing physical assumptions. The addition of such a regularization term is interpreted, in the context of Bayesian probabilities, as the assumption adopted here that the power spectral density of the estimated shape of the mirror decays according to a power law, of order -2 to -4, for example. This assumption was chosen to correspond to specific polishing conditions of the surface of the mirror M whose shape we seek to estimate. Any regularization criterion relating to the power spectral density that allows the problem to be regularized can be chosen.

[0036] Note that under the assumption that the spectral power density of the estimated geometry of the mirror surface M is uniform, the regularization term R ( ϕ ) becomes R ( ϕ ) = l || ϕ || 2< .

[0037] THE figures 3A to 3D illustrate the effectiveness of integrating the slope field P by equation E.

[0038] There figure 3A This represents a map of the geometry of a two-meter diameter mirror M, estimated by deflectometric measurement using deflectometer device 1 and equation E. The screen S is an uncalibrated 19-inch display with a resolution of 1280x1024 pixels, and the camera D has a 1.3-megapixel sensor and a frame rate of 30 FPS. The slope field P was integrated based on equation E by minimizing the criterion ∥ Dϕ ∓ P a - P ∥ 2< + µ ∥ div P a ∥ 2< + λ ∥ D'ϕ ∥ 2< by sparse matrix-type methods and with µ = 10 9< mm 2< and λ = 4.4 . 10 -10< mm 2. The regularization coefficients µ And l are defined empirically here.

[0039] The illustrated map at the figure 3B differs from the one illustrated in the figure 3A in that the geometry of mirror M is estimated by a deflectometric measurement using Southwell's equation, namely φ̂ = arg min(∥ Dϕ - P ∥ 2< ), and not that of equation E.

[0040] While the map illustrated at the figure 3A exhibits a high-frequency defect of approximately 147 nm (nanometers) in root mean square deviation, the map illustrated in the figure 3B exhibits a high-frequency defect of approximately 1175 nm in root mean square deviation. High-frequency defects thus appear much less visible on the figure 3B that on the figure 3A .

[0041] The illustrated map at the figure 3C differs from the one illustrated in the figure 3A in that the geometry of mirror M is estimated by a phase-shift interferometric measurement known as "PSI" (Phase Shift Interferometry). The map illustrated in the figure 3C exhibits a high-frequency defect of approximately 128 nm in root mean square deviation, in other words, relatively close to that observed at the figure 3A The actual high-frequency defects of the mirror thus appear just as visible on the figure 3C that on the figure 3A .

[0042] The map of the 3D figure , also called "Residuals", represents the difference between the deflectometric measurement using equation E illustrated in the figure 3A and the interferometric measurement illustrated at the figure 3C The measurement concerning the high spatial frequencies of mirror M, the first thirty-six Legendre polynomials were subtracted from each of the maps of figure 3A And3C The difference between the two measurements is approximately 47.5 nm in squared error.

[0043] Therefore, deflectometric measurement using equation E is effective and allows for a significant increase in the measurement dynamics of deflectometry, particularly on aspherical mirrors.

[0044] Of course, the invention is not limited to the embodiment described but encompasses any variant falling within the scope of the invention as defined by the claims.

[0045] Although here the slope field P has been integrated based on equation E using sparse matrix methods, other methods can be used to minimize the criterion ∥ Dϕ + P a - P ∥ 2< + C ( P a ) + R ( ϕ ) : gradient descent, quasi-Newton, simulated annealing, matrix...

[0046] Any form of regularization is acceptable. The term regularization R ( ϕ ) can notably be chosen independently of any measure.

[0047] For example, we could have R ( ϕ ) = λ ∥ D'ϕ ∥ n< with n≥2.

Claims

1. Method for estimating a geometry of a reflective surface, comprising the steps of measuring (10) a slope field (P) of the surface by a deflectometric device (1) connected to a measurement processing computer, then of integrating (20), via the measurement processing computer, the slope field by modelling the presence of a rotational component (A) in the field, and by jointly searching for a gradient component (ϕ) of said field and rotational components of alignment errors (Pa) of the deflectometric device.

2. Method according to claim 1, wherein the integration step is carried out by using the following equation (E): ϕ ^ = arg min D → ϕ ∓ P a → − P → 2 + C P a → where: - ϕ̂ is the estimated geometry of the reflective surface; - ϕ is the gradient component of the slope field P; - D is a derivation matrix modelling the measurement of the slope field P; - C(Pa) is a term of penalisation based on the alignment errors Pa of the deflectometric device 1; and with C(Pa) = µ∥div Pa∥2 where µ is a regularisation coefficient.

3. Method according to claim 1, wherein the integration step is carried out by using the following equation (E): ϕ ^ = arg min D → ϕ ∓ P a → − P → 2 + C P a → + R ϕ where: - ϕ̂ is the estimated geometry of the reflective surface; - ϕ is the gradient component of the slope field P; - D is a derivation matrix modelling the measurement of the slope field P; - C(Pa) is a term of penalisation based on the alignment errors Pa of the deflectometric device 1; - R(ϕ) is a term of regularisation; and with C(Pa) = µ∥divPa∥2 where µ is a first regularisation coefficient.

4. Method according to claim 3, wherein the term of regularisation R(ϕ) is based on a power spectral density of the estimated geometry of the reflective surface.

5. Method according to claim 4, wherein R(ϕ) = λ∥D'ϕ|n with n≥2 and where D' is a differentiation matrix and λ is a second regularisation coefficient.

6. Method according to claim 5, wherein R(ϕ)=λ∥ϕ∥2.

7. Method according to any one of the preceding claims, wherein the reflective surface is a mirror (M).

8. Method according to claim 3, wherein the mirror (M) is an aspherical mirror.