Deflectometric method for surface shape reconstruction from its gradient

Antonin Miks; Jiri Novak

doi:10.1364/OE.20.028341

1. Introduction

Todays, it is paid a large attention to the development of methods for surface topography measurements, which find a wide application area in many parts of science, engineering, and biomedicine. The existing measurement and evaluation methods are based on different physical principles. Generally, it is possible to divide these methods into contact [1,2] and noncontact techniques [2–8]. One type of optical methods is based on deflectometry [7,9–15], which is used frequently for noncontact surface measurements.

In this work we present a deflectometric approach to solving a 3D surface reconstruction problem, which is based on measurements of a surface gradient of optically smooth surfaces. It is shown that a description of this problem leads to a nonlinear partial differential equation, from which the surface shape can be reconstructed numerically. The reconstruction process is presented on an example of 3D aspheric shape reconstruction.

2. Differential equation for surface reconstruction

Consider an optical surface, mathematically described by the formula $z = f (x, y)$ . The principle of the gradient deflectometric method is shown in Fig. 1 , where O is the origin of a chosen coordinate system and P(x,y,z) is a point at the surface.

Fig. 1 Principle of gradient deflectometric method

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Further, assume that the axis z is perpendicular to the plane ξ, which is located at the distance a from the plane xy. The ray described by its unit direction vector s, which is parallel to the z axis, intersects the plane ξ in the point C and the measured surface in the point P(x,y,z). The incidence angle ε is defined as an angle between the direction vector s and the unit vector n of the surface normal in the point P(x,y,z). After the reflection from the surface the ray intersects the plane ξ in the point Q, which is located in the distance t from the point C. The distance t = t(x,y) depends on the position of the point P(x,y,z) at the measured surface. The direction of the reflected ray is characterized by the unit vector s' and the angle ε' with respect to the unit surface normal vector n.

Corresponding to the law of reflection the vectors s, s' and n as well as the points P, C and Q lie in one plane. It holds due to the reflection law [16,17]

s^{'} = s - 2 n (s n)

and

\cos ε = - s n, ε^{'} = ε .

We obtain for the normal vector [17]

n = \frac{s - s^{'}}{\sqrt{2 [1 - (s s^{'})]}} .

Corresponding to Fig. 1 we can write

\frac{t}{\bar{PC}} = \frac{t}{a - z} = \tan 2 ε = \frac{2 \tan ε}{1 - \tan^{2} ε} .

Moreover, we denote the derivatives of the function

z = f (x, y)

as

p = \frac{\partial z}{\partial x}, q = \frac{\partial z}{\partial y} .

Then, it is well known that the unit normal vector to the surface

z = f (x, y)

can be expressed as [18,19]

n = [- \frac{p}{N}, - \frac{q}{N}, \frac{1}{N}],

where

N = \sqrt{1 + p^{2} + q^{2}}

. Assuming that the direction vector s of the incident ray is identical with the z axis direction, then

s = (0, 0, - 1) .

It holds

\cos ε = - s n = 1 / N .

Using the following relationship [18,19]

\tan ε = \pm \frac{\sqrt{1 - \cos^{2} ε}}{\cos ε} = \pm \sqrt{p^{2} + q^{2}},

we obtain from Eqs. (4) and (6)

\tan 2 ε = \pm \frac{2 \sqrt{p^{2} + q^{2}}}{1 - p^{2} - q^{2}} .

By substitution of previous formula into Eq. (4) we have

K = \frac{t}{a - z} = \pm \frac{2 \sqrt{p^{2} + q^{2}}}{1 - p^{2} - q^{2}} .

Equation (11) can be also written in the form

K (1 - p^{2} - q^{2}) = \pm 2 \sqrt{p^{2} + q^{2}} .

By squaring the previous equation we obtain

{(1 - p^{2} - q^{2})}^{2} - \frac{4}{K^{2}} (p^{2} + q^{2}) = 0.

After simplification of Eq. (13) we can write the final equation

{(p^{2} + q^{2})}^{2} - 2 b (p^{2} + q^{2}) + 1 = 0,

where we denoted

b = (1 + 2 / K^{2}) .

The solution of Eq. (14) has the following form

p^{2} + q^{2} = b \pm \sqrt{b^{2} - 1} .

Equation (16) can be generally written in the form

{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} = F (x, y, z),

where

F (x, y, z) = b \pm \sqrt{b^{2} - 1} = 1 + 2 / K^{2} \pm 2 \sqrt{1 + 1 / K^{2}} / K .

The nonlinear partial differential equation of the first order (17) is a general solution of the problem of the reconstruction of the surface

z = f (x, y)

using the deflectometric method. Due to the fact, that the general solution of Eq. (17) cannot be found in an explicit form, Eq. (17) must be solved numerically. Several methods exists how to solve this nonlinear differential equation [18–20]. An universal method is to write the equation of surface as a sum of certain basis functions and then to find the coefficients in the sum in order to satisfy the differential Eq. (17) as well as possible. To find a solution

z = f (x, y)

of Eq. (17), the unknown function

z = f (x, y)

can be expressed in the form of two-dimensional polynomials, for example

z = f (x, y) = \sum_{m = 0}^{M} \sum_{n = 0}^{N} c_{m n} x^{m} y^{n} .

By substitution of derivatives (5) of the chosen function

z = f (x, y)

into differential Eq. (17), we obtain the optimization problem instead of the solution of the partial differential equation. The measured area of the surface can be discretized into a regular grid of sampling points

(x_{i}, y_{k})

and the value

t (x_{i}, y_{k})

is measured using the detector (

i = 1, 2, ..., I

,

k = 1, 2, ..., K

). It is also possible to discretize the measured area by an irregular grid. By substitution of values

(x_{i}, y_{k})

and

t (x_{i}, y_{k})

into Eq. (5) and Eq. (19) we obtain values

p_{i k} = p (x_{i}, y_{k})

,

q_{i k} = q (x_{i}, y_{k})

, and

z_{i k} = z (x_{i}, y_{k})

. These values of derivatives are substituted into the following function

M = {(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} - F (x, y, z) .

We obtain values

M_{i k} (c_{00}, c_{10}, c_{01}, c_{20}, c_{11}, c_{02}, .....)

and we can create a merit functions Φ for the formulation of the optimization problem in the form:

Φ = \sum_{i = 1}^{I} \sum_{k = 1}^{K} [M_{i k} (c_{00}, c_{10}, c_{01}, c_{20}, c_{11}, c_{02}, .....)]^{2} .

The goal is to find such values of unknown coefficients

c_{m n}

, in order the function (21) was minimized. Different optimization techniques can be used of the mentioned optimization problem [21,22]. Knowing coefficients

c_{m n}

one can calculate coordinates

(x_{i}, y_{k}, z_{i k})

of the measured surface for each measurement point using Eq. (19). We transformed the problem of solving the partial differential equation to the optimization problem. Generally, the convergence of optimization techniques is dependent on initial values of coefficients

c_{m n}

. In the case of testing spherical and aspheric surfaces, which are fabricated, for example, in optics industry, one knows the nominal shape of the test surface very well and good initial parameters for the optimization procedures can be easily found, which ensure a very good convergence.

3. Example and simulation

As an example we present a numerical simulation of the described reconstruction method for the case of an aspheric surface.

Figure 2 presents a principal scheme of a possible technical realization of the sensor head for the measurement of distance t. The ray with the direction vector s propagates through the semitransparent mirror M and reflects at the measured surface $S (x, y, z) = 0$ in the point P. The reflected ray, which is characterized by the direction vector s′, is reflected by the mirror M and intersects the plane of the two dimensional array CCD sensor in the point Q. The position of the point Q is evaluated and the distance t of the point Q from the center of the sensor C can be determined (see Fig. 1). During the measurement the surface is scanned and the distance t(x,y) is measured for different scanned points at the surface. Consider now the case of the shape reconstruction of a rotationally symmetric aspheric surface, given by the following formula [5]

z = f (x, y) = \frac{c (x^{2} + y^{2})}{1 + \sqrt{1 - c^{2} (x^{2} + y^{2})}} + \sum_{s = 1}^{S} A_{s + 1} {(x^{2} + y^{2})}^{s + 1},

where

c = 1 / R

is the surface vertex curvature, R is the vertex radius, and

A_{i}

are aspheric coefficients. Using Eq. (5) and Eq. (22) we obtain values of the surface gradient. As an example we have chosen an aspheric surface has the vertex radius

R = - 133.2126 mm

(vertex curvature

c = - 7, 5067974 \cdot 10^{-3}

), diameter

D = 20 mm

and aspheric coefficients

A_{2} = 2,6934596 \cdot 10^{-7}

,

A_{3} = -4,0803344 \cdot 10^{-10}

. As the starting point for the optimization we choose a flat surface with the following coefficients:

c = 0

,

A_{2} = A_{3} = A_{4} = ... = 0

. We obtained by the solution of the Eq. (17) using the proposed method for the approximation coefficients:

c = - 7, 5067974 \cdot 10^{-3}

,

A_{2} = 2,6934596 \cdot 10^{-7}

,

A_{3} = -4,0803344 \cdot 10^{-10}

. Figure 3 presents an approximation error of the shape using the coefficients obtained from the optimization procedure Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [21,22]. We can see that the approximation error is very small, in the order of 10⁻¹³ mm and the proposed mathematical technique for the solution of Eq. (17) is working very well. It is evident from the previous example that a very good surface shape approximation can be found from an initial optimization point (approximation coefficients), which is really different from the optimal solution. We performed the analysis on several examples of spherical and aspheric surfaces with similar results, which prove the robustness of the described method. The proposed measurement method and the approach to find the solution of the partial differential Eq. (17), which describes theoretically the problem of surface reconstruction, give very good results and can be principally used for measurements of flat, spherical and aspheric surfaces.

Fig. 2 Principal scheme of measuring sensor.

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Fig. 3 Approximation error of shape of reconstructed aspheric surface

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We also performed an analysis of the influence of random errors in coordinates x,y using a computer simulation, where we considered the same uncertainty in both coordinates. The root-mean-square (RMS) of approximation errors for the given aspheric surface example is 0.1 μm for the uncertainty 3 μm, 0.4 μm for the uncertainty 10 μm, and 0.75 μm for the uncertainty 20 μm. These results prove the robustness of the proposed method.

4. Conclusion

The previous theoretical analysis had shown that a general solution of the 3D shape reconstruction of the optical surface is given by a nonlinear partial differential equation of the first order, which is a completely original mathematical approach in surface topography. The shape of the measured surface can be numerically calculated from the derived equation. We performed a numerical simulation of the presented method, which confirmed a robustness of the reconstruction method and a good possibility for measurements of spherical and aspheric specular surfaces. To our knowledge such a mathematical method of reconstruction of the surface shape has not been published yet.

Acknowledgment

This work has been supported by the Ministry of Industry and Trade of the Czech Republic the grant FR−TI2/074.

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Deflectometric method for surface shape reconstruction from its gradient

Abstract

1. Introduction

2. Differential equation for surface reconstruction

3. Example and simulation

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (3)

Equations (22)

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