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摘要: 采用基于拉普拉斯算符聚焦形貌恢复方法，提出了模拟目标深度测量的数值模型。数值模拟的核心是基于通过几何光学预测的理想图像的卷积与透镜广义孔径函数的多色点扩散函数，即用聚焦误差替代抛物线圆柱形貌或高斯函数。该模型可以使用基于聚焦形貌恢复方法的传感器真实组件参数、光源光谱、光学系统离差、相机的光谱灵敏度。提出了光学系统离差（消球差、消色差、色差）对确定目标表面形貌的精确度和可靠性的影响。结果表明，该模型可以有效提高实验效率，缩短时滞，降低成本。Abstract: We propose a numerical model for simulation of an object depth measurement by means of a shape from focus method using Laplacian operator. The core of the simulation is based on convolution of an ideal image (predicted by the geometrical optics) with polychromatic point spread functions of a generalized aperture function of lens including focus error instead of more exploited the pillbox shape or the Gaussian functions. The model allows to employ parameters of real components of the sensor based on the method, a light source spectrum, dispersion of an optical system and spectral sensitivity of a camera. The influence of dispersion of optical systems (aberration-free, achromatic and with chromatic aberration) on accuracy and reliability of the determination of the object's surface topography is presented. It is indicated that this model can increase the experiment effectively and decrease time lag with the reducing of operating expenses.
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Figure 2. The depth z (x, y) of the 3D object and intensity distribution I (x′, y′) corresponding to its ideal image. The image is divided into the limited regions A, B, …, C (gray areas) , in which the spatially invariant hi, j (x′, y′) is computed for j=1, 2, …, N, while N denotes number of the regions. The depth z (x, y) changes in discrete increments to, j=to (height of a single step) . For simplicity, we assume imaging 1: 1, therefore widths wo, j of individual steps are the same as the widths wi, j of the appropriate regions A, B, …, C
Figure 3. Scheme for simulation of the shape from focus method. The pyramidal object is placed at the distance d1 from the optical system. The image of the object produced by the optical system is observed at the distance d2 by means of the detector. Light source is situated at the distance s from the object
Figure 4. (a) A one-dimensional profile z (x) of the object under test and (d) a resulting intensity distribution I (x′) , computed by summation of (b) the intensity distribution I (x′) with (c) the fluctuation Ifluct (x′)
Figure 5. (a) An ideal case of the depth map z (x, y) (matrix of 451×451 pixels) of the object represented by the pyramid with 5 levels with total height 735 μm, and height of the individual level of the pyramid is 147 μm (b) a cross section of the depth map z (x, y) from (a) at a position y=225, (c) the depth map z (x, y) of the object acquired via simulation of the shape from focus method using the aberration-free optical system (d) a cross section of the depth map z (x, y) from (c) at a position y=225
Figure 6. (a) A depth map z (x, y) of the object under test acquired via simulation of the shape from focus method using the optical system with chromatic aberration (b) a cross section of the depth map z (x, y) at a position y=225
Figure 7. (a) A depth map z (x, y) of the object under test acquired via simulation of the shape from focus method using the achromatic optical system (b) a cross section of the depth map z (x, y) at a position y=225
Figure 8. Sum of modified Laplacian function F (i, j) computed by Eq. (14) for i=225, j=125 (the point on the third level of the pyramidal object) as a function of d1 for aberration-free (ideal) optical system, optical system with chromatic aberration and achromatic optical system. The total object height is 735 μm
Figure 9. (a) A one-dimensional profile z (x) of the object under test, (b) a depth map z (x, y) of the object under test acquired via simulation of the shape from focus method using the achromatic optical system (c) a cross section of the depth map z (x, y) at a position y=225. The total object′s height is 1470 μm
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