# Generic and Model-Based Calibration Method for Spatial Frequency Domain Imaging with Parameterized Frequency and Intensity Correction

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multispectral SFDI Setup

#### 2.2. Calibration Model

#### 2.3. Calibration Routine

#### 2.3.1. Geometric Calibration

#### 2.3.2. Parametrized 3D Point Estimation

#### 2.3.3. Calculating Normals and Angles for Spatial Frequency Correction

**Figure 4.**(

**a**) Schematic representation of a sinusoidal pattern with spatial frequency ${f}_{0}$ projected obliquely onto a plane surface in the local coordinate system of a point $\overrightarrow{P}$. (

**b**) Both a height shift of Δz and (

**c**) an inclination of the sample relative to the reference plane by $\Delta \mathsf{\theta}$ lead to scaling of the local spatial frequency f observed on the sample surface.

#### 2.3.4. Intensity Calibration

#### 2.4. Phantom Measurements

## 3. Results and Discussion

#### 3.1. Geometric Model of the SFDI Setup in Global Coordinates

#### 3.2. Phase-Distance Conversion

#### 3.3. Calculating the Projection and Detection Angles

#### 3.4. Look-Up Table for Intensity Reference

#### 3.5. Determining Multispectral Optical Properties of a Hemispherical Phantom

## 4. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic drawing of (

**a**) the multispectral spatial frequency domain imaging (SFDI) setup consisting of a projection unit with a digital micromirror device (DMD), a tunable LED lightsource and a sCMOS camera to detect the diffuse reflect light. (

**b**) Pixel-wise demodulation for spatial frequency $\mathrm{f}$ yields the offset ${I}_{DC}$, the modulation amplitude ${I}_{AC}$ and the unwrapped phase.

**Figure 2.**Schematic illustration of the pinhole camera model for a projector–camera system with intrinsic and extrinsic parameters.

**Figure 3.**Schematic illustration with an overview of the processing steps for the geometric calibration.

**Figure 5.**Flowchart showing an overview of the calibration routines and processing steps with spatial frequency and intensity correction applied to an SFDI measurement.

**Figure 6.**(

**a**–

**c**) 2D representation of the virtual camera and projector center, the illuminated area and the image area in the global coordinate system as they result from the geometric calibration. (

**d**) The corresponding 3D illustration with the outer corner rays of both the camera (blue) and projector (red).

**Figure 7.**For a single pixel, e.g., in the center of the CCD, ${\mathsf{\varphi}}_{glob}$ can be described as a function of the distance ${\mathrm{l}}_{(\mathrm{u},\mathrm{v})}$ between the sample surface and the camera (blue dots). Fitting a third degree polynomial gives the analytical phase–distance conversion (black line). (

**b**) shows three planes with a distance of 10 mm, centered around the reference plane, with color coding showing ${\mathsf{\varphi}}_{glob}$. The black arrow corresponds to the profile shown in (

**a**) while the red dashed line marks ${\mathsf{\varphi}}_{glob}=0$ in each case. (

**c**) For validation, we measured the 3D topography of a flat reference target using the presented phase–distance conversion. The mean surface deviation was determined by the regression of a plane as $15\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$.

**Figure 8.**(

**a**) 3D topography of an optical phantom with a centered spherical cap showing the normal vectors (black arrows) and color-coded z-coordinate. (

**b**) Detailed view with the additional marking of the incident projector direction (red arrows) and the camera detection (blue arrows). (

**c**,

**d**) show the polar angles $({\mathsf{\theta}}_{Cam},{\mathsf{\theta}}_{Pro})$ and $({\mathsf{\varphi}}_{Cam},{\mathsf{\varphi}}_{Pro})$, respectively, with the mean angles indicated by a white bar in the colorbar.

**Figure 9.**(

**a**) Three-dimensional representation of the DC images for three intensity reference measurements, each shifted by $\Delta \mathrm{z}=10\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ in the z-direction, with an exemplary camera ray marked in the center of the image (black arrow). (

**b**) shows for this selected ray the reflectance resulting from the intensity calibration, which is plotted in 2D, color-coded against spatial frequency and distance. The right plot shows an example of the change in reflectance at $\mathrm{f}=0\phantom{\rule{0.166667em}{0ex}}{\mathrm{mm}}^{-1}$, $\mathrm{f}=0.4\phantom{\rule{0.166667em}{0ex}}{\mathrm{mm}}^{-1}$, and $\mathrm{f}=0.8\phantom{\rule{0.166667em}{0ex}}{\mathrm{mm}}^{-1}$ for different distances as solid lines, and the bottom plot shows the MTF in the range between $0\phantom{\rule{0.166667em}{0ex}}{\mathrm{mm}}^{-1}$ and $1\phantom{\rule{0.166667em}{0ex}}{\mathrm{mm}}^{-1}$ for distances 310 mm, 320 mm and 325 mm as dashed lines.

**Figure 10.**(

**a**) Measurement of a hemispherical optical phantom with a radius of curvature of 40 mm in three different z-positions using false colors to display the geometrically corrected spatial frequency for a pattern with $\mathrm{f}=0.45\phantom{\rule{0.166667em}{0ex}}{\mathrm{mm}}^{-1}$ in the reference plane. (

**b**) Averaged reflectances measured at a wavelength of 521 nm after post-processing considering (circles) and neglecting (squares) frequency and intensity correction and compared to a forward calculation. (

**c**) Spectrally resolved effective scattering coefficient ${\mathsf{\mu}}_{s}^{\prime}$ and absorption coefficient ${\mathsf{\mu}}_{a}$ determined after post-processing, taking into account (circles) and neglecting (squares) the frequency and intensity correction compared to an integrating sphere measurement.

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**MDPI and ACS Style**

Lohner, S.A.; Nothelfer, S.; Kienle, A.
Generic and Model-Based Calibration Method for Spatial Frequency Domain Imaging with Parameterized Frequency and Intensity Correction. *Sensors* **2023**, *23*, 7888.
https://doi.org/10.3390/s23187888

**AMA Style**

Lohner SA, Nothelfer S, Kienle A.
Generic and Model-Based Calibration Method for Spatial Frequency Domain Imaging with Parameterized Frequency and Intensity Correction. *Sensors*. 2023; 23(18):7888.
https://doi.org/10.3390/s23187888

**Chicago/Turabian Style**

Lohner, Stefan A., Steffen Nothelfer, and Alwin Kienle.
2023. "Generic and Model-Based Calibration Method for Spatial Frequency Domain Imaging with Parameterized Frequency and Intensity Correction" *Sensors* 23, no. 18: 7888.
https://doi.org/10.3390/s23187888