constraints

The Optical Resolution Limit at ~3 Meters¶

The fundamental limit of any camera-based metrology system (like DIC) or laser-based system (like LDV or Shearography) at a distance $D$ is defined by the angular resolution of the optics. For a camera system, the projected size of a single pixel on the target surface, known as the Ground Sample Distance (GSD) or object-space pixel size ($S_{pix}$), determines the minimum resolvable feature.

\[ S_{pix} = \frac{D \cdot p}{f} \]

Where:

$D$ is the working distance (~3000 mm).
$p$ is the physical pixel pitch of the sensor (typically $3.45 \mu m$ for modern CMOS sensors).
$f$ is the focal length of the lens.

To achieve sub-millimetric measurement, specifically to resolve a deformation of $0.1$ mm with high confidence, the system typically requires a sampling density where the deformation magnitude spans at least 3 to 5 pixels. This implies a desired object-space pixel size of approximately $0.02$ mm to $0.03$ mm.

If we were to use a standard machine vision lens with a focal length of $50$ mm:

\[ S_{pix} = \frac{3000 \text{ mm} \cdot 0.00345 \text{ mm}}{50 \text{ mm}} \approx 0.207 \text{ mm} \]

A pixel size of $0.2$ mm is too coarse to reliably detect sub-millimetric deformation, as a $0.1$ mm movement would correspond to only half a pixel shift. While sub-pixel interpolation algorithms can detect shifts down to $0.01$ pixels, relying heavily on interpolation introduces noise and reduces accuracy.

Therefore, a critical hardware requirement for this application is the use of telephoto optics . Increasing the focal length to $200$ mm improves the resolution significantly:

\[ S_{pix} = \frac{3000 \cdot 0.00345}{200} \approx 0.051 \text{ mm} \]

With a $200$ mm lens, a $0.1$ mm deformation spans roughly 2 pixels, which allows for robust detection and quantification. This dictates that any instrument purchased must accept interchangeable C-mount or F-mount lenses to accommodate high-focal-length glass, or possess a built-in super-zoom capability.

The Geometric Dilution of Precision (GDOP) in Stereo Vision¶

For 3D measurement, which is necessary to capture the out-of-plane "bending" of the plate, a stereo-vision approach is required. This involves triangulating the position of a surface point $(X, Y, Z)$ using lines of sight from two cameras. The accuracy of the depth measurement ($Z$, the direction away from the camera) is heavily dependent on the angle between the two cameras, known as the stereo angle ($\theta$).

The uncertainty in depth ($\sigma_z$) is related to the uncertainty in image coordinate matching ($\sigma_{img}$) and the geometric baseline ($B$) between the cameras:

\[ \sigma_z \approx \frac{D^2}{B \cdot f} \sigma_{img} \]

Since the baseline $B$ is related to the stereo angle by $B \approx 2 D \tan(\theta/2)$, a small stereo angle results in a small baseline and a massive amplification of error.

The Single-Window Trap:

If the user attempts to look through a "small glass window" with both cameras (or a single stereoscopic sensor), the baseline $B$ is physically limited by the diameter of the window. For a typical $100$ mm diameter viewport at a distance of $3000$ mm, the maximum baseline is roughly $80$ mm.

\[ \theta \approx 2 \cdot \arctan\left(\frac{40}{3000}\right) \approx 1.5^{\circ} \]

A stereo angle of $1.5^{\circ}$ is essentially useless for 3D triangulation. The error ellipsoid would be extremely elongated along the Z-axis, meaning the system could measure in-plane ($X, Y$) deformation accurately, but would be nearly blind to the out-of-plane bending ($Z$) or would report massive noise.

The Multi-Window Solution: The user's note that "several other windows [are] available to have other points of view" is the single most important variable in this feasibility study. By placing Camera 1 at Window A and Camera 2 at Window B , the baseline $B$ can be increased from centimeters to meters. If the windows are separated by 1 meter, the stereo angle becomes $\approx 19^{\circ}$, which is within the optimal range (typically $15^{\circ}-30^{\circ}$) for high-precision DIC. This geometric arrangement reduces the depth uncertainty by a factor of 10 to 20, making sub-millimetric 3D measurement possible.

Refractive Distortion through Vacuum Interfaces¶

The presence of a thick glass window between the sensor and the object breaks the pinhole camera model used in standard photogrammetry. Light rays do not travel in straight lines from the object to the sensor; they undergo refraction at the air-glass and glass-vacuum interfaces according to Snell's Law.

\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]

For a window of thickness $t$ and refractive index $n$, a ray entering at an angle $\alpha$ is laterally shifted by an amount $\Delta$:

\[ \Delta = t \sin(\alpha) \left( 1 - \frac{\cos(\alpha)}{\sqrt{n^2 - \sin^2(\alpha)}} \right) \]

This shift ($\Delta$) is not constant across the image. It is zero at the optical center (where rays are perpendicular to the glass) and increases non-linearly toward the edges of the field of view. This creates a phenomenon known as radial refractive distortion , which compresses the image radially.

Magnitude: For a 20 mm thick window viewed at $20^{\circ}$ incidence, the lateral shift can be several millimeters.
Consequence: In a stereo setup, if uncorrected, this shift interprets the object as being closer or further than it actually is, and induces artificial curvature into flat surfaces (the "virtual surface" effect).

Any instrument selected for this task must utilize software algorithms capable of modeling and correcting for these refractive effects. Standard "calibration target" methods are insufficient because the calibration target cannot be placed inside the "unaccessible" chamber to capture the distortion empirically.

The Physics of the Measurement Chain¶

To understand the complexity of achieving 0.2 mm precision in this scenario, one must deconstruct the optical path. Light travels at velocity $v = c/n$, where $c$ is the vacuum speed of light and $n$ is the refractive index of the medium.

Air Medium ($n_{air} \approx 1.00027$): The instrument is situated in air.
Window Medium ($n_{glass} \approx 1.5$): The beam passes through a viewport, undergoing refraction (bending) and retardation.
Vacuum Medium ($n_{vac} \equiv 1.00000$): The beam travels 3 meters to the target.

Most commercial Time-of-Flight (ToF) and Phase-Shift instruments are calibrated for air. If an instrument measures the time of flight to the plate through a vacuum, the pulse will return faster than expected (since $v_{vac} > v_{air}$). Uncorrected, the instrument will interpret this shorter time as the object being closer.

The magnitude of this error $\Delta d$ over a distance $D=3000 \, \text{mm}$ is approximated by:

\[ \Delta d \approx D \cdot (n_{air} - n_{vac}) = 3000 \cdot (1.00027 - 1.00000) \approx 0.81 \, \text{mm} \]

This systematic error of 0.81 mm is four times the allowable tolerance of 0.2 mm . Consequently, the selection of hardware is secondary to the selection of a software framework capable of applying a composite Refractive Index (RI) correction or a "Vacuum Scale Factor."

Defining "Main Deformation"¶

The requirement focuses on "main deformation" rather than vibration. This implies a quasi-static measurement regime where the sampling frequency can be low (e.g., 1 Hz or even per-minute intervals). The deformation is thermal, suggesting it will manifest as:

Absolute expansion / position shift: Absolute repositioning / scaling of the plate dimensions.
Out-of-Plane Warping: Bending or buckling due to thermal gradients or constrained boundary conditions (e.g., clamped edges). To capture "main deformation," the metrology system must be capable of mapping the 3D surface topology or tracking specific discrete points to calculate strain vectors. A single distance measurement to the center of the plate is likely insufficient to characterize "deformation" (which implies shape change), necessitating a system with scanning or multi-point capability.

The Critical Role of Software and Environmental Correction¶

The hardware selected (Laser Radar or Laser Tracker) provides the raw data, but the 0.2 mm precision is achieved only through software processing. The vacuum environment introduces a systematic error that must be mathematically removed.

Refractive Index Correction (Vacuum Scaling)¶

As established in Section 1.1, the vacuum path induces an apparent distance error of $\approx 0.81 \, \text{mm}$.

Correction Workflow: In metrology software like SpatialAnalyzer (SA) , the user must apply a "Scale Factor" to the observations.
Calculation: The Refractive Index of air is $n_{air} \approx 1.00027$. The scale factor $S$ to convert an air-calibrated measurement to vacuum is the ratio of indices:

$$ S = \frac{n_{vac}}{n_{air}} = \frac{1.00000}{1.00027} \approx 0.99973 $$ * Implementation: In SA, the user selects the instrument or the point group and applies a specific Temperature/Pressure correction. By setting the "Environment" pressure to 0 (Vacuum), the software automatically adjusts the speed of light calculation, removing the 0.81 mm bias.

Transparent Material Refraction (Window Shift)¶

Passing through the glass window shifts the beam laterally and changes the optical path length.

Lateral Shift: For a window thickness $t$ and incidence angle $\alpha$:

$$ \Delta_{lat} \approx t \cdot \sin(\alpha) \cdot \left(1 - \frac{\cos \alpha}{\sqrt{n^2 - \sin^2 \alpha}}\right) $$

This shift causes the metal plate to appear distorted (bowed) because the incidence angle $\alpha$ changes as the laser scans from the center of the window (normal incidence) to the edges. * Software Solution: SpatialAnalyzer and PolyWorks utilize "Refraction Correction" or "Transparent Material" definitions.

Step 1: The user creates a CAD plane or surface representing the physical window in the software model.
Step 2: The user assigns material properties (Thickness, Index of Refraction $\approx 1.5$) to this surface.
Step 3: The software performs ray-tracing for every measurement point, calculating the refraction vector and correcting the coordinate to its true position in the vacuum chamber.

USMN (Unified Spatial Metrology Network)¶

For the highest confidence, SpatialAnalyzer's USMN feature can be employed. USMN uses uncertainty fields to bundle measurements. By defining the uncertainty of the window thickness and the vacuum index, USMN computes the "most probable" true surface of the plate and provides a statistical confidence interval (e.g., "Deformation is 0.15 mm $\pm 0.02$ mm"). This rigorously proves compliance with the 0.2 mm requirement.

The "Small Window" Advantage¶

The APDIS utilizes a focused laser beam steered by precision galvanometers and a simplified optical path.

Coaxial Design: The transmit and receive paths are coaxial. This allows the system to measure through a "keyhole" aperture effectively. As long as the clear aperture of the window is larger than the beam diameter (typically $< 5 \, \text{mm}$), the system can scan the plate behind it without vignetting, provided the scan angle does not exceed the window's field of view.
Window Correction: The APDIS ecosystem is deeply integrated with SpatialAnalyzer (SA) and PolyWorks Inspector . Both software suites contain specific "Transparent Material" modules designed to correct the refractive shift caused by measuring through thick glass.

Technology Class II: Direct Scanning Laser Trackers¶

Laser trackers have traditionally been contact measurement devices requiring a Spherically Mounted Retroreflector (SMR). However, the evolution of the Leica Absolute Tracker ATS600 has introduced "Direct Scanning" capabilities that compete directly with Laser Radar for non-contact applications.

Leica ATS600 Technology¶

The ATS600 combines a traditional Absolute Distance Meter (ADM) for reflector tracking with a Wave-Form Digitizer (WFD) for non-contact scanning.

Reflectorless Accuracy: The ATS600 specifies a reflectorless accuracy (MPE) of $\pm 50 \, \mu\text{m} + 10 \, \mu\text{m/m}$.
Application to User Scenario: At 3 meters:

$$ E_{dist} = 50 + (10 \times 3) = 80 \, \mu\text{m} \quad (0.08 \, \text{mm}) $$ * Result: While less precise than the Laser Radar ($35 \, \mu\text{m}$), the ATS600 is still solidly within the 0.2 mm ($200 \, \mu\text{m}$) requirement, offering a safety factor of 2.5x.

Scanning Mechanism and Window Interaction

Mechanical Steering: Unlike the Laser Radar's galvanometers, the ATS600 steers the beam by rotating the entire tracker head (azimuth and elevation axes). This results in a slower scanning process, which is perfectly acceptable for "slowly changing" thermal deformations.
Overview Camera (OVC): The ATS600 features a wide-angle Overview Camera (OVC) that provides a live video feed of the measurement volume.
Operational Benefit: The operator can view the vacuum chamber window on the control laptop, zoom in, and draw a "scan region" polygon directly on the image of the metal plate seen through the window. This "point-and-scan" simplicity is highly advantageous for remote setups where the object is inaccessible.

The "Small Window" Problem¶

DIC typically requires Stereoscopic Vision (two cameras) to resolve 3D out-of-plane deformation (Z-axis). To calculate depth, the cameras must have a separation angle (stereo baseline).

The Constraint: Viewing a plate 3 meters away through a single small window forces the two cameras to be positioned very close together. This results in a very narrow stereo angle ($< 5^\circ$).
The Impact: A narrow baseline drastically degrades Z-axis accuracy. While X-Y (in-plane) expansion can be measured accurately, the "bulging" of the plate towards the window (a common thermal deformation mode) becomes difficult to distinguish from simple magnification changes.
Monocular Solution: A single-camera Video Extensometer (like Imetrum Video Gauge ) can be used if the deformation is strictly 2D. Using a high-quality telephoto lens (e.g., 300mm+ focal length) can resolve the plate at 3m. However, correcting for the refractive distortion of the window across the 2D image field is mathematically complex compared to correcting a single laser beam.

Second-Order Insights and Risks¶

The "Thermal Lens" Effect¶

Second-order analysis suggests that if the metal plate is significantly heated, it radiates heat to the viewport window. The window itself may develop a radial thermal gradient ($dn/dT$). This transforms the flat window into a weak gradient-index lens. While likely a sub-micron effect for moderate temperatures, for high-temperature tests ($>200^\circ\text{C}$), this could introduce a "thermal lensing" error. Mitigate this by blowing cool air on the outside of the window to maintain thermal equilibrium.