Visual hull method for tomographic PIV measurement of flow around moving objects

Abstract

Tomographic particle image velocimetry (PIV) is a recently developed method to measure three components of velocity within a volumetric space. We present a visual hull technique that automates identification and masking of discrete objects within the measurement volume, and we apply existing tomographic PIV reconstruction software to measure the velocity surrounding the objects. The technique is demonstrated by considering flow around falling bodies of different shape with Reynolds number ~1,000. Acquired image sets are processed using separate routines to reconstruct both the volumetric mask around the object and the surrounding tracer particles. After particle reconstruction, the reconstructed object mask is used to remove any ghost particles that otherwise appear within the object volume. Velocity vectors corresponding with fluid motion can then be determined up to the boundary of the visual hull without being contaminated or affected by the neighboring object velocity. Although the visual hull method is not meant for precise tracking of objects, the reconstructed object volumes nevertheless can be used to estimate the object location and orientation at each time step.

Appendix: Formulae derivations for obscured regions of a simple object shape and standard four-camera configuration

Consider the camera arrangement and object orientation as shown in Fig. 17. We assume the lines of sight are parallel for each camera, and the object is a cuboid. The inclination angles of Cameras 1–4 are α₁, α₂, α₃, and α₄ respectively. The dimensions of the cuboid are a (length) × b (width) × c (depth), where a and b are parallel to the laser sheet, and the side faces of the object are aligned with the camera angles. The distance from the rear of the object (relative to the cameras) to the rear edge of the laser sheet is given by t.

The fully obscured region (green in Fig. 17) is not optically accessible by any camera. For the given object orientation, this region can be either trapezoidal or pyramidal in shape, depending on the laser sheet thickness, object location, and object dimension. Mathematically, the following relations must be satisfied for a trapezoidal-shaped fully obscured region:

$$ \tan \left( {\alpha_{1} } \right) + \tan \left( {\alpha_{3} } \right) < \frac{b}{t} $$

(6)

$$ \tan \left( {\alpha_{2} } \right) + \tan \left( {\alpha_{4} } \right) < \frac{a}{t} $$

(7)

Otherwise, the fully obscured region is pyramidal.

The partially obscured region (blue in Fig. 17), which is accessible by less than 3 cameras, can be calculated by subtracting the fully obscured volume from the object “shadow” (i.e. a × b × t).

1.1 Trapezoidal fully obscured region

First, we consider the trapezoidal fully obscured volume (Fig. 18) which can be split into a cuboid (blue), 4 half—cuboids (red) and 4 pyramids (green). We can obtain the total volume by summing the 9 parts (Fig. 19).

$$ a^{\prime } = a - t \times \tan \left( {\alpha_{2} } \right) - t \times \tan \left( {\alpha_{4} } \right) $$

(8)

$$ b^{\prime } = a - t \times \tan \left( {\alpha_{1} } \right) - t \times \tan \left( {\alpha_{3} } \right) $$

(9)

Fig. 18

Schematic representation of the a trapezoidal fully obscured region, and b an exploded view of the various subvolumes

Fig. 19

Schematic drawing and dimensions of a cuboid (blue), half cuboid (red) and pyramid (green) that make up the trapezoid

(1) Cuboid

$$ \begin{aligned} {\text{Volume}}\, ( {\text{Cuboid)}} & = a^{\prime } \times b^{\prime } \times t = {\text{abt}} - t^{2} \left[ {a\left( {\tan \left( {\alpha_{1} } \right) + \tan \left( {\alpha_{3} } \right)} \right) + b\left( {\tan \left( {\alpha_{2} } \right) + \tan \left( {\alpha_{4} } \right)} \right)} \right] \\ & \quad + t^{3} \left( {\tan \left( {\alpha_{1} } \right) + \tan \left( {\alpha_{3} } \right)} \right)\left( {\tan \left( {\alpha_{2} } \right) + \tan \left( {\alpha_{4} } \right)} \right) \\ \end{aligned} $$

(10)

(2) Total Volume of 4 Half Cuboids

$$ \begin{aligned} {\text{Volume}}\,({\text{Half}}\, {\text{Cuboids}}) & = \frac{1}{2}\left[ {a^{\prime } t\left( {\left( {\tan \left( {\alpha_{1} } \right) + \tan \left( {\alpha_{3} } \right)} \right) + b^{\prime } t\left( {\tan \left( {\alpha_{2} } \right) + \tan \left( {\alpha_{4} } \right)} \right)} \right)} \right] \\ & = \frac{{t^{2} }}{2}\left[ {a\left( {\tan (\alpha_{1} ) + \tan (\alpha_{3} )} \right) + b(\tan (\alpha_{2} ) + \tan (\alpha_{4} ))} \right] \\ & \quad - t^{3} (\tan (\alpha_{1} ) + \tan (\alpha_{3} ))(\tan (\alpha_{2} ) + \tan (\alpha_{4} )) \\ \end{aligned} $$

(11)

(3) Total Volume of 4 Pyramids

$$ \begin{aligned} {\text{Volume}}\,({\text{Pyramids}}) & = \frac{1}{3} \times t \times [t \times (\tan (\alpha_{2} ) \times \tan (\alpha_{4} ) + \tan (\alpha_{4} ) \times \tan (\alpha_{3} ) \\ & \quad + \tan (\alpha_{3} ) \times \tan (\alpha_{2} ) + \tan (\alpha_{2} ) \times \tan (\alpha_{1} ) ) ] \\ \end{aligned} $$

(12)

$$ \begin{aligned} {\text{Volume}}\, ( {\text{Fully}}\,{\text{Obscured)}} & = {\text{Volume\,(Cuboid)}} + {\text{Volume\,(Half}}\,{\text{Cuboids)}} + {\text{Volume\,(Pyramids)}} \\ & = {abt} - \frac{{t^{2} }}{2} \left[ {a\left( {\tan (\alpha_{1} ) + \tan (\alpha_{3} )} \right) + b\left( {\tan (\alpha_{2} ) + \tan (\alpha_{4} )} \right)} \right] \\ & \quad + \frac{{t^{3} }}{3}\left( {\tan (\alpha_{1} ) + \tan (\alpha_{3} )} \right)\left( {\tan (\alpha_{2} ) + \tan (\alpha_{4} )} \right) \\ \end{aligned} $$

(13)

The partially obscured region is given by:

$$ \begin{gathered} {\text{Volume}}\,({\text{Partially}}\,{\text{Obscured}}) = {abt} - {\text{Volume}}\, ( {\text{Fully}}\,{\text{Obscured)}} \hfill \\ = t^{2} \left[ {\frac{a}{2}(\tan (\alpha_{1} ) + (\tan (\alpha_{3} ) + \frac{b}{2}(\tan (\alpha_{2} ) + (\tan (\alpha_{4} )) - \frac{t}{3}((\tan (\alpha_{1} ) + (\tan (\alpha_{3} ))(\tan (\alpha_{2} ) + (\tan (\alpha_{4} ))} \right] \hfill \\ \end{gathered} $$

(14)

To simplify Eqs. (13) and (14) further, we consider a symmetric camera arrangement (i.e., α₁ = α₂ = α₃ = α₄ = α) and a cubic object (i.e. a = b = c).

$$ {\text{Volume}}\, ( {\text{Fully }}\,{\text{Obscured)}} = a^{2} t - 2at^{2} \tan \left( \alpha \right) + \frac{{4t^{3} }}{3}\tan^{2} \left( \alpha \right) $$

(15)

$$ {\text{Volume}}\,({\text{Partially }}\,{\text{Obscured}}) = 2at^{2} \tan \left( \alpha \right) - \frac{{4t^{3} }}{3}\tan^{2} \left( \alpha \right) $$

(16)

where the assumptions for trapezoidal-shape (see Eqs. 6 and 7) both reduce to

$$ \tan (\alpha ) < \frac{a}{2t} $$

(17)

1.2 Pyramidal fully obscured region (simplified with α₁ = α₂ = α₃ = α₄ = α and a = b = c)

If $\tan (\alpha ) \ge \frac{a}{2t}$, the fully obscured volume will converge to a pyramidal shape (see green in Fig. 20)

Fig. 20

Schematic diagram of a specific “4-corner” camera arrangement where t is large such that tan(α) > a/2t. The white region is optically accessible by all cameras

The volume of the green pyramid (fully obscured region) is given by:

$$ {\text{Volume}}\,({\text{Fully}}\,{\text{Obscured}}) = \frac{1}{3} \times a \times a \times h = \frac{{a^{3} }}{6\tan \left( \alpha \right)} , $$

(18)

where the height h is found easily by trigonometry.

The partially obscured region is now given by:

$$ {\text{Volume\,(Partially}}\,{\text{Obscured)}} = {abt} - {\text{Volume}}\, ( {\text{Fully }}\,{\text{Obscured)}} - {\text{Volume}}\,({\text{Fully}}\,{\text{Obscured}}) \times \left( {\frac{t - h}{t}} \right)^{3} $$

(19)

where the last term is the white optically accessible region in Fig. 20. Note, however, that this equation, when simplified, reduces to the same form as Eq. (16).