4dcloudscatteringtomography/full.md

4D Cloud Scattering Tomography

Roi Ronen, Yoav Y. Schechner
Viterbi Faculty of Electrical & Computer Eng.
Technion - Israel Institute of Technology
Haifa, Israel

ronen.ROI@gmail.com

yoav@ee.technion.ac.il

Eshkol Eytan
Department of Earth & Planetary Sciences
The Weizmann Institute of Science
Rehovot, Israel

eshkol.eytan@weizmann.ac.il

Abstract

We derive computed tomography (CT) of a time-varying volumetric scattering object, using a small number of moving cameras. We focus on passive tomography of dynamic clouds, as clouds have a major effect on the Earth's climate. State of the art scattering CT assumes a static object. Existing 4D CT methods rely on a linear image formation model and often on significant priors. In this paper, the angular and temporal sampling rates needed for a proper recovery are discussed. Spatiotemporal CT is achieved using gradient-based optimization, which accounts for the correlation time of the dynamic object content. We demonstrate this in physics-based simulations and on experimental real-world data.

1. Introduction

Computed tomography (CT) aims to recover the inner structure of three dimensional (3D) volumetric heterogeneous objects [15, 16]. CT has extensive use in many domains. These include medicine [14, 41], sensing of atmospheric pollution [2], geophysics [49] and fluid dynamics [27, 51, 52]. CT requires multi-view imaging [3, 22]. In nearly all CT approaches, the object is considered static during the multi-view acquisition. However, often the object changes while views are acquired sequentially [8, 53]. Thus, effort has been invested to generalize 3D CT to four-dimensional (4D) spatiotemporal CT, particularly in the computer vision and graphics communities [42, 52, 53]. This effort has focused on linear-CT modalities. Linear CT is computationally easier to handle, thus common for decades, mainly in medical imaging [19]. Medical CT often exploits the periodic temporal nature of organ dynamics, to synchronize sequential acquisitions [41].

This paper deals with a more complicated model: scattering CT. It is important to treat this case for scientific, societal and practical reasons. The climate is strongly affected

Figure 1. Multiple moving sensors image a time-varying object (cloud) from multiple-views. Tomography seeks the inner content.

by interaction with clouds [13]. To reduce major errors in climate predictions, this interaction requires a much finer understanding of cloud physics. Current models are based on remote sensing data that is analyzed under the assumption that the atmosphere and clouds are made of very broad and uniform layers. This leads to errors in climate understanding. To overcome this problem, 3D scattering CT has been suggested as a way to study clouds [29, 30, 46].

Scattering CT of clouds requires high resolution multiview images from space. There are spaceborne and high-altitude systems that may provide such data, such as AirM-SPI [6], MAIA [4], HARP [38], AirHARP [35] and the planned CloudCT formation [43]. These systems are so expensive, that it is unrealistic to deploy them in large numbers to simultaneously image the same clouds from many angles. Therefore, in practice, platforms move above the clouds: a sequence of images is taken, in order to span and sample a wide angular breadth (Fig. 1), but the cloud evolves meanwhile. Hence there are important reasons to derive 4D scattering CT of clouds.

We pose conditions for performing this task. These relate to temporal sampling and angular breadth, in relation to the correlation time of the evolving object. Then, we generalize prior 3D scattering CT, to spatiotemporal recovery using data taken by moving cameras. We present an optimization-based method to reach this. The method is demonstrated both in rigorous simulations and on real data.

2. Theoretical Background

2.1. Cloud Microphysical Parameters

Warm clouds are composed of water droplets. The droplet size distribution is typically parameterized by an effective radius $r^{\mathrm{e}}$ and a dimensionless droplet variance $v^{\mathrm{e}}$ (see [17] for details). These parameters vary with spatial location $\mathbf{x}$ , and assumed here to be uniform in a voxel around $\mathbf{x}$ . An additional characterization is the liquid water content (LWC), $\mathcal{L}$ , which increases linearly with the number-density and volume of droplets in a voxel [17, 30].

A common approximation in convective cloud models is that $\mathcal{L}$ and $r^{\mathrm{e}}$ tend to increase with altitude inside a cloud [32, 47, 54]. Let $z_0$ be the cloud base. For adiabatic convection, at $z > z_0$

$$\begin{array}{cccc}& r^{\mathrm{e}}\approx {\xi }_{\mathrm{r}}(z-z_0{)}^{\frac{1}{3}}+r_0^{\mathrm{e}},\mathcal{L}\approx {\xi }_{\mathcal{L}}(z-z_0),& & \text{(1)}\end{array}r\; ^\; \{\backslash mathrm\; \{e\}\}\; \backslash approx\; \backslash xi\_\; \{\backslash mathrm\; \{r\}\}\; (z\; -\; z\; \_\; \{0\})\; ^\; \{\backslash frac\; \{1\}\{3\}\}\; +\; r\; \_\; \{0\}\; ^\; \{\backslash mathrm\; \{e\}\},\; \backslash quad\; \backslash mathcal\; \{L\}\; \backslash approx\; \backslash xi\_\; \{\backslash mathcal\; \{L\}\}\; (z\; -\; z\; \_\; \{0\}),\; \backslash tag\; \{1\}$$

where $r_0^{\mathrm{e}}$ is the effective radius at the cloud base and $\xi_{\mathrm{r}},\xi_{\mathcal{L}} > 0$ . The error of this approximation grows as the cloud mixes with its surroundings air at the cloud shell, mainly at its top. The values of $r^{\mathrm{e}},v^{\mathrm{e}}$ tend to be rather uniform per altitude, while $\mathcal{L}$ can change significantly in 3D [21].

Overall, the vector $\nu_{t} = [\mathcal{L}{t}, r{t}^{\mathrm{e}}, v_{t}^{\mathrm{e}}]$ characterizes a voxel at time $t$ . Concatenating these parameters across all spatial voxels results in a vector $\pmb{\nu}_{t}$ , which expresses the cloud structure at time $t$ .

2.2. The Forward Model

The interaction of radiation with a scattering volumetric object is modelled by 3D radiative transfer, which includes multiple scattering. Define $\mathcal{M}_{\lambda}[\cdot ]$ as a differentiable operator that maps microphysical parameters to macroscopic optical parameters corresponding to wavelength $\lambda$ , using Mie theory [12, 30]. Let

$$\begin{array}{cccc}& L(\mathbf{x},\mathbf{\Omega },t)=\mathrm{RT}({\mathcal{M}}_{\lambda }[{\mathit{\nu }}_t])& & \text{(2)}\end{array}L\; (\backslash mathbf\; \{x\},\; \backslash boldsymbol\; \{\backslash Omega\},\; t)\; =\; \backslash operatorname\; \{R\; T\}\; \backslash left(\backslash mathcal\; \{M\}\; \_\; \{\backslash lambda\}\; [\; \backslash boldsymbol\; \{\backslash nu\}\; \_\; \{t\}\; ]\backslash right)\; \backslash tag\; \{2\}$$

be the radiance resulting from radiative transfer at each spatial location $\mathbf{x}$ and each direction $\Omega$ . There are various algorithms to implement $\mathrm{RT}(\mathcal{M}_{\lambda}[\nu_t])$ , including Monte-Carlo [32, 34] and the spherical harmonic discrete ordinate method (SHDOM). We use the latter, as it is considered trustworthy by the scientific community [9] and has an open-source online code [28].

A camera observes the scene from a specific location, while each pixel in a camera samples a viewing direction

$\Omega$ . Hence, imaging (forward model) amounts to sampling the output of 3D radiative transfer at the camera locations and the lines of sight of the pixels. This sampling integrates over the camera exposure time and spectral bands. Camera sampling is denoted by a projection operator $P_{\mathbf{x},\Omega}$ .

The forward image formation model $\mathcal{F}(\pmb{\nu}_t)$ yields the expected graylevel at a pixel, at time $t$ :

$$\begin{array}{cccc}& I_{\mathbf{x},\mathbf{\Omega },t}=\mathcal{F}\left({\mathit{\nu }}_t\right)\approx {\gamma }^{\mathrm{c}\mathrm{a}\mathrm{m}}{P}_{\mathbf{x},\mathbf{\Omega }}\left\{\mathrm{R}\mathrm{T}({\mathcal{M}}_{\lambda }[{\mathit{\nu }}_t])\right\}\mathrm{.}& & \text{(3)}\end{array}I\; \_\; \{\backslash mathbf\; \{x\},\; \backslash boldsymbol\; \{\backslash Omega\},\; t\}\; =\; \backslash mathcal\; \{F\}\; \backslash left(\backslash boldsymbol\; \{\backslash nu\}\; \_\; \{t\}\backslash right)\; \backslash approx\; \backslash gamma^\; \{\backslash mathrm\; \{c\; a\; m\}\}\; P\; \_\; \{\backslash mathbf\; \{x\},\; \backslash boldsymbol\; \{\backslash Omega\}\}\; \backslash left\backslash \{\backslash mathrm\; \{R\; T\}\; \backslash left(\backslash mathcal\; \{M\}\; \_\; \{\backslash lambda\}\; [\; \backslash boldsymbol\; \{\backslash nu\}\; \_\; \{t\}\; ]\backslash right)\; \backslash right\backslash \}.\; \backslash tag\; \{3\}$$

Here $\gamma^{\mathrm{cam}}$ expresses camera properties, including the lens aperture area, exposure time, spectral band, quantum efficiency and lens transmissivity. Eq. (3) assumes that the exposure time is sufficiently short, such that within this time, the scene and the camera pose vary insignificantly.

Empirical measurements include random noise [2,6,48]. The noise mainly originates from the discrete nature of photons and electric charges, which yields a Poisson process. There are additional noise sources, and their parameters can be extracted from the sensor specifications. Denote incorporation of noise into the expected signal by the operator $\mathcal{N}$ . Then, a raw measurement is

$$\begin{array}{cccc}& y_{\mathbf{x},\mathrm{\Omega },t}=\mathcal{N}\left\{I_{\mathbf{x},\mathrm{\Omega },t}\right\}\mathrm{.}& & \text{(4)}\end{array}y\; \_\; \{\backslash mathbf\; \{x\},\; \backslash Omega\; ,\; t\}\; =\; \backslash mathcal\; \{N\}\; \backslash left\backslash \{I\; \_\; \{\backslash mathbf\; \{x\},\; \backslash Omega\; ,\; t\}\; \backslash right\backslash \}.\; \backslash tag\; \{4\}$$

Per $t$ , all multi-view, multi-pixel measurements are concatenated into a vector $\mathbf{y}_t$ . Concatenating $\mathbf{y}_t$ over all $t$ yields the vector $\mathbf{y}$ .

2.3. Scattering 3D Tomography of Clouds

This section expresses the state of the art in 3D scattering tomography [32,44,47], in which $\nu_{t}$ is assumed to be invariant to $t$ . Hence, $t$ is generally dropped from the derivations here. Estimation of $\pmb{\nu}$ is done by minimization of a cost $\mathcal{E}$ , which penalizes the discrepancy between $\pmb{y}$ and the forward model,

$$\begin{array}{cccc}& \widehat{\nu }=\mathcal{E}[\mathit{y},\mathcal{F}(\mathit{\nu })]\mathrm{.}& & \text{(5)}\end{array}\backslash hat\; \{\backslash nu\}\; =\; \backslash underset\; \{\backslash boldsymbol\; \{\backslash nu\}\}\; \{\backslash operatorname\; \{a\; r\; g\; m\; i\; n\}\}\; \backslash mathcal\; \{E\}\; [\; \backslash boldsymbol\; \{y\},\; \backslash mathcal\; \{F\}\; (\backslash boldsymbol\; \{\backslash nu\})\; ].\; \backslash tag\; \{5\}$$

Eq. (5) can be solved efficiently by gradient-based methods. By setting

$$\begin{array}{cccc}& \mathcal{E}[\mathit{y},\mathcal{F}(\mathit{\nu })]=\frac{1}{2}\mathrm{\parallel }\mathit{y}-\mathcal{F}(\mathit{\nu }){\mathrm{\parallel }}_2^2,& & \text{(6)}\end{array}\backslash mathcal\; \{E\}\; [\; \backslash boldsymbol\; \{y\},\; \backslash mathcal\; \{F\}\; (\backslash boldsymbol\; \{\backslash nu\})\; ]\; =\; \backslash frac\; \{1\}\{2\}\; \backslash |\; \backslash boldsymbol\; \{y\}\; -\; \backslash mathcal\; \{F\}\; (\backslash boldsymbol\; \{\backslash nu\})\; \backslash |\; \_\; \{2\}\; ^\; \{2\},\; \backslash tag\; \{6\}$$

the gradient of Eq. (5) with respect to $\pmb{\nu}$ is

$$\begin{array}{cccc}& \mathbf{g}(\mathit{\nu })\equiv \frac{\mathrm{\partial }\mathcal{E}[\mathit{y},\mathcal{F}(\mathit{\nu })]}{\mathrm{\partial }\mathit{\nu }}=\frac{\mathrm{\partial }\mathcal{F}(\mathit{\nu })}{\mathrm{\partial }\mathit{\nu }}[\mathcal{F}(\mathit{\nu })-\mathit{y}]\mathrm{.}& & \text{(7)}\end{array}\backslash mathbf\; \{g\}\; (\backslash boldsymbol\; \{\backslash nu\})\; \backslash equiv\; \backslash frac\; \{\backslash partial\; \backslash mathcal\; \{E\}\; [\; \backslash boldsymbol\; \{y\}\; ,\; \backslash mathcal\; \{F\}\; (\backslash boldsymbol\; \{\backslash nu\})\; ]\}\{\backslash partial\; \backslash boldsymbol\; \{\backslash nu\}\}\; =\; \backslash frac\; \{\backslash partial\; \backslash mathcal\; \{F\}\; (\backslash boldsymbol\; \{\backslash nu\})\}\{\backslash partial\; \backslash boldsymbol\; \{\backslash nu\}\}\; [\; \backslash mathcal\; \{F\}\; (\backslash boldsymbol\; \{\backslash nu\})\; -\; \backslash boldsymbol\; \{y\}\; ].\; \backslash tag\; \{7\}$$

Gradient-based optimization performs per iteration $k$

$$\begin{array}{cccc}& \mathit{\nu }(k+1)=\mathit{\nu }(k)-\eta \mathbf{g}[\mathit{\nu }(k)]& & \text{(8)}\end{array}\backslash boldsymbol\; \{\backslash nu\}\; (k\; +\; 1)\; =\; \backslash boldsymbol\; \{\backslash nu\}\; (k)\; -\; \backslash eta\; \backslash mathbf\; \{g\}\; [\; \backslash boldsymbol\; \{\backslash nu\}\; (k)\; ]\; \backslash tag\; \{8\}$$

where $\eta$ is a step size. In scattering CT, computing the Jacobian $\partial \mathcal{F}(\pmb{\nu}) / \partial \pmb{\nu}$ is complex. However, there are approximations to the Jacobian of 3D RT, which can be computed efficiently [29, 31, 32], making recovery tractable.

We stress that traditional CT methods cannot apply to cloud recovery. To see this, let $L^{\mathrm{sun}}$ be the solar irradiance. Let $a^{\mathrm{ground}} \approx 0.05$ be the ground albedo. The range of the optical depth $T^{\mathrm{cloud}}$ of warm clouds is typically 10-100. The albedo of warm clouds is $a^{\mathrm{cloud}} \approx 0.5$ . The radiance directly transmitted from the ground through a cloud to a sensor above is

$$\begin{array}{cccc}& D\sim L^{\mathrm{s}\mathrm{u}\mathrm{n}}{a}^{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}\exp (-{\mathrm{T}}^{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{u}\mathrm{d}})\mathrm{.}& & \text{(9)}\end{array}D\; \backslash sim\; L\; ^\; \{\backslash mathrm\; \{s\; u\; n\}\}\; a\; ^\; \{\backslash mathrm\; \{g\; r\; o\; u\; n\; d\}\}\; \backslash exp\; (-\; \backslash mathrm\; \{T\}\; ^\; \{\backslash mathrm\; \{c\; l\; o\; u\; d\}\}).\; \backslash tag\; \{9\}$$

Sunlight reflected above by a cloud has radiance

$$\begin{array}{cccc}& S\sim L^{\text{s u n}}{a}^{\text{c l o u d}}\mathrm{.}& & \text{(10)}\end{array}S\; \backslash sim\; L\; ^\; \{\backslash text\; \{s\; u\; n\}\}\; a\; ^\; \{\backslash text\; \{c\; l\; o\; u\; d\}\}.\; \backslash tag\; \{10\}$$

From these orders of magnitude, $D \ll S$ . The measured signal is dominated by $S$ . Suppose a naive approach following traditional CT, associating a measured signal with direct transmission i.e., $S \approx D$ . Then from Eqs. (9,10), CT estimates $\hat{\mathrm{T}}^{\mathrm{cloud}} \approx \log(a^{\mathrm{ground}} / a^{\mathrm{cloud}})$ . From these orders of magnitude, $\hat{\mathrm{T}}^{\mathrm{cloud}} < 0$ , which is not-physical. Thus, traditional CT is irrelevant for cloud tomography in visible light.

2.4. Temporal Sampling of a Random Object

A temporal sample indexed $l$ corresponds to continuous time $t_l^{\prime}$ . The time interval between consecutive samples is $T = |t_{l + 1}^{\prime} - t_l^{\prime}|$ . Consider a continuously varying object $\beta_{t}$ . A temporal sample is denoted $\beta_{t'}^{\mathrm{sample}}$ . The Nyquist sampling theorem [40] relates to objects whose time-spectrum is limited to temporal frequencies $\omega$ satisfying $|\omega| < B$ , where $B$ is a cutoff frequency. Then, time domain samples satisfying $T \leq (2B)^{-1}$ can yield reconstruction of $\beta_{t}$ using a linear superposition:

$$\begin{array}{cccc}& {\beta }_t(\mathbf{x})\sim \sum _{t^{\mathrm{\prime }}}{w}_t(t^{\mathrm{\prime }}\mid T){\beta }_{t^{\mathrm{\prime }}}^{\text{s a m p l e}}(\mathbf{x})\mathrm{.}& & \text{(11)}\end{array}\backslash beta\_\; \{t\}\; (\backslash mathbf\; \{x\})\; \backslash sim\; \backslash sum\_\; \{t\; ^\; \{\backslash prime\}\}\; w\; \_\; \{t\}\; \backslash left(t\; ^\; \{\backslash prime\}\; \backslash mid\; T\backslash right)\; \backslash beta\_\; \{t\; ^\; \{\backslash prime\}\}\; ^\; \{\backslash text\; \{s\; a\; m\; p\; l\; e\}\}\; (\backslash mathbf\; \{x\}).\; \backslash tag\; \{11\}$$

There, the superposition uses $w_{t}(t^{\prime}|T) = \mathrm{sinc}[(t - t^{\prime}) / T]$ .

There is a generalization, however, to cases where the object $\beta_{t}$ is random and not strictly band-limited. The temporal auto-correlation of $\beta_{t}$ is

$$\begin{array}{cccc}& \alpha (\tau \mathrm{\mid }\sigma )={⟨{\beta }_t(\mathbf{x}),{\beta }_{t-\tau }(\mathbf{x})⟩}_{t,\mathbf{x}}\mathrm{.}& & \text{(12)}\end{array}\backslash alpha\; (\backslash tau\; |\; \backslash sigma)\; =\; \backslash left\backslash langle\; \backslash beta\_\; \{t\}\; (\backslash mathbf\; \{x\}),\; \backslash beta\_\; \{t\; -\; \backslash tau\}\; (\backslash mathbf\; \{x\})\; \backslash right\backslash rangle\_\; \{t,\; \backslash mathbf\; \{x\}\}.\; \backslash tag\; \{12\}$$

The function $\alpha(\tau|\sigma)$ generally decays with $|\tau|$ , where $\sigma$ is the effective decay time of $\alpha$ , termed correlation time. Two limiting cases are illustrative. For $\sigma \to \infty$ , we have $\alpha(\tau|\sigma) \to \infty$ constant. This means that the object $\beta$ is effectively static. In contrast, for $\sigma \to 0$ , we have $\alpha(\tau|\sigma) \to \delta(t - t')$ , i.e., a Dirac delta function. This means that the object $\beta$ varies so fast, that at any time $t$ its state is uncorrelated to the state at other times.

Once again, $\beta_{t}$ can be linearly reconstructed from temporal samples using Eq. (11), but $w_{t}(t^{\prime}|T)$ can be general. Any sampling rate and reconstruction kernel can be used,

Figure 2. Illustration of Eq. (13) for Gaussian or triangular spectra.

but then there is a reconstruction error. The reconstruction mean squared error (MSE) has a bound [36]. Let $\Lambda (\omega)$ be the time-spectrum of the random object, i.e., the Fourier transform of $\alpha (\tau |\sigma)$ . The MSE bound [36] is then

$$\begin{array}{cccc}& \mathrm{MSE}(T)=\frac{T}{2\pi }{\int }_{-\pi }^{\pi }[\mathrm{\Lambda }(\omega )-\frac{\sum _{q\in \mathbb{Z}}{\mathrm{\Lambda }}^2[\omega -(2\pi \mathrm{/}T)q]}{\sum _{q\in \mathbb{Z}}\mathrm{\Lambda }[\omega -(2\pi \mathrm{/}T)q]}]d\omega \mathrm{.}& & \text{(13)}\end{array}\backslash operatorname\; \{M\; S\; E\}\; (T)\; =\; \backslash frac\; \{T\}\{2\; \backslash pi\}\; \backslash int\_\; \{-\; \backslash pi\}\; ^\; \{\backslash pi\}\; \backslash left[\; \backslash Lambda\; (\backslash omega)\; -\; \backslash frac\; \{\backslash sum\_\; \{q\; \backslash in\; \backslash mathbb\; \{Z\}\}\; \backslash Lambda^\; \{2\}\; [\; \backslash omega\; -\; (2\; \backslash pi\; /\; T)\; q\; ]\}\{\backslash sum\_\; \{q\; \backslash in\; \backslash mathbb\; \{Z\}\}\; \backslash Lambda\; [\; \backslash omega\; -\; (2\; \backslash pi\; /\; T)\; q\; ]\}\; \backslash right]\; d\; \backslash omega\; .\; \backslash tag\; \{13\}$$

For illustration, Fig. 2 plots $\mathrm{MSE}(T)$ for objects that have Gaussian or triangular spectra. When $T < \sigma$ , the error is negligible, but error accumulates significantly as the sampling interval $T$ increases beyond $\sigma$ . Hence, to keep reconstruction error small, an efficient temporal sampling interval should satisfy $T \approx \sigma$ .

3. Clouds: Correlation Time and Sampling

Warm convective clouds are governed by air turbulence of decameter scale. In these scales [13], the correlation time of content in a voxel is about 20 to 50 seconds. This indicates that $4D$ spatiotemporal clouds can be recovered well using $4D$ spatiotemporal samples, if the temporal samples are about 25 seconds apart. The lifetime of a warm convective cloud is typically measured in minutes.

Consider a cloud simulation, described in detail in Sec. 6. The cloud evolves for about 10 minutes. For each cloud voxel, we calculated the temporal auto-correlation of $\mathcal{L}t$ . Similarly, temporal auto-correlations were derived for horizontally-averaged $r_t^{\mathrm{e}}$ and $v{t}^{\mathrm{e}}$ . The auto-correlation functions of $\mathcal{L}t$ and $r_t^{\mathrm{e}}$ are plotted in Fig. 3a (Eq. 12). The autocorrelation function of $r_t^{\mathrm{e}}$ behaves similarly to that of $v{t}^{\mathrm{e}}$ . Clearly, the correlation times of $r_t^{\mathrm{e}}$ and of $v_{t}^{\mathrm{e}}$ are very long, comparable to the lifetime of a cloud, and longer than the typical time it takes to acquire multi-view data of clouds. Hence, when recovering microphysical parameters $r^{\mathrm{e}}$ and $v^{\mathrm{e}}$ , we neglect temporal variations.

On the other hand, $\mathcal{L}_t$ has a short correlation time: $\sigma \sim 25\mathrm{sec}$ . Hence, 4D recovery is necessary for $\mathcal{L}_t$ , if data is sparsely sampled in time. Following the conclusion of Sec. 2.4, it is advisable to sample warm convective clouds at temporal sampling interval of $T \approx \sigma \approx 25\mathrm{sec}$ .

From Sec. 2.4, at an arbitrary $t$ , reconstruction of $\mathcal{L}_t$

(a)

(b)
Figure 3. (a) Auto-correlation of cloud field microphysics. The auto-correlation of LWC and $r^{\mathrm{e}}$ decrease to 0.5 after 25 sec and 185 sec, respectively. Typical cloud multi view sensing is about a minute, during which the LWC auto-correlation decreases to 0.26, while the auto-correlation of $r^{\mathrm{e}}$ is 0.82. (b) MSE (Eq. 13) of LWC and $r^{\mathrm{e}}$ .

from samples would use a temporal kernel whose effective width is $\sigma$ . In CT, however, we do not have direct access to the object samples. We only have noisy projections of radiance scattered by the object. Thus, reconstruction does not involve direct application of the optimal kernel [36], which achieves the bound in Eq. (13). Reconstruction involves a tomographic process, which we describe in Sec. 5.

4. Tomographic Angular Extent

Section 2.4 dealt with sampling of an object, as if 4D measurements are done in-situ. However, in CT, we have no direct access to $\pmb{\nu}_t$ : we only measure projections $\pmb{y}_t$ . As we discuss now, projections must have a wide angular breadth, while object evolution is small.

Consider an extreme case. Let a cloud be temporally constant and reside only in a single voxel, over the ocean. Viewed from space by two cameras simultaneously, cloud recovery here amounts to triangulation. In triangulation, the best cloud-localization resolution is obtained if the angular range between the two cameras is $90^{\circ}$ . At small baselines, localization decreases linearly with a decreasing angular extent. When more than two cameras operate, the trend is similar. Consider an error measure that has been used in 3D cloud scattering CT [20, 29, 30, 32],

$$\begin{array}{cccc}& \epsilon =\frac{\mathrm{\parallel }{\mathcal{L}}^{\text{t r u e}}-\widehat{\mathcal{L}}{\mathrm{\parallel }}_1}{\mathrm{\parallel }{\mathcal{L}}^{\text{t r u e}}{\mathrm{\parallel }}_1}\mathrm{.}& & \text{(14)}\end{array}\backslash varepsilon\; =\; \backslash frac\; \{\backslash |\; \backslash boldsymbol\; \{\backslash mathcal\; \{L\}\}\; ^\; \{\backslash text\; \{t\; r\; u\; e\}\}\; -\; \backslash hat\; \{\backslash boldsymbol\; \{\backslash mathcal\; \{L\}\}\}\; \backslash |\; \_\; \{1\}\}\{\backslash |\; \backslash boldsymbol\; \{\backslash mathcal\; \{L\}\}\; ^\; \{\backslash text\; \{t\; r\; u\; e\}\}\; \backslash |\; \_\; \{1\}\}.\; \backslash tag\; \{14\}$$

Figure 4. A static heterogeneous cloud and a single-voxel "cloud" (having size $20\mathrm{m}\times 20\mathrm{m}\times 20\mathrm{m}$ ) are recovered from nine viewpoints using [30]. The plots are of errors defined in Eq. (14).

Here $\hat{\mathcal{L}}$ is the estimated 3D cloud LWC field. Fig. 4 plots $\varepsilon$ when CT attempts to recover a single-voxel of a static cloud (extending $20\mathrm{m}$ ), when 9 cameras surround it from $500\mathrm{km}$ away, while the true $r^{\mathrm{e}}$ , $v^{\mathrm{e}}$ are used. Above $\approx 60^{\circ}$ total angular extent, recovery reaches a limiting excellent quality, but quality is very poor at narrow angle spans.

In general, objects have multiple voxels. In linear-CT (as in medical X-ray CT), information loss due to limited-angle imaging is known as the missing cone of frequencies [1,33]. In scattering CT, with the exception of very sparse objects, the missing cone linear theory does not apply. While a pure theory for nonlinear CT does not exist, the implication of angular span can be assessed numerically. Let us consider a static cloud in a single state, simulated as in Sec. 6. There are no dynamics. We can then see how angular sampling alone affects the quality of recovery. Results are shown in Fig. 4. There is a marked degradation of quality if the angular extent is narrow.

So far, this section dealt with static clouds. Clouds are considered nearly static between times $t, t'$ if $|t - t'| < \sigma$ . The viewing angular extent covered in those times (and in intermediate times) is denoted $\Theta(t, t')$ , in radians. So, within time span approximately equal to $\sigma$ , good recovery can be achieved only if $2\Theta(t, t') / \pi$ is large. If it is low, then spatial (altitude) resolution in CT recovery is lost. Most CT systems cover wide angular extent, eventually. So, quality is set by the angular rate. Define a dimensionless figure

$$\begin{array}{cccc}& \rho =\frac{2\mathrm{\Theta }(t,t^{\mathrm{\prime }})}{\pi }\frac{\sigma }{\mid t-t^{\mathrm{\prime }}\mid }\mathrm{.}& & \text{(15)}\end{array}\backslash rho\; =\; \backslash frac\; \{2\; \backslash Theta\; \backslash left(t\; ,\; t\; ^\; \{\backslash prime\}\backslash right)\}\{\backslash pi\}\; \backslash frac\; \{\backslash sigma\}\{\backslash left|\; t\; -\; t\; ^\; \{\backslash prime\}\; \backslash right|\}.\; \backslash tag\; \{15\}$$

Good 4D recovery requires $\rho \gtrsim 1$ , while $T \lesssim \sigma$ (Sec. 3). The more these conditions are violated, the worse 4D CT is expected to perform.

5. 4D Scattering Tomography

5.1. Estimation of the Liquid Water Content

We now generalize Eq. (8) to 4D CT. Data is captured sequentially at the time set $\mathcal{T} = {t_1, t_2, \dots, t_{N^{\mathrm{state}}}}$ , while the object evolves. At each $t' \in \mathcal{T}$ , the object is viewed simultaneously from a set of viewpoints $\mathcal{C}{t'}$ , yielding a concatenated data vector $\boldsymbol{y}{t'}$ . At that time, the modelled LWC

is the vector $\mathcal{L}{t'}$ . The set of LWC values in all sampled times is denoted $\mathcal{B} = {\mathcal{L}{t'}}_{t' \in \mathcal{T}}$ . At optimization iteration $k$ , the set of all modelled LWC values is $\mathcal{B}(k)$ .

Let $w_{t}(t^{\prime}|\sigma)$ be a normalized weighting function. Consider the vector

$$\begin{array}{cccc}& {\mathbf{g}}_t(\mathcal{B})=\sum _{t^{\mathrm{\prime }}\in \mathcal{T}}{w}_t(t^{\mathrm{\prime }}\mid \sigma )\frac{\mathrm{\partial }\mathcal{F}\left({\mathcal{L}}_{t^{\mathrm{\prime }}}\right)}{\mathrm{\partial }{\mathcal{L}}_{t^{\mathrm{\prime }}}}[\mathcal{F}\left({\mathcal{L}}_{t^{\mathrm{\prime }}}\right)-{\mathit{y}}_{t^{\mathrm{\prime }}}],& & \text{(16)}\end{array}\backslash mathbf\; \{g\}\; \_\; \{t\}\; (\backslash mathcal\; \{B\})\; =\; \backslash sum\_\; \{t\; ^\; \{\backslash prime\}\; \backslash in\; \backslash mathcal\; \{T\}\}\; w\; \_\; \{t\}\; \backslash left(t\; ^\; \{\backslash prime\}\; \backslash mid\; \backslash sigma\backslash right)\; \backslash frac\; \{\backslash partial\; \backslash mathcal\; \{F\}\; \backslash left(\backslash boldsymbol\; \{\backslash mathscr\; \{L\}\}\; \_\; \{t\; ^\; \{\backslash prime\}\}\backslash right)\}\{\backslash partial\; \backslash boldsymbol\; \{\backslash mathscr\; \{L\}\}\; \_\; \{t\; ^\; \{\backslash prime\}\}\}\; \backslash left[\; \backslash mathcal\; \{F\}\; \backslash left(\backslash boldsymbol\; \{\backslash mathscr\; \{L\}\}\; \_\; \{t\; ^\; \{\backslash prime\}\}\backslash right)\; -\; \backslash boldsymbol\; \{y\}\; \_\; \{t\; ^\; \{\backslash prime\}\}\; \backslash right],\; \backslash tag\; \{16\}$$

and an iteration move to assess $\mathcal{L}_t$ at arbitrary time $t$

$$\begin{array}{cccc}& {\mathcal{L}}_t(k+1)={\mathcal{L}}_t(k)-\eta {\mathbf{g}}_t[\mathcal{B}(k)]\mathrm{.}& & \text{(17)}\end{array}\backslash boldsymbol\; \{\backslash mathcal\; \{L\}\}\; \_\; \{t\}\; (k\; +\; 1)\; =\; \backslash boldsymbol\; \{\backslash mathcal\; \{L\}\}\; \_\; \{t\}\; (k)\; -\; \backslash eta\; \backslash mathbf\; \{g\}\; \_\; \{t\}\; [\; \backslash mathcal\; \{B\}\; (k)\; ].\; \backslash tag\; \{17\}$$

We use Eqs. (16,17) iteratively for 4D scattering tomography. We use $w_{t}(t^{\prime}|\sigma)$ which decays in effective time $\sigma$ .

Sec. 2.4 serves as a guideline for the kernel properties. However, as said in Sec. 3, we do not have access to the LWC, but to noisy images. Hence we approximate the optimal kernel using a cropped Gaussian,

$$\begin{array}{cccc}& w_t(t^{\mathrm{\prime }}\mid \sigma )=s\exp (-\frac{{\mid t-t^{\mathrm{\prime }}\mid }^2}{2{\sigma }^2})\mathrm{.}& & \text{(18)}\end{array}w\; \_\; \{t\}\; \backslash left(t\; ^\; \{\backslash prime\}\; \backslash mid\; \backslash sigma\backslash right)\; =\; s\; \backslash exp\; \backslash left(-\; \backslash frac\; \{\backslash left|\; t\; -\; t\; ^\; \{\backslash prime\}\; \backslash right|\; ^\; \{2\}\}\{2\; \backslash sigma^\; \{2\}\}\backslash right).\; \backslash tag\; \{18\}$$

Here $s$ is a normalization factor, set so $\sum_{t' \in \mathcal{T}} w_t(t'| \sigma) = 1$ while $w_t(t' \notin \mathcal{T}|\sigma) = 0$ . In the limiting case where $w_{t'}(t|\sigma) \to 1 / N^{\mathrm{state}}$ , the cloud is considered static (correlation time is very long). Then, Eqs. (16,17) degenerate to Eqs. (7,8). That is, recovery of $\mathcal{L}t$ is tightly related to ${\mathcal{L}{t'}}_{t' \in \mathcal{T}}$ . More generally, the correlation time $\sigma$ has a finite value. Thus, when estimating $\mathcal{L}t$ , there is gradually lower information carried by $\mathcal{L}{t'}$ , as $|t' - t|$ increases, and particularly as $|t' - t| > \sigma$ . Thus, an iteration to refine an estimate of $\mathcal{L}t$ should give less weight to $\mathcal{L}{t'}$ . Eqs. (16,17) provide this capacity in a natural way.

Eq. (16) is equivalent to a gradient of a cost function. This interpretation is detailed in the Supplementary material. The complexity of Eqs. (16,17) is similar to static 3D CT (5), as discussed in the Supplementary material. We performed iterations using L-BFGS-B [55]. Following [30], prior to iterations, the set of voxels to estimate is bounded using space-carving [25]. Space-carving bounds a 3D shape by back-projecting multi-view images. A voxel is labeled as belonging to the object, if the number of back-projected rays that intersect this voxel is greater than a threshold. We adapt this bounding to dynamic scenarios using two ways: $(i)$ by setting a coarse spatial grid for carving and $(ii)$ using a low threshold for labeling voxels as potentially being part of a cloud.

5.2. Estimation of the Effective Radius

We exploit simplifications with regards to $r^{\mathrm{e}}$ and $v^{\mathrm{e}}$ . As discussed in Sec. 3, we may treat $r^{\mathrm{e}}$ and $v^{\mathrm{e}}$ as time-invariant per voxel, in the scale of minutes. Furthermore, by Sec. 2.1, $r^{\mathrm{e}}$ can be approximated as laterally uniform (varying vertically) [21], using a parametric model (1) [47]. Moreover, it can be often assumed that $v^{\mathrm{e}} = 0.1$ [30].

Hence, we focus here on time-invariant estimation of $r^{\mathrm{e}}$ , using the model in Eq. (1), namely, estimating the global parameters $\xi_{\mathrm{r}}, r_{0}^{\mathrm{e}}, z_{0}$ . This is done intermittently, among optimization of the LWC (Sec. 5.1). Overall we estimate both LWC and $r^{\mathrm{e}}$ using the following steps:

{i} Perform exhaustive search on $\xi_{\mathrm{r}},\xi_{\mathcal{L}},r_0^{\mathrm{e}},z_0$ , in a coarse grid of values, to minimize Eq. (6), assuming time invariance. This sets the initial values of $r^{\mathrm{e}}$ ${\mathcal{L}t}{t\in \mathcal{T}}$
${\mathrm{ii}}$ Hold $r^{\mathrm{e}}$ temporally fixed. Run the gradient-based optimization described in Sec. 5.1, to estimate ${\mathcal{L}t}{t\in \mathcal{T}}$ in 4D. Here 10 iterations are run.
${\mathrm{iii}}$ Hold ${\mathcal{L}t}{t\in \mathcal{T}}$ fixed. Perform exhaustive line-search on $\xi_{\mathrm{r}}$ , in a fine grid of values, to minimize Eq. (6).
{iv} Return to step {ii}.

6. Simulations

We now test the feasibility of 4D cloud scattering tomography. The tests demonstrate the effect of varying $\sigma$ , the kernel parameter in Eq. (18), and the importance of the angular breadth. The evolving concentration of cloud water droplets is the main unknown we sense and seek.

For realistic complexity, we use a rigorous simulation based on cloud physics. Clouds are simulated using the System of Atmospheric Modeling (SAM) [24], which is a non-hydrostatic, inelastic large eddy simulator (LES) [18, 37, 50]. It describes the turbulent atmosphere using equations of momentum, temperature, water mass balance and continuity. We couple SAM to a spectral (bin) microphysical model (HUJI SBM) [11, 23] of the droplets' size. It propagates the evolution of the droplet size distribution, by solving the equations for nucleation, diffusional growth, collision-coalescence and break-up. This is done on a logarithmic grid of 33 size bins in the domain $[2\mu \mathrm{m}, 3.2\mathrm{mm}]$ .

The simulation runs according to the BOMEX case [45] of trade wind cumulus clouds near Barbados. Humidity and potential temperature profiles are used as initial conditions, while the surface fluxes and large-scale forcing are constant. The mean horizontal background wind is zero. The horizontal boundary condition is cyclic. The domain is $5.12\mathrm{km}$ long (cloud diameter is $\approx 800\mathrm{m}$ ) at $10\mathrm{m}$ resolution. The vertical resolution is $10\mathrm{m}$ from sea level to $3\mathrm{km}$ , coarsening to $50\mathrm{m}$ above. Cloud tops reach $2\mathrm{km}$ . The simulation expresses an hour, of which 30 minutes includes the cloud's lifetime. The temporal resolution is 0.5 sec.

We present results using two different time-varying clouds: Cloud (i) has size $43 \times 30 \times 45$ voxels (see Fig. 5). Cloud (ii) has size $60 \times 40 \times 45$ (see Supplementary material). A voxel size is $10\mathrm{m} \times 10\mathrm{m} \times 10\mathrm{m}$ .

Figure 5. Cloud (i). Results of recovery by the Baseline and Setup A are compared to the ground-truth by a 3D presentation and scatter plots that use $20%$ of the data points, randomly selected for display clarity. The Baseline and Setup A scatter plot correlations are 0.9 and 0.86, respectively.

6.1. Rendered Measurements

The scene is irradiated by the sun, whose illumination angle changes in time, relative to the Earth's coordinates, while cameras overfly the evolving cloud. The solar trajectory in Earth coordinates corresponds to Feb/03/2013 at 13:54:30 - 14:01:00 local time, around 38N 123W. We tested several types of imaging setups :

Setup A: Three satellites orbit at $500\mathrm{km}$ altitude, one after the other. Their velocity is $7.35\mathrm{km/s}$ . The orbital arc-length between nearest- neighboring satellites is $500\mathrm{km}$ . At mid-time of the simulation, $t = (t_1 + t_{N\text{state}})/2$ , the setup is symmetric around the nadir direction. Then, the setup spans an angular range of $114^{\circ}$ . Each satellite carries a perspective camera. The camera resolution is such that at nadir view, a pixel corresponds to $10\mathrm{m}$ at sea level. Images are taken every 10 sec, during 60 sec, i.e., $N^{\text{state}} = 7$ . This setup is illustrated in Fig. 1.

Figure 6. Illustration of Setup C. A domain is viewed at 21 pushbroom angles, sequentially.

Baseline: The baseline uses all the accumulated 21 viewpoints of Setup A. However, all viewpoints here have perspective cameras that simultaneously acquire the cloud. In other words, this baseline is not prone to errors that stem from temporal sampling. The baseline is used for recovery only at time $t = (t_1 + t_{N\mathrm{state}}) / 2$ .

Setup B: This setup is similar to Setup A, but it uses only two satellites. Thus, at mid-time of the simulation, the setup spans a $57^{\circ}$ angular range.

Setup C: A single camera, similar to the Multi-angle Spectro-Polarimeter Imager (AirMSPI) [6], is mounted on an aircraft flying $154^{\circ}$ relative to North at $20\mathrm{km}$ altitude. Imaging has a pushbroom scan geometry, having $10\mathrm{m}$ spatial resolution at Nadir view. AirMSPI scans view angles in a step-and-stare mode [6]. Based on AirMSPI PODEX campaign [5], we set 21 viewing angles along-track: $\pm 65^{\circ}$ , $\pm 62^{\circ}$ , $\pm 58^{\circ}$ , $\pm 54^{\circ}$ , $\pm 44^{\circ}$ , $\pm 38^{\circ}$ , $\pm 30^{\circ}$ , $\pm 21^{\circ}$ , $\pm 11^{\circ}$ off-nadir and $0^{\circ}$ (nadir). For example, three sample angles are illustrated in Fig. 6. It takes $\approx 1$ sec to scan a cloud domain in any single view angle, during which the cloud and solar directions are assumed constant. Dynamics are noticeable between view angles.

A spherical harmonic discrete ordinate method (SHDOM) code [10] provides the numerical forward model $\mathcal{F}$ . Simulated measurements ${\pmb{y}t}{t\in \mathcal{T}}$ include noise. The noise model follows the AirMSPI sensors parameters [6, 48]. There, the sensor full-well depth is 200,000 photo-electrons, readout noise has a standard deviation of 20 electrons, and the overall readout is quantized to 9 bits.

6.2. Results of 4D Tomography of the LWC

The rendered and noisy images, in the spectral band of $\lambda = 660\mathrm{nm}$ , served as input to 4D tomographic reconstruction. The voxel size in the recovery was set to $10\mathrm{m} \times 10\mathrm{m}$ horizontal, $25\mathrm{m}$ vertical and 10 sec resolution. For parallelization, optimization ran on a computer cluster, where each computer core was dedicated to rendering a modelled image from a distinct angle.

In this section, we recover only the cloud LWC. Hence, we set $r^{\mathrm{e}} = 10\mu \mathrm{m}$ and $v^{\mathrm{e}} = 0.1$ to be uniform constants during optimization (though they were not uniform in the

Figure 7. Cloud (i). The errors $\varepsilon_{t}$ are marked by colored circles, whose saturation decays the farther the sampling time is from $(t_{1} + t_{N\text{state}}) / 2$ . The measure $\varepsilon$ is marked by solid or dashed lines, with corresponding colors. The setting $\sigma = \infty$ refers to the solution by the state of the art, i.e. 3D static scattering tomography.

Figure 8. Setup C. The error $\varepsilon$ of Cloud (i), for different acquisition inter-angular temporal intervals, $T$ . The setting $\sigma = \infty$ refers to the solution by the state of the art, i.e. 3D static scattering tomography.

simulated data). The LWC optimization was initialized by ${\mathcal{L}t}{\tau \in \mathcal{T}} = 0.01\mathrm{g / m}^3$ . Convergence was reached in several dozen iterations. Depending on the number of input images, it took between minutes to a couple of hours to converge.

For result assessment, we generalize Eq. (14) to the whole sample set $t\in \mathcal{T}$ by

$$\begin{array}{cccc}& \epsilon =\frac{1}{N^{\text{s t a t e}}}\sum _{t\in \mathcal{T}}{\epsilon }_t,\text{w h e r e},{\epsilon }_t=\frac{{\parallel {\mathcal{L}}_t^{\text{t r u e}}-{\widehat{\mathcal{L}}}_t\parallel }_1}{{\parallel {\mathcal{L}}_t^{\text{t r u e}}\parallel }_1}\mathrm{.}& & \text{(19)}\end{array}\backslash varepsilon\; =\; \backslash frac\; \{1\}\{N\; ^\; \{\backslash text\; \{s\; t\; a\; t\; e\}\}\}\; \backslash sum\_\; \{t\; \backslash in\; \backslash mathcal\; \{T\}\}\; \backslash varepsilon\_\; \{t\}\; ,\; \backslash text\; \{w\; h\; e\; r\; e\},\; \backslash varepsilon\_\; \{t\}\; =\; \backslash frac\; \{\backslash left\backslash |\; \backslash boldsymbol\; \{\backslash mathcal\; \{L\}\}\; \_\; \{t\}\; ^\; \{\backslash text\; \{t\; r\; u\; e\}\}\; -\; \backslash hat\; \{\backslash boldsymbol\; \{\backslash mathcal\; \{L\}\}\}\; \_\; \{t\}\; \backslash right\backslash |\; \_\; \{1\}\}\{\backslash left\backslash |\; \backslash boldsymbol\; \{\backslash mathcal\; \{L\}\}\; \_\; \{t\}\; ^\; \{\backslash text\; \{t\; r\; u\; e\}\}\; \backslash right\backslash |\; \_\; \{1\}\}\; .\; \backslash tag\; \{19\}$$

From Sec. 2.4, we assess that a value $\sigma \sim 20$ sec is natural. Indeed, this is supported numerically in the plots of $\varepsilon_{t}, \varepsilon$ for Cloud (i) (Fig 7). A naive solution may only use measurements captured at each sampled time $t$ , to solve $\mathcal{L}t$ , independently of other times. This solution is reached by $\sigma = 0$ and presented in Figs. 7 and 8. The 3D tomographic results of Cloud (i) at $t = (t_1 + t{N^{\mathrm{state}}}) / 2$ using Setup A are shown in Fig. 5. Recovery used $\sigma = 20$ sec. In the Supple

(a)
(b)
Figure 9. Cloud (i), joint recovery of $r^{\mathrm{e}}$ , $\mathcal{L}$ . (a) The errors $\varepsilon_{t}$ of the LWC are marked by colored circles, whose saturation decays the farther the sampling time is from $(t_1 + t_{N\mathrm{state}}) / 2$ . The measure $\varepsilon$ is marked by solid or dashed lines, with corresponding colors. The setting $\sigma = \infty$ refers to solution by the state of the art, i.e. 3D static scattering tomography. (b) The estimated $r^{\mathrm{e}}$ is presented in blue dashed line. The true horizontally-averaged $r_t^{\mathrm{e}}$ are presented in solid lines, for different time samples.

mentary material we present analogous plots for Cloud (ii) and additional results using Setup B.

Setup C uses a single platform, which is challenging. Results depend significantly on how fast the aircraft flies, i.e., how long it takes to capture the cloud from a variety of angles (up to 21 angles). Fig. 8 compares the results for inter-angle time interval of 5 sec, 10 sec and 20 sec. As expected, quality $(\varepsilon)$ improves with velocity. Moreover, if the camera moves slowly (long time interval between angular samples), results improve by using a longer temporal support, observing the cloud from a wider angular range, despite its dynamics.

6.3. Microphysics Estimation

In this section, we recover both LWC and $r^{\mathrm{e}}$ as described in Sec. 5.2. We use Setups A, B and Baseline, as described in Sec. 6.1, with an additional spectral band at $\lambda = 865 \mathrm{~nm}$ . Fig. 9 shows the results for Cloud (i).

(b)

(a)

(c)

(d)

Figure 10. (a) Recovered 3D LWC field using real data. (b) A raw AirMSPI nadir image. Corresponding rendered views of a cloud, that was estimated using data that had excluded the nadir, either by our 4D CT approach (c) or current static 3D CT (d). Gamma correction was applied on (b,c,d) for display clarity. (e) A scatter plot of rendered vs. raw AirMSPI images at nadir. The scatter plot correlations of our solution and the static solution are 0.862 and 0.656, respectively.

7. Experiment: Real World AirMSPI Data

We follow the experimental approach of [29], and use real-world data acquired by JPL's AirMSPI, which flies on board NASA's ER-2. The geometry is exactly as described in Setup C in Sec. 6.1, including location and time. An atmospheric domain of size $1.5\mathrm{km} \times 2\mathrm{km} \times 2\mathrm{km}$ in the East-North-Up coordinates is examined. We discretized the domain to $80 \times 80 \times 80$ voxels. Because $N^{\mathrm{states}} = 21$ , the total number of unknowns is 10,752,000.

The inter-angle time interval in this experiment is around 20 sec. Based on Fig. 8, we set here $\sigma = 60$ sec in Eq. (18). We want to focus on dynamic tomography of the evolving cloud, and not on global motion due to wind in the cloud field. Hence, we used the pre-processing approach of [29] to align the cloud images. Additionally, the ground albedo is estimated to be 0.04. The pre-processing and albedo estimation are described in the Supplementary material.

A recovered volumetric reconstruction for one time instant is displayed in Fig. 10. We have no ground-truth for the cloud content in this case. Hence we check for consistency using cross-validation. For this, we excluded the nadir image (Fig. 10b) from the recovery process. Thus tomography used 20 out of the 21 raw views. Afterward, we placed the recovered cloud in SHDOM physics-based ren

	+54° view	nadir view	-54° view
Static solution	1.73	0.94	0.61
Ours	0.96	0.38	0.24

Table 1. Analysis of empirical data in different view angles. Quantitative fit (6) of our 4D result to the data, as compared to the error of state-of-the-art static 3D CT.

derer [10], to generate the missing nadir view. The result is then compared to the ground-truth missing view. Fig. 10 compares the result of this process for two solutions: our 4D tomographic solution, and the state-of-the-art, i.e., 3D static scattering tomography.

The same cross-validation process was repeated for the $\pm 54^{\circ}$ view angles. Quantitatively, we measure the fitting error using Eq. (6). The results are summarized in Table 1.

8. Discussion

We derive a framework for 4D CT of dynamic objects that scatter, using moving cameras. The natural temporal evolution of an object indicates the temporal and angular sampling needed for a good reconstruction. Given these conditions, 4D CT recovery can be done, even with a small number of cameras. Fig. 7 and specifically Setup A may indicate that 4D CT may be achieved using $\sigma = 0$ and strong priors. This possibility should be a welcome topic for further computer vision research. In a sense, our work also uses a cloud prior, which is temporal correlation in clouds (Sec. 3): the correlation is analyzed using signal processing tools, and implemented by gradient weights $w_{t}$ .

The model introduced in Sec. 5 to account for temporal correlations is independent of the differentiable renderer. Eqs. (16-18) can be combined with any differentiable renderer that can compute the derivative term $\partial \mathcal{F} / \partial \mathcal{L}$ in Eq. (16). This can be either a differentiable renderer based on SHDOM, or a Monte Carlo differentiable renderer like those of [15, 39]. Additionally, some of our findings may be helpful in other fields. Bio-medical CT [7] and flow imaging [26] already have tools exploiting controlled illumination. Elements of this work may add to that toolkit.

Acknowledgements: We thank Aviad Levis and Tali Treibitz for their advice, and Johanan Erez, Ina Talmon and Daniel Yagodin for technical support. Yoav Schechner is the Mark and Diane Seiden Chair in Science at the Technion. He is a Landau Fellow supported by the Taub Foundation. His work was conducted in the Ollendorff Minerva Center. Minvera is funded through the BMBF. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (CloudCT, grant agreement No. 810370).

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