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1. Bibliographic Information

Title: Simulating Anisoplanatic Turbulence by Sampling Inter-modal and Spatially Correlated Zernike Coefficients
Authors: Nicholas Chimitt (Student Member, IEEE) and Stanley H. Chan (Senior Member, IEEE). Affiliations are not explicitly stated, but the acknowledgements suggest a connection to Purdue University (where Prof. Chan is faculty) and collaborations with the Air Force Research Lab.
Journal/Conference: The paper is available as a preprint on arXiv. The acknowledgements mention that a shorter version was presented at the IEEE International Conference on Computational Photography (ICCP), 2020. ICCP is a reputable conference focusing on the intersection of computer graphics, computer vision, and optics.
Publication Year: 2020
Abstract: The paper presents a computationally efficient, propagation-free method for simulating imaging through atmospheric turbulence. The core innovation is a new technique to draw Zernike coefficients that are both inter-modally and spatially correlated. The authors establish an equivalence between two existing theoretical models—angle-of-arrival correlation (Basu et al., 2015) and multi-aperture correlation (Chanan, 1992)—to derive a covariance matrix for these coefficients. They propose fast and scalable sampling strategies based on this finding. This approach transforms the computationally intensive wave propagation problem into a much faster sampling problem. Experimental results demonstrate that the simulator's output closely matches both theoretical predictions and real-world turbulence data.
Original Source Link: The paper is available at https://arxiv.org/abs/2004.11210v2 (PDF: http://arxiv.org/pdf/2004.11210v2). It is a preprint version of a conference paper.

2. Executive Summary

Background & Motivation (Why):
- Core Problem: Simulating atmospheric turbulence is crucial for developing and testing turbulence mitigation algorithms and for training modern deep learning models. However, existing methods present a difficult trade-off. High-fidelity simulators, like those based on split-step wave propagation, are physically accurate but computationally prohibitive, taking minutes to generate a single image frame. On the other hand, fast methods used in computer graphics often lack physical realism, failing to model the spatially varying nature (anisoplanatism) of turbulence, especially the correlated geometric distortions (tilts) across a wide field of view.
- Gap in Prior Work: There is a need for a simulator that balances physical accuracy, computational speed, and interpretability. This is particularly important for generating large datasets required by learning-based methods.
- Innovation: This paper introduces a propagation-free simulator that directly models the statistical properties of turbulence in the Zernike polynomial domain. Instead of simulating the wave's journey through the atmosphere, it "compresses" the physics into a problem of sampling from a carefully constructed statistical distribution.
Main Contributions / Findings (What):
- A Novel Propagation-Free Simulation Framework: The paper proposes a fast simulator that decouples wavefront distortions into two components:
  1. Geometric Tilts: Spatially correlated geometric shifts across the entire image.
  2. High-Order Aberrations: Spatially varying blurs.
- Theoretical Equivalence for Spatial Correlation: The central technical contribution is a new theoretical link between the angle-of-arrival correlation model of Basu et al. and the multi-aperture correlation model of Chanan. This equivalence allows the authors to derive an analytical expression for the spatial covariance of Zernike tilt coefficients, which was previously challenging.
- Fast and Scalable Sampling: Based on the derived covariance, the authors develop an efficient sampling method using the Fast Fourier Transform (FFT) to generate spatially correlated tilt fields, avoiding the construction of massive covariance matrices.
- Validation: The simulator's output is rigorously validated against established theoretical models (long/short exposure PSFs, tilt statistics) and shows strong visual similarity to both high-fidelity split-step simulators and real-world turbulence imagery.

Foundational Concepts:
- Atmospheric Turbulence: The Earth's atmosphere is not a uniform medium. Pockets of air with different temperatures and pressures create random fluctuations in the refractive index. As light propagates through these pockets, its wavefront gets distorted, leading to image degradation.
- Anisoplanatism vs. Isoplanatism: In an isoplanatic system, the image degradation (blur) is the same across the entire field of view (spatially invariant). In long-range imaging through turbulence, the degradation is anisoplanatic, meaning it varies depending on the viewing angle. An object on the left side of the scene may experience a different blur and shift than an object on the right.
- Point Spread Function (PSF) and Optical Transfer Function (OTF): The PSF describes the response of an imaging system to a point source of light. In turbulence, the PSF is a random, spatially varying blur kernel. The OTF is the Fourier transform of the PSF and is often more convenient for mathematical analysis.
- Zernike Polynomials: A sequence of polynomials that are orthogonal on a unit disk. They are widely used in optics to represent wavefront aberrations (distortions) over a circular aperture because each polynomial term corresponds to a specific type of optical aberration (e.g., piston, tilt, defocus, astigmatism).
- Structure Function: A statistical tool used to describe random processes with stationary increments, like the phase of a light wave in turbulence. For two points separated by a vector $\mathbf{r}$ , the structure function $D_{\phi}(\mathbf{r})$ is the mean squared difference of the phase at those points: $D_{\phi}(\mathbf{r}) = \mathbb{E}[(\phi(\mathbf{x}) - \phi(\mathbf{x}+\mathbf{r}))^2]$ .
- Fried Parameter ( $r_0$ ): A key parameter that quantifies the strength of atmospheric turbulence. It represents the diameter of a circular area over which the wavefront is considered reasonably flat. A smaller $r_0$ indicates stronger turbulence and more severe image degradation.
Previous Works & Differentiation: The paper positions its work in the middle ground between high-precision, slow simulators and low-precision, fast simulators.

该图像是一个流程示意图，展示了不同精度层次的湍流模拟方法，左侧为高精度复杂模型，中间为本文提出的中等精度简单模型，右侧为低精度快速生成数据的模型。图中突出显示了本文方法的优势和应用场景。
- High Precision, Complex Models:
  - Split-Step Propagation: These simulators (e.g., [8], [9]) numerically solve the wave propagation equation by dividing the atmospheric path into multiple "thin screens." Each screen applies a random phase distortion. This method is highly accurate and can model complex phenomena but is extremely slow due to repeated Fourier transforms. Hardie et al. [9] report runtimes of ~120 seconds for a single $256 \times 256$ image.
  - Ray Tracing: Methods like [11] trace light rays through the turbulent medium. They are also accurate but computationally intensive.
- Low Precision, Naive Models:
  - Graphics Rendering: These methods [1], [4] often use generic image processing techniques like non-rigid deformations to create visually plausible "heat haze" effects. While fast, they do not adhere to the physical statistics of atmospheric turbulence.
  - Simple Propagation-Free Simulators:
    - Fraunhofer Institute's simulator [7] assumes tilts are independent and identically distributed (i.i.d.), which is physically inaccurate as real tilts are spatially correlated.
    - Potvin et al. [5] model the PSF as a simple bivariate Gaussian, which is inadequate for moderate to strong turbulence.
- The Proposed Method's Niche: The work by Chimitt and Chan provides a "moderate precision, simple model." It is propagation-free, making it much faster than split-step methods. Unlike naive models, it is built on the physical principles of turbulence (Kolmogorov statistics, Zernike decomposition) and explicitly models both inter-modal (between different Zernike modes) and spatial correlations. This makes it ideal for applications like generating realistic training data for machine learning and performing large-scale evaluations of reconstruction algorithms.

4. Methodology (Core Technology & Implementation)

The core idea is to simulate turbulence by directly sampling Zernike coefficients from a distribution that captures their known statistical correlations, thus bypassing wave propagation simulation. The process is decoupled into handling high-order aberrations (blur) and tilts (geometric distortion).

Principles: Zernike Polynomials for Phase Distortions The random phase distortion $\phi$ $ϕ$ over a circular aperture of radius $R$ $R$ is represented as a sum of Zernike polynomials $Z_j$ $Z_{j}$ : $\phi ( R \pmb { \rho } ) = \sum _ { j = 1 } ^ { N } a _ { j } Z _ { j } ( \pmb { \rho } ) .$
- $\pmb{\rho}$ is the normalized polar coordinate on the unit disk.
- $a_j$ are the Zernike coefficients, which are random variables.
- Crucially, $a_2$ and $a_3$ correspond to horizontal and vertical tilts, while $a_j$ for $j \geq 4$ correspond to higher-order aberrations like defocus, astigmatism, etc., which collectively create blur.

This step generates the spatially varying blur component of the turbulence.

Inter-modal Correlation: The coefficients $a_j$ are not independent. Their correlation is described by the covariance matrix $C$ , where $[C]_{j, j'} = \mathbb{E}[a_j^* a_{j'}]$ . Noll [15] derived an analytical expression for this covariance based on Kolmogorov turbulence statistics. The paper leverages this result.
Key Derivation (Lemma 2): The paper shows that the covariance, originally defined using the phase autocorrelation function, can be equivalently expressed using the phase structure function $D_{\phi}$ . This is a subtle but important theoretical step that connects the covariance calculation directly to Fried's well-known structure function formula: $\mathcal{D}_{\phi}(|\mathbf{f}-\mathbf{f}'|) = 6.88 (|\mathbf{f}-\mathbf{f}'|/r_0)^{5/3}$
Sampling Procedure:
1. Construct the Noll covariance matrix $C$ for the desired number of Zernike modes (e.g., $j=4$ to 36 for blur).
2. Perform a Cholesky decomposition of the matrix: $C = RR^T$ .
3. Draw a vector $\mathbf{b}$ of independent, zero-mean, unit-variance Gaussian random numbers.
4. Generate the correlated Zernike coefficients $\mathbf{a}$ using the transformation: $\mathbf{a} = \sqrt{(D/r_0)^{5/3}} R \mathbf{b}$ The scaling factor $(D/r_0)^{5/3}$ adjusts the variance of the coefficients based on the turbulence strength.
5. These coefficients are used to generate a random PSF. This process is repeated independently for different blocks of the image.

4.2 Sampling Tilts with Spatial Correlations

This is the most innovative part of the paper, generating the global, correlated geometric warp.

The Challenge: To model anisoplanatism correctly, one needs the spatial correlation of the tilt coefficients, i.e., $\mathbb{E}[a_j(\pmb{\theta})a_j(\pmb{\theta}')]$ for $j \in \{2, 3\}$ , where $\pmb{\theta}$ and $\pmb{\theta}'$ are two different lines of sight (angles-of-arrival).
The Key Insight (Lemma 3): The authors establish an equivalence between two physical models:
1. Angle-of-Arrival Model (Basu et al. [16]): Describes the correlation of wavefronts arriving from two different angles, $\pmb{\theta}$ and $\pmb{\theta}'$ .
2. Multi-Aperture Model (Chanan [17]): Describes the correlation of wavefronts measured at two spatially separated apertures.
  
  By making a first-order Taylor approximation on the integral defining the angle-of-arrival structure function, the authors show that it can be reformulated into the same mathematical form as the multi-aperture structure function. This links the angular separation $L(\pmb{\theta}-\pmb{\theta}')$ in the object plane to a spatial separation $D\pmb{\xi}$ in the aperture plane, where $D\pmb{\xi} = L(\pmb{\theta}-\pmb{\theta}')$ .
Deriving the Spatial Correlation Function: This equivalence allows the use of Chanan's analytical result for the spatial correlation of tilt coefficients: $\mathbb { E } [ a _ { j } ^ { * } a _ { j } ( D \pmb { \xi } ) ] = \frac { c _ { 2 } } { 2 ^ { 5 / 3 } } \left( \frac D { r _ { 0 } } \right) ^ { 5 / 3 } [ I _ { 0 } ( s ) \mp \cos 2 \psi _ { 0 } I _ { 2 } ( s ) ]$
- $j \in \{2, 3\}$ for horizontal/vertical tilt.
- $[s, \psi_0]$ are the polar coordinates of the separation vector $\pmb{\xi}$ .
- $I_0(s)$ and $I_2(s)$ are specific integrals involving Bessel functions that can be computed numerically. This equation gives the continuous, anisotropic correlation function for the tilts.
  
  $Fig. 10: Spatial correlation of the horizontal and vertical tilts. Shown in this figure are the normalized correlation coefficients (a) $C _ { 2 } ( \\pmb { \\xi } )$ and (b) $C _ { 3 } ( \\pmb { \\xi }…$ 该图像是论文中的对比示意图，展示了不同方法模拟大气湍流对同一场景图像的影响。左至右依次为：(a)清晰图像，(b)基于Split-Step方法，(c)作者提出的方法，(d)对应的倾斜图。倾斜图显示了各区域波前倾斜的大小和方向。
- vs. Schwartzman et al. [10]: The comparison in Figure 18 shows that the proposed method produces similar geometric distortions. A key advantage is that the proposed simulator also includes spatially varying blur, which is absent in the Schwartzman et al. simulation shown.
  
  ![Fig. 18: Comparison with Schwartzman [10]. The optical parameters are $C _ { n } ^ { 2 } = 3 . 6 \\times 1 0 ^ { - 1 3 } \\mathrm { m ^ { - 2 / 3 } }$ . $L = 2 0 0 0 \\mathrm { m } ,$ $d = 0 . 3 \\mathrm…](/files/papers/68ef1c0358c9cb7bcb2c7f60/images/18.jpg) *该图像是三幅对比图，展示了通过不同方法模拟各向异性湍流对图像的影响。(a)为原始清晰图像，(b)展示了参考文献[10]的方法结果，(c)为本文提出的仅倾斜分量模拟效果，体现了湍流引起的图像波动。* * **vs. Real Data:** The simulated images show a high degree of visual realism when compared to actual field data from the NATO dataset, capturing the look and feel of turbulence at different strength levels. * **Runtime Performance** Table 2 shows the runtime of the simulator. For a$ 512 \times 512 $image, the total time ranges from ~4 seconds (with an$ 8 \times 8 $grid for blur PSFs) to ~22.5 seconds (with a$ 32 \times 32 $grid). This is a significant speedup compared to the ~120 seconds reported for a smaller$ 256 \times 256 $image using a split-step method [9]. The speed makes it practical for generating large datasets. **Table 2: Average run time for processing one simulated 512\times512 pixel frame. (Manual Transcription)** | Component | Grid Size | Run time (s) | :--- | :--- | :--- | Zernike PSF generation | 8\times8 | 1.11 | | 16\times16 | 3.31 | | 32\times32 | 11.16 | Tilt generation and warp | 512\times512 | 1.84 | Spatial variant convolution | 8\times8 | 1.13 | | 16\times16 | 3.10 | | 32\times32 | 9.53 | **Total (w/o GPU)** | **8\times8** | **4.08** | | **16\times16** | **8.25** | | **32\times32** | **22.53** # 7. Conclusion & Reflections * **Conclusion Summary:** The paper successfully develops a fast, physically-grounded simulator for anisoplanatic turbulence. By decoupling tilts and high-order aberrations and introducing a novel method for sampling spatially correlated tilts, it transforms an expensive wave-propagation problem into an efficient statistical sampling problem. The simulator's accuracy is rigorously confirmed through theoretical validations, and its outputs are visually comparable to both high-fidelity simulators and real-world data. Its speed and physical realism make it a valuable tool for evaluating turbulence mitigation algorithms and generating training data for learning-based approaches. * **Limitations & Future Work:** The authors acknowledge several limitations: * The model is primarily aimed at **ground-to-ground imaging**, where the turbulence strength ($ C_n^2 $) can be assumed constant along the path. It may not be directly applicable to astronomical imaging where$ C_n^2 $varies with altitude. * The current simulator focuses only on **spatial correlations**. It does not model **temporal correlations** (how turbulence evolves over time), which is left for future work. * The results are derived for **incoherent imaging**, which is common in passive imaging systems but may not apply to coherent systems like LIDAR. * The use of independent blur kernels for different image blocks is an **approximation**, though well-justified by the short correlation length of high-order aberrations. * **Personal Insights & Critique:** * The paper's primary strength is its elegant theoretical contribution: connecting the angle-of-arrival and multi-aperture models to create a practical and efficient sampling algorithm. This is a clever piece of physics-based modeling that directly addresses a major computational bottleneck. * The decision to decouple tilts from blurs is highly effective. Tilts contain most of the wavefront energy and are responsible for the large-scale geometric distortions, while blurs are a more localized effect. Handling them with different statistical models and spatial scales is both physically intuitive and computationally efficient. * The work is a significant step forward for the image processing and computer vision communities. By providing an open-source, fast, and realistic simulator, it lowers the barrier to entry for researchers working on turbulence mitigation and empowers the development of data-hungry deep learning models. * An interesting area for future exploration would be to extend this sampling-based framework to incorporate non-constant$ C_n^2$ profiles and temporal dynamics (e.g., using Taylor's frozen flow hypothesis), which would broaden its applicability even further.