【论文精读系列】之《Turbulence-Induced 2D Correlated Image Distortion》其一

论文地址：《Turbulence-Induced 2D Correlated Image Distortion》

Abstract（摘要）

Due to atmospheric turbulence, light randomly refracts in three dimensions (3D), eventually entering a camera at a perturbed angle. Each viewed object point thus has a distorted projection in a two-dimensional (2D) image. Simulating 3D random refraction for all viewed points via complex simulated 3D random turbulence is computationally expensive. We derive an efficient way to render 2D image distortions, consistent with turbulence. Our approach bypasses 3D numerical calculations altogether. We directly create 2D random physics-based distortion vector fields, where correlations are derived in closed form from turbulence theory. The correlations are nontrivial: they depend on the perturbation directions relative to the orientation of all object-pairs, simultaneously. Hence, we develop a theory characterizing and rendering such a distortion field. The theory is turned to a few simple 2D operations, which render images based on camera and atmospheric properties.

由于大气湍流，光线会在三个维度（3D）上随机折射，最终以受扰动的角度进入相机。因此，每个被相机观看的对象点在二维（2D）图像中的投影都失真。通过复杂模拟的3D随机湍流，为所有这些对象点模拟3D随机折射的计算量很大。我们推导了一种有效的方法，来造成与湍流带来的失真效果一致的2D图像失真。我们的方法完全绕过了3D数值计算。我们直接创建二维随机的、基于物理学的失真矢量场，其中相关性是根据湍流理论以封闭形式推导出的。这里的相关性很重要：它们同时取决于相对于所有对象对的方向的扰动方向。因此，我们开发了表征和造成这种失真场的理论。该理论转变为一些简单的2D操作，这些操作基于相机和大气特性来产生图像。

1 Introduction（介绍）

Imaging through refractive media is of interest both for rendering and scene analysis. It is studied in computer vision and graphics, as the importance of participating media is acknowledged. Complex, random refraction is created by turbulent media, often encountered in long range observations through the atmosphere and ground-based astronomy. There, random perturbations of the refractive index follow a complicated fractal multiscale structure of eddies in the three dimensional (3D) domain. This structure, in turn, creates highly complex refraction of propagating light, in the 5D plenoptic domain (space and direction). This leads to random perturbations of all light rays passing through this medium. Finally, the complexly refracted light is projected by a camera, forming a distorted image of the scene.

通过折射介质成像对于渲染和场景分析都很有意义。由于人们已经认识到参与成像的介质的重要性，因此在计算机视觉和图形学方面对其进行了研究。复杂的随机折射是由湍流介质产生的，这种介质通常在经过大气层的远距离观测和地基天文学中经常遇到。在那里，折射率的随机扰动在三维空间（3D）中遵循一种复杂的、分形多尺度的涡流结构。反过来，这种结构在5D全景区域（空间和方向）中造成正在传播的光线的高度复杂的折射。这导致穿过该介质的所有光线的随机扰动。最终，复杂折射的光线投影在相机中，从而形成场景的失真图像。

Now, suppose one seeks to render images that are distorted as if they are taken through atmospheric turbulence. A motivation for rendering can be computer graphics. Another motivation is to form a database on which to test and develop recovery algorithms, to correct for turbulence-induced distortion. A database can also train a learning system to recognize objects via turbulence. From the description above, apparently, rendering should include a series of computationally complex steps: Simulate a huge 3D turbulent random field (scale of kilometers) at a 3D resolution that is relevant to optics; Raytrace refraction through this 3D medium, from an object point to the camera; Repeat this propagation process for all resolvable points in the field of view. In large scales, such a rendering approach poses a computational burden. It is also unnecessary.

现在，假设有人试图生成类似通过大气湍流拍摄的图像一样失真的图像。生成的动机可能是计算机图形学的需要。另一个动机是形成一个数据库，来测试和开发用以校正湍流引起的失真的恢复算法。数据库还可以训练学习系统以识别湍流引起成像失真的物体。显然，根据上面的描述，生成类似通过大气湍流拍摄的图像一样失真的图像应包括一系列计算复杂的步骤： 以与光学相关的3D分辨率模拟巨大的3D湍流随机场（以千米为单位）；跟踪从对象点到相机、经过这个3D介质的光线的折射；对视场中的所有可分辨的点重复此传播过程。在大规模上，这种生成方法带来了计算负担。这也是不必要的。

We believe that for some applications, there is no need for 3D simulations, in order to render turbulence-induced image distortion. There is no need to simulate a fractal random refractive index 3D field, or ray-trace through such a field. Basically, image distortion is an operation in just two dimensions (2D). The input is a 2D image, free of random distortion, i.e. the view in the absence of turbulence. The output is a 2D distorted image. Rendering boils down, thus, to creating a 2D distortion operation, which is consistent with distortion that turbulence can induce.

我们认为对于某些应用，无需为了生成湍流引起的图像失真而进行3D仿真。不需要模拟分形随机折射率的3D场，也不需要模拟穿过该场的光线轨迹。基本上，图像失真只是二维（2D）的操作。输入是2D图像，没有随机失真，即没有湍流时的视图。输出为2D失真图像。因此，生成归结为创建一种2D失真操作，该操作与湍流可能引起的失真一致。

Random 3D turbulence eventually leads to a randomly distorted 2D projection. The random distorted projection must be drawn from a distribution that is characterised by a covariance function. The covariance function determines how the distortion of any pixel is correlated to any other pixel, and what the variance is. The covariance function of turbulence-induced distortion is defined by physics. In other words, the physics of turbulence in 3D (a random process), and 5D refraction in it, dictate the image distortion covariance function, in 2D. The probability distribution of distortion had already been derived using the theory of turbulence, for pairs of object points (not full-field images). This pair-wise function had also been verified empirically, using field experiments, where correlations between image points were measured. We thus use the pair-wise covariance function of turbulence-induced image distortion, to create 2D distortion fields.

随机3D湍流最终导致随机失真的2D投影。随机失真的2D投影肯定是由以协方差函数为特征的分布刻画的。协方差函数确定任一像素的失真是如何与其他任一像素相关联的、以及它们的方差是什么。湍流引起的失真的协方差函数由物理学定义。换句话说，在3D（随机过程）中的湍流物理学以及其中的5D折射决定了2D中的图像失真协方差函数。对于一对目标点（不是全场图像），已经使用湍流理论得出了失真的概率分布。这种成对函数也已通过现场实验进行了实证验证，实验中测量了图像点之间的相关性。因此，我们使用湍流引起的图像失真的成对协方差函数来创建2D失真场。

Transferring the physics-based pair-wise covariance function to a full distortion field is nontrivial in turbulence. Distortion is a vector-valued spatial field. The covariance of this field is a matrix-valued function. It is a function of relative coordinates that vary for each pair of pixels, including the relative orientation of each pair, and the distortion orientation in each pixel. We derive theoretically the solution: a full-field covariance of a 2D distortion field, based on the physics-based pair-wise orientation-sensitive covariance. We then give a recipe how to render random 2D distortion fields that satisfy the physical model. The recipe is composed of several simple 2D image operations.

在湍流中，将基于物理学的成对协方差函数转移到完整的失真场是很重要的。失真是向量值的空间场。该场的协方差是矩阵值函数。它是一个每对像素都不同的、像素对之间相对坐标的函数，包括每对像素的相对方向和每个像素中的失真方向。我们从理论上得出解决方案：在基于物理学的成对方向敏感的协方差的基础上，构建2D失真场的全场协方差。然后，我们给出如何生成满足物理模型的随机2D失真场的方法。该方法由几个简单的2D图像操作组成。

2 Related Work（相关工作）

2.1 Phase Screen Propagation（相位屏传播）

A common method for simulating imaging through turbulence is based on light propagation through multiple random 2D phase screens, approximating a 3D turbulent medium. Phase-shifting layers are generated using either fast Fourier transform (FFT), the Zernike polynomial method or the fractal interpolation method. This approach requires simulating a 3D refractive field and light propagation in 3D.

一种模拟经过湍流的成像的普遍方法是光传播，这种光传播通过多个随机的2D相位屏，以近似3D湍流介质。使用快速傅里叶变换（FFT）、Zernike多项式方法或分形插值方法生成相移层。这种方法需要模拟3D折射场和3D中的光传播。

2.2 3D Ray Tracing for Graphics（图形学的三维光线跟踪）

There are rendering techniques that trace rays through a 3D randomly refractive media. Physically based simulation of atmospheric phenomena is done in. Rendering complicated lighting effects through various refractive objects is presented in. These 3D methods often require specialized hardware such as GPU and extensive computation time.

有生成技术可以跟踪通过3D随机折射介质的光线。该技术完成了基于物理学的大气现象的模拟，呈现了通过各种折射对象而生成的复杂照明效果。这些3D方法通常需要专用硬件（例如GPU）和大量的计算时间。

2.3 2D Image Distortion Simulation（二维图像失真仿真）

2D image distortion is simpler to implement, and does not require extensive computational power. Usually, parametric models are used. These models are based on analysis of real empirical distortions observed through various atmospheric conditions. Other methods, as in, use simple Gaussian random functions to generate image distortion fields. The results resemble turbulence distortion. However, these methods do not use a physical model and are not set to have physically consistent spatial correlations.

2D图像失真更易于实现，并且不需要大量的计算能力。通常使用基于对实际经验性模型的分析的参数模型，其中经验性模型可以通过各种大气状况观察到。例如，其他方法使用简单的高斯随机函数来生成图像失真场。结果类似于湍流失真。但是，这些方法不使用物理模型，并且没有被设置为具有物理上一致的空间相关性。

参考链接：
采用Zernike多项式法计算模拟大气相位屏
什么是分形
分形
MATLAB分形插值