Defocus Aberration Theory Confirms Gaussian Model in Most Imaging Devices

TL;DR

The paper investigates depth estimation from defocus and argues that the defocus operator is well approximated by a Gaussian in conventional, diffraction-limited imaging systems. By deriving the monochrome Optical Transfer Function (OTF) for defocused imaging, using an analytic approximation, and integrating over ambient illumination, the authors show that the defocus transfer can be modeled as a Gaussian $H_{def}^{σ}(\rho)=\exp(-σ^{2} ρ^{2}/2)$ with a device-dependent variance $σ$, across realistic camera parameters. They validate this Gaussian model across thousands of camera configurations, identifying a substantial subset of conventional cameras where the mean absolute error is $<1\%$, supporting practical depth-from-defocus pipelines. The work offers a fast, analytic alternative to learning-based depth estimation by exploiting the Gaussian defocus model, while noting limitations such as neglecting lens aberrations and chromatic dispersion and pointing to future hardware/estimation developments.

Abstract

Over the past three decades, defocus has consistently provided groundbreaking depth information in scene images. However, accurately estimating depth from 2D images continues to be a persistent and fundamental challenge in the field of 3D recovery. Heuristic approaches involve with the ill-posed problem for inferring the spatial variant defocusing blur, as the desired blur cannot be distinguished from the inherent blur. Given a prior knowledge of the defocus model, the problem become well-posed with an analytic solution for the relative blur between two images, taken at the same viewpoint with different camera settings for the focus. The Gaussian model stands out as an optimal choice for real-time applications, due to its mathematical simplicity and computational efficiency. And theoretically, it is the only model can be applied at the same time to both the absolute blur caused by depth in a single image and the relative blur resulting from depth differences between two images. This paper introduces the settings, for conventional imaging devices, to ensure that the defocusing operator adheres to the Gaussian model. Defocus analysis begins within the framework of geometric optics and is conducted by defocus aberration theory in diffraction-limited optics to obtain the accuracy of fitting the actual model to its Gaussian approximation. The results for a typical set of focused depths between $1$ and $100$ meters, with a maximum depth variation of $10\%$ at the focused depth, confirm the Gaussian model's applicability for defocus operators in most imaging devices. The findings demonstrate a maximum Mean Absolute Error $(\!M\!A\!E)$ of less than $1\%$, underscoring the model's accuracy and reliability.

Figures (9)

• Figure 1: Image Formation in geometric optics. Both triples $(f,d_i,d_f)$ and $(f,d_i+\Delta d_i,d_f+\Delta d_f)$ are described by the lens law.
• Figure 2: OTF for the defocused imaging system with $A_{R}/\lambda$ as a parameter. Circular pupil with the diameter $A=2R$.
• Figure 3: OTF for the defocused imaging system with$A_{R}/\lambda$ as a parameter Cross section of OTF for square pupil of width $2R$ along the $f_x$ axis JWGo2005.
• Figure 4: Defocusing OTF for defocused imaging system at fixed wave length $\lambda$ and with$A_{R}/\lambda$ as parameter.
• Figure 5: Defocusing OTF for defocused imaging system with the pixel width $P=5.6\mu m$ at ambient illumination and maximum coherent cut off frequency $\rho_{M}=\frac{R}{\lambda_{min} d_i } \text{ for } f=15mm, d_f=1m \text{ and } f_n=\frac{f}{2R}=1.4$.

Theorems & Definitions (2)

• Theorem C.1
• proof