Event-Based Double Integral Model

• The EDI model is a computational framework that integrates blurred frame formation with event-driven log-intensity dynamics to recover latent sharp images.
• It mathematically couples physical blur with a double-integral relation of events, reducing deblurring to a scalar optimization problem for precise imaging.
• Extensions like mEDI offer temporal consistency and real-time acceleration, and the model serves as an analytical prior in modern neural and 3D reconstruction methods.

The Event-Based Double Integral (EDI) model is a computational framework for joint image deblurring and high frame rate video reconstruction using event cameras. EDI exploits the temporal precision of asynchronous event streams, as provided by Dynamic and Active-pixel Vision Sensor (DAVIS)-style hardware, to recover sharp intensity images and temporally dense videos from blurred low-frame-rate intensity frames. By mathematically coupling the physical blur-formation process with the logarithmic intensity increments encoded by events, EDI establishes a double-integral relation connecting the latent sharp image, the observed blur, and the integrated event stream. The model has been extended to temporally consistent multi-frame settings (mEDI), accelerated for real-time robotics, and incorporated as an analytical prior in contemporary neural and 3D scene reconstruction methods.

1. Mathematical Foundations of the EDI Model

The EDI model jointly models classic camera blur integration and event-driven log-intensity dynamics. For a pixel location $(x, y)$, let $I(x,y,t)$ denote the instantaneous latent image at time $t$, and $B(x, y)$ the observed blurry frame formed over an interval $[t_s, t_e]$ of duration $T$:

$$B(x, y) = \frac{1}{T} \int_{t_s}^{t_e} I(x, y, t) \, dt.$$

In parallel, an event camera triggers an event $(x, y, t, \sigma)$ whenever the log-intensity increment exceeds the threshold $c$:

$$|\log I(x, y, t) - \log I(x, y, t_{\text{prev}})| \geq c, \quad \sigma = \operatorname{sign}\left(\log I(x, y, t) - \log I(x, y, t_{\text{prev}})\right).$$

These events form a stream $e_{x, y}(t) = \sum_k \sigma_k \delta(t - t_k)$, where $\delta$ is the Dirac delta, $\sigma_k$ is the event polarity, and $t_k$ is the timestamp.

By integrating the event stream, the cumulative log-intensity change from reference $f$ to time $t$ is:

$E(t) = \int_{f}^{t} e(s) \, ds,$

yielding the log-intensity propagation relation:

$\log I(x, y, t) = \log I(x, y, f) + c\, E(t).$

Substituting into the blur formation model results in the EDI double-integral formula:

$$B(x, y) = \frac{I(x, y, f)}{T} \int_{t_s}^{t_e} \exp[c\, E(t)] \, dt.$$

Solving for the latent image at reference time $f$:

$I(x, y, f) = \frac{B(x, y)}{J(c)},$

where $$J(c) \equiv \frac{1}{T} \int_{t_s}^{t_e} \exp[c\, E(t)] \, dt$$. This operation “deconvolves” the blur by dividing out the integrated effect of log-intensity changes tracked by events (Pan et al., 2019, Pan et al., 2018, Deng et al., 14 Apr 2025, Lin et al., 2023).

2. Model Inversion and Scalar Optimization

The accuracy of EDI deblurring relies on the event camera’s contrast threshold $c$, which is not known a priori and may exhibit modest spatiotemporal variation in real sensors. The model reduces the blind deblurring task to a 1D scalar optimization. A typical approach is to minimize the residual between the modeled blurred frame $B(c)$—re-obtained by synthesizing the blur from an EDI-reconstructed latent sequence—and the original observed blur:

$E(c) = \|B(c) - B_{\text{obs}}\|_2^2.$

This minimization is conducted via golden-section or Fibonacci line search, exploiting the fact that $E(c)$ is empirically near-unimodal. For robustness in the presence of noisy events or weak texture, Pan et al. introduced regularization terms based on total-variation (TV) smoothing and edge map alignment:

• TV regularizer: $\varphi_{TV}(c) = \|\nabla I(f;c)\|_1$
• Edge-alignment: cross-correlation between Sobel-filtered event edge-map and Sobel-filtered reconstruction

The scalar objective is thus

$$c^* = \arg\min_c \left[\varphi_{TV}(c) + \lambda\, \varphi_{edge}(c)\right],$$

with $\lambda < 0$ balancing the two priors (Pan et al., 2019, Pan et al., 2018).

3. Multi-Frame EDI (mEDI) and Temporal Consistency

The mEDI model extends EDI to jointly deblur a temporal sequence of $N$ blurred frames $\{B_i\}$ and their corresponding event subsequences, mitigating per-frame flicker and improving temporal coherence. For each frame centered at $f_i$, the model forms

$$\log B_i = \log L_i + \log J_i(c), \quad \log L_{i+1} - \log L_i = c\, b_i,$$

where $b_i$ is the integrated event count between centers $f_i$ and $f_{i+1}$, and $L_i$ is the latent sharp image at $f_i$. This results in a tridiagonal linear system in $\{\log L_i\}$, efficiently solved via LU decomposition exploiting the structure of the normal equations. The cost per pixel is $O(N)$, with overall complexity $O(P \times N \times \log(1/\varepsilon))$ for $P$ pixels and search accuracy $\varepsilon$ (Pan et al., 2019).

4. Algorithmic Implementations and Real-Time Acceleration

The original EDI method is computationally intensive due to nested integrals across event streams and exposure. For robotics applications, the “fast EDI” algorithm restructures computation to achieve real-time performance on single-core CPUs. Key strategies include:

• Online accumulation of event-driven log-intensity increments
• Summarization of the exponential gain at each event arrival
• List-based Riemann sum for the outer time integral, indexed by event counter
• Replacement of precise timestamp intervals with uniform event “ticks” for normalization

This reduces the computational complexity from $O(N_{px} \cdot N_{ev})$ (per-frame, pixel-major) to $O(N_{ev})$, removing explicit dependence on image resolution. Empirically, fast EDI achieves event processing rates up to $13$ million events per second, with robust deblurring and significant improvements on tasks including feature detection and SLAM in low-light, high-speed settings (Lin et al., 2023).

Implementation	Complexity	Max Event Rate	Empirical Speedup
Original (offline)	$O(N_{px} N_{ev})$	$\sim$49kEv/s	Baseline
Fast EDI (real-time)	$O(N_{ev})$	$13$MEv/s	$260\times$

5. Integration as Analytical Priors in Modern Learning Frameworks

The EDI model’s analytical inversion properties have motivated its adoption as a physically grounded prior in neural and hybrid scene reconstruction. In EBAD-Gaussian, which jointly estimates 3D scene radiance (via Gaussian Splatting) and camera motion during exposure, the EDI formula is used to generate deblurred reference images and as a consistency constraint. For each synthetic or real blurry frame and its event stream, EDI-derived latent images at chosen subintervals serve as hard supervision for sharp image predictions produced by the generative model. The EDI prior loss combines $\ell_1$ error and structural similarity (SSIM) between EDI reconstructions and model outputs:

$$\mathcal{L}_{EDI} = (1 - \lambda_{SSIM}) \mathcal{L}_1 + \lambda_{SSIM} \mathcal{L}_{SSIM}, \ \lambda_{SSIM}=0.2$$

This enforces that the learned 3D representation respects the true physics of blur formation and event-driven latent intensity changes (Deng et al., 14 Apr 2025).

6. Empirical Performance, Limitations, and Extensions

Routine empirical validation demonstrates substantial superiority of EDI and mEDI over APS-only and event-only techniques. On synthetic datasets derived from high-speed GoPro videos, the single-image EDI method achieves SSIM ≈ $0.943$ and PSNR ≈ $29.06$ dB, with video reconstructions surpassing $0.92$ SSIM and $28.49$ dB PSNR. On real-world sequences—characterized by high motion, low-light, and abrupt intensity transitions—EDI restores sharper edges, temporally consistent structure, and detail that conventional or learning-based methods fail to recover (Pan et al., 2018).

Primary limitations include:

• Event noise and spatially varying thresholds ($c_+, c_-$); a global scalar $c$ is a compromise for tractability
• Flicker in single-frame EDI (addressed by mEDI)
• Violation of the constant-$c$ assumption in cases of sudden global intensity change
• Sensitivity to event sparsity in low-texture or static regions (mitigated by TV/edge priors)
• Applicability to global-shutter architectures; rolling-shutter imaging requires model modification (Pan et al., 2019, Lin et al., 2023)

7. Summary and Impact

The Event-Based Double Integral model is a foundational, analytically derived framework in event-based vision, coupling the integration of blurred frames and asynchronous log-intensity jumps to recover latent sharp images and videos at extremely high frame rates. Its key attributes—single-parameter model inversion, efficient solvers, compatibility with modern learning and 3D methods, and real-time feasible implementations—have positioned it as a central technique for deblurring, high-resolution reconstruction, and as a physical prior in complex vision pipelines (Pan et al., 2019, Pan et al., 2018, Deng et al., 14 Apr 2025, Lin et al., 2023).