InDI: Direct Denoising Inversion

• InDI is an inversion framework that leverages neural network denoisers to directly invert forward or generative models through iterative updates.
• It applies fixed-point and gradient-free methods to solve inverse problems in imaging, tomography, and diffusion models, bypassing traditional hand-crafted priors.
• The approach offers efficient convergence, reduced computational load, and adaptability to complex applications while ensuring stability through careful parameter tuning.

Inversion by Direct Denoising (InDI) encompasses methodologies that address the inversion of physically, statistically, or algorithmically defined mapping processes (typically forward or generative models) by leveraging the denoising capabilities of neural networks. These approaches are distinguished by their direct, often iterative, application of a learned or explicit denoiser as the means of inversion, circumventing classical hand-crafted priors or explicit modeling of the degradation. InDI has been adopted for a range of tasks, including image restoration, large-scale inverse tomography, encoder/decoder inversion in diffusion models, and control applications where structured priors are unavailable or intractable. The following sections survey foundational theory, algorithmic classes, convergence results, practical implications, and recent empirical advances.

1. Fundamental Principles and Mathematical Formulations

At the core of InDI methodologies lies the modification or reinterpretation of classical inversion and regularization formulas to incorporate denoising operators. A canonical example is the Regularization by Denoising (RED) fixed-point formulation: $$\nabla g(x^*) + \tau \left( x^* - D_{\sigma}(x^*) \right) = 0,$$ where $g(x)$ is a data fidelity term tied to the forward physics (e.g., $g(x) = \frac{1}{2} \|Ax - y\|^2$), and $D_\sigma$ is a trained (typically deep) denoiser acting as a learned proximal operator or prior, weighted by $\tau$ (Wu et al., 2019). The inversion problem seeks $x^*$ that is consistent with both the measurements and the denoising-induced prior.

For process inversion problems (e.g., in image restoration and diffusion models), fixed-point or iterative updates are crafted: $$x^{(k)} \leftarrow x^{(k-1)} - \gamma \left(\nabla g(x^{(k-1)}) + \tau (x^{(k-1)} - D_{\sigma}(x^{(k-1)})) \right)$$ for explicit RED-style updates (Wu et al., 2019), or gradient-free fixed-point iterations for decoder inversion in diffusion models: $x^{(k+1)} = x^{(k)} - \rho (f(x^{(k)}) - y),$ where $f$ is a decoder or forward process, and $y$ is a measured or target image (Hong et al., 27 Sep 2024).

In learned iterative inversion (Delbracio et al., 2023), the problem is decomposed into a series of incremental steps, where at each iteration, a denoiser/refiner $F_t$ is applied to "partially denoised" intermediates: $$\hat x_{t-\delta} = \frac{\delta}{t} F_t(\hat x_t, t) + \left(1 - \frac{\delta}{t}\right) \hat x_t,\quad \hat x_1 = y,$$ where $x_t$ linearly interpolates between the degraded and clean signal, and the continuous limit yields an ODE interpretation.

2. Algorithmic Classes and Representative Implementations

InDI-aligned methods fall broadly into the following classes:

InDI Algorithm	Domain/Application	Denoiser Type
RED/SIMBA	Optical tomography, physics-based inversion	Deep CNN denoiser
Invertible Networks	Denoising, artifact removal	Invertible blocks
Fixed-Point Iteration	Decoder/Diffusion model inversion	UNet/decoder
Gradient-Free Forward Step	Decoder inversion	Direct residual op.
Iterative Denoising	Image restoration (learned)	U-Net-based

• SIMBA/RED: Merges physics (e.g., imaging system forward model) with deep denoisers in large-scale tomographic inversion, with updates executed via minibatch gradient steps and a RED correction term. Fixed-point convergence is established under the nonexpansiveness of the denoiser (Wu et al., 2019).
• Invertible Denoising Networks: Partition the latent space into clean and noisy/null manifolds and perform denoising by manipulation of the latent code (e.g., InvDN, hierarchical disentanglement) (Liu et al., 2021, Du et al., 2023). Denoising occurs by substituting the noise-encoding portions in the inverse mapping.
• Denoising-based Iterative Inversion: Directly iterates the inversion process by "undoing" incremental degradations using denoising neural networks trained on paired examples, without requiring an explicit analytical degradation model (Delbracio et al., 2023).

3. Convergence Guarantees and Theoretical Results

Fixed-point convergence in InDI largely depends on the contraction or cocoercivity properties of the composite operator combining data fidelity and denoising. For instance, in the RED/SIMBA regime, if the denoiser $D_\sigma$ is nonexpansive and $g$ is convex and Lipschitz, the operator $G(x) = \nabla g(x) + \tau(x - D_\sigma(x))$ becomes cocoercive. Under appropriate step-size selection: $$\mathbb{E}\left[ \frac{1}{t} \sum_{k=1}^t \|G(x^{(k-1)})\|_2^2 \right] \le (L+2\tau) \left( \frac{\|x^0 - x^*\|_2^2}{\gamma t} + \frac{\gamma \nu^2}{B} \right),$$ ensuring convergence to a fixed point as $t\to\infty$ (Wu et al., 2019).

Gradient-free decoder inversion relies on the cocoercivity of the residual operator $T(x) = f(x) - y$. For step-size $\rho \in (0, 2\beta)$, iterates satisfy: $$\|x^{(k+1)} - x^*\|^2 \le \|x^{(k)} - x^*\|^2 - \rho(2\beta - \rho)\|T(x^{(k)})\|^2,$$ which guarantees convergence to $f(x) = y$ (Hong et al., 27 Sep 2024).

4. Empirical Performance and Computational Considerations

Implementation of InDI approaches consistently demonstrates reduced memory and computation in inverse imaging. For example, SIMBA's minibatch approach reduces memory usage from ~75 GB (full batch) to ~11 GB for $1024 \times 1024 \times 25$ volumes and achieves per-iteration speed gains (0.31 s vs 0.52 s) compared to batch RED (Wu et al., 2019).

In decoder inversion for latent diffusion models, gradient-free fixed-point iterations yield a reduction of runtime from 9.5 seconds (gradient-based) to 1.89 seconds and peak GPU memory from 64.7 GB to 7.13 GB in video inversion scenarios, with comparable normalized MSE (Hong et al., 27 Sep 2024).

Empirical studies in invertible denoising networks show superior peak SNR and structural similarity (SSIM), with InvDN achieving better scores and 25× parameter reduction over comparable architectures, and capacity for rapid, memory-efficient inference (Liu et al., 2021). Hierarchical disentangled approaches yield superior PSNR-B for artifact removal and superior visual restoration, with lower model complexity (Du et al., 2023).

5. Extensions and Adaptations Across Domains

The InDI philosophy extends beyond classical inverse problems into generative modeling (e.g., diffusion model inversion), watermarking, and real-time control:

• Latent Diffusion Models: Inversion involves direct fixed-point iteration (forward step, inertial KM) on the decoder, facilitating watermark detection and editing with lower memory and computation (Hong et al., 27 Sep 2024).
• Accelerated Diffusion Inversion: Advanced strategies such as Anderson acceleration or blended guidance improve robustness and speed of inversion, especially at low step counts typical in fast editing applications (Pan et al., 2023).
• Image Restoration with Blind and Non-Blind Degradations: Coupling INNs with diffusion models enables direct inversion of complex, unknown degradation processes—with the invertible net trained to simulate the forward degradation and guide the denoising trajectory (You et al., 23 Jan 2025).
• Practical Control and Robotics: InDI underpins robust dynamic inversion for aerial systems, with neural-augmented InDI compensating for residual forces in quadrotors, improving robustness to payload variations and external disturbances (Hachem et al., 13 Jan 2025, Cobo-Briesewitz et al., 12 Mar 2025).

6. Limitations, Challenges, and Remedies

While InDI methods achieve substantial gains, limitations remain:

• Approximation Error: In iterative direct denoising (e.g., DDIM inversion), early steps exhibit substantial noise prediction errors for low-variation image regions, biasing the latent space and diminishing editability. An effective remedy is to substitute early inversion steps with a forward diffusion process, thus decorrelating the latent encoding and improving edit/interpolation quality (Staniszewski et al., 31 Oct 2024).
• Convergence Sensitivity: For strong classifier-free guidance or large step sizes, fixed-point iteration stability must be ensured by explicit step-size tuning or using gradient-based backward Euler corrections (Hong et al., 2023).
• Noise-Editability Trade-Off: Excessively accurate inversion may overfit, hampering downstream manipulations (e.g., text-driven edits). Regularization strategies during iterative inversion, such as edit-enhancement losses and noise correction, balance fidelity and flexibility (Garibi et al., 21 Mar 2024).
• Degradation Model Mismatch: Blind restoration in complex, non-analytic settings demands explicit modeling or estimation of latent degradation parameters; performance hinges on the capacity of the invertible neural network and degradation estimator (You et al., 23 Jan 2025).

7. Prospects and Emerging Directions

InDI techniques have demonstrated particular strength in large-scale, high-fidelity inversion where explicit modeling is infeasible but rich data is available for training. The trend toward using fixed-point, acceleration, and hybrid guided-gradient methods, as well as integrating invertible neural components, continues to broaden the applicability of InDI—from physics-based imaging to deep generative analytics and real-time control. Ongoing work focuses on improving the invertibility of mappings, developing better contraction/composition analyses for stability, and devising initialization/aggregation schemes that further lessen computational burden (e.g., state aggregation in EasyInv (Zhang et al., 9 Aug 2024)). As generative architectures diversify and the need for high-throughput, editable, and data-consistent inversion grows, InDI will likely remain a foundational tool in inverse modeling and restoration.