DepthAny-thing: Unified Models for Computational Depth

• DepthAny-thing is a multi-disciplinary framework uniting theoretical constructs, computational depth measures, and foundation models to enable robust depth estimation.
• It leverages large-scale datasets, transformer architectures, and teacher-student paradigms to achieve state-of-the-art performance on diverse depth prediction benchmarks.
• The framework extends to remote sensing, amodal depth reasoning, and statistical data analysis, offering scalable and adaptable solutions across various applications.

DepthAny-thing encompasses a constellation of methodologies and foundational models centered on the concept of "depth," both as applied in computational complexity theory and in modern machine learning, computer vision, and statistical data analysis. The term spans diverse meanings: algorithmic sequence depth in complexity theory, depth as a measure of computational history or structure in natural and artificial systems, and depth maps as representations of spatial geometry. The latest works under titles such as "Depth Anything" and its derivatives extend these notions by developing large-scale, generalizable, and cross-domain models for geometric and semantic inference, most notably in monocular depth estimation, scientific remote sensing, and probabilistic modeling. This article provides a comprehensive survey of DepthAny-thing, tracing its theoretical roots, architectural underpinnings, algorithmic strategies, applications, and research outlook.

1. Theoretical Foundations: Notions of Depth in Complexity Theory

The origins of "depth" as a technical concept trace to algorithmic information theory, particularly as formalized in Bennett’s logical depth and its generalizations. Here, depth is defined not as an intrinsic property of a sequence, but as a function of classes of "observers" (algorithms) attempting to interpret or compress a sequence $S$ (0906.3186). The formal framework is as follows:

Let $G$ and $G'$ be classes of algorithms (e.g., computable compressors, polynomial-time predictors), and let $$\mathrm{Perf}: (G \cup G') \times \{0,1\}^* \to [0,1]$$ be a performance function (such as compression efficiency). A sequence $S$ is $(G, G')$-deep if

$$\forall A \in G, \ \exists A' \in G', \ m \in M: \quad \forall^\infty n \in \mathbb{N}, \quad \mathrm{Perf}(A', S[1..n]) - \mathrm{Perf}(A, S[1..n]) \geq \frac{m(n)}{n}$$

where $M$ is a family of lower bound "margin" functions (e.g., $O(\log n)$).

This abstract framework subsumes:

• Bennett's Logical Depth: Depth via incompressibility gap between a computable compressor and the universal (uncomputable) compressor.
• Recursive and Polynomial-Time Depth: Replace G′ with the same class as G for computable compressors; restrict G/G′ to polynomial-time predictors or distinguishers.
• Finite-State Depth: Limit G/G′ to finite-state transducers with performance measured by information-lossless compression.

Classical results include the "slow growth law," prohibiting the creation of deep sequences from shallow ones via simple reductions, and the established shallowness of both computable and random sequences.

2. Computational Depth in Natural and Artificial Systems

Outside classical complexity theory, "depth" also characterizes the degree of embedded computation within a system’s dynamics (Machta, 2011). Here depth is defined as the minimal parallel time required to sample a typical system state from simple initial conditions: $$D = \min_{\mathcal{A}} T_{\text{parallel}}(\mathcal{A})$$ where the minimum is over all efficient (e.g., randomized) parallel algorithms $\mathcal{A}$ that generate the target distribution.

Unlike entropy-based measures (e.g., excess entropy, statistical complexity), which quantify stored information or correlations, computational depth is maximal: the system's depth is dominated by its deepest, most computationally irreducible subsystem, not additive across independent subsystems. For example, a biosphere embedded in a simple environment retains its large depth, undiluted by averaging. Depth is only large in systems with nontrivial embedded computation (e.g., diffusion-limited aggregation), not merely memory or long-range correlations.

3. Foundation Models for Depth Prediction

Recent advances have seen the emergence of large-scale "Depth Anything" models (Yang et al., 19 Jan 2024), which leverage enormous datasets of automatically pseudo-labeled images (on the order of $60$ million) to train monocular depth estimation architectures with broad generalization:

• Architecture: Transformer backbone (e.g., ViT pretrained with DINOv2), DPT (Dense Prediction Transformer) decoder.
• Training Regimen: Teacher-student paradigm: a teacher model is trained on $\sim${1.5M} labeled images with scale-invariant loss for relative depth; then, student models are trained on both labeled and pseudo-labeled data via strong augmentations (e.g., CutMix), enhancing robustness.
• Auxiliary Supervision: Semantic feature alignment enforces that depth features inherit coarse semantic priors via cosine similarity loss with frozen semantic encoders.
• Zero-Shot and Fine-Tuning Capabilities: The models achieve strong generalization on six public datasets (KITTI, NYUv2, Sintel, DDAD, ETH3D, DIODE), with improved results upon fine-tuning on metric-labeled data (state-of-the-art on NYUv2, KITTI). Enhanced depth predictions also provide superior conditioning for downstream image synthesis tasks (e.g., ControlNet).

These models demonstrate that leveraging large, diverse unlabeled datasets with self-supervised and auxiliary semantic signals yields robust, general-purpose depth representations, establishing a universal prior for geometric perception across domains.

4. Integrative and Task-Conditioned Depth Synthesis

"Depth Anything with Any Prior" (Wang et al., 15 May 2025) advances the paradigm by proposing a coarse-to-fine pipeline to merge incomplete metric priors (e.g., sparse depth from sensors) with the dense but relative predictions of monocular depth estimation:

• Metric Alignment: For each pixel with measured depth, fill missing points by solving for local scale and shift that optimally aligns the relative prediction to the metric prior via distance-weighted k-nearest neighbor fitting.
• Conditioned Refinement: A conditioned MDE model ingests the original image, pretrained prediction, and filled prior, using a zero-initialized convolutional integration. This allows for fine reweighting of metric and geometric cues, with normalization ensuring arbitrary prior swapping at test time.
• Generalization: Demonstrated strong zero-shot transfer to tasks including depth completion, super-resolution, and inpainting across 7 datasets. Notably, the approach remains robust with challenging, mixed, and sparse priors; modularity enables seamless test-time upgrades as new MDE models become available.

This suggests a flexible architecture capable of adapting to both the precision of metric sensors and the coverage of monocular predictors, facilitating accuracy/efficiency trade-offs in real-world scenarios.

5. Cross-Domain Adaptation: Remote Sensing and Scientific Applications

Depth foundation models now enable applications in domains beyond conventional vision benchmarks. In "Depth Any Canopy" (Cambrin et al., 8 Aug 2024), the authors fine-tune the Depth Anything v2 architecture for global canopy height mapping using remote sensing imagery:

• Methodology: Retrain the model to output canopy height maps (CHMs) using EarthView high-resolution aerial imagery paired with LiDAR-derived CHM labels. Image-quality filtering removes artifacts to improve reliability. Training is shallow (3 epochs), cost- and carbon-efficient (< $2, < 0.24$ kg CO₂).
• Performance: The model delivers state-of-the-art mean absolute error and IoU with an order-of-magnitude reduction in parameters and computational cost over baselines.
• Significance: The approach provides scalable alternatives to expensive LiDAR surveys, enabling forest monitoring, conservation, and biomass estimation at unprecedented computational and environmental efficiency.

The successful transfer underscores the adaptability of large-scale monocular depth architectures for specialized, scientific computer vision tasks with minimal modification.

6. Amodal Depth and Occluded Geometry Reasoning

The challenge of predicting the geometry of occluded or invisible scene regions ("amodal depth") has been addressed in "Amodal Depth Anything" (Li et al., 3 Dec 2024):

• Dataset Construction: The ADIW dataset is constructed by compositing objects onto backgrounds and generating scene-consistent amodal (whole-object) masks, aligning pre-trained depth model outputs over visible/occluded regions using scale-and-shift minimization.
• Modeling Frameworks:
  • Amodal-DAV2: Adapts the DAV2 (Depth Anything v2) model by introducing parallel convolutional pathways for observed monocular depth and amodal mask guidance, initialized to preserve pre-trained weights.
  • Amodal-DepthFM: A generative model integrating conditional flow matching, synthesizing multiple plausible amodal depth completions guided by latent representations and observation cues.
• Results: Achieves 69.5% improvement over previous SoTA on ADIW; ablation confirms importance of guiding signals, object-level supervision, and scale-and-shift alignment.
• Implications: Enables more complete scene understanding critical for 3D reconstruction, AR, and robotics. This framework generalizes to wild, diverse occlusion settings, overcoming prior reliance on synthetic datasets and narrow metric formulations.

7. Statistical Data Depth and Loss-Based Depth Measures

DepthAny-thing has also catalyzed developments in statistical data analysis. "Data Depth as a Risk" (Castellanos et al., 11 Jul 2025) reinterprets classical data depth (e.g., Tukey depth) as the minimum risk (loss) attained by a classifier distinguishing a point from the rest of the data distribution:

$$D_l(z|Q) = \inf_{f \in \mathcal{H}} \mathbb{E}_{(x,y) \sim \mathscr{L}} [l(f(x),y)]$$

where $\mathcal{H}$ is a hypothesis class (e.g., linear, SVM, logistic regression), $l$ is a loss, and $\mathscr{L}$ is the artificial labelling with $z$ as negative and all other data as positive.

This approach enables:

• Efficient computation in high dimensions by leveraging existing optimization algorithms (e.g., QP for SVMs).
• Theoretical guarantees: statistical convergence at $O(n^{-1/2})$ rate via Rademacher complexity.
• Interpretability: Links anomaly detection to classifier risk; outliers are those for which simple classifiers attain low loss.
• Flexibility: The depth definition naturally adjusts to the complexity of the hypothesis class, reflecting intrinsic data structure and classifier simplicity.

This framework unifies robust nonparametric statistics and supervised machine learning, opening new directions in high-dimensional analysis, anomaly detection, and the interpretability of classifier-based depth scores.

8. Outlook and Research Directions

DepthAny-thing has catalyzed a shift toward data-driven, foundation model-based depth perception that is cross-modal, cross-domain, and cross-task. Technical and scientific directions that promise further advances include:

• Scaling Foundation Models: Development of even larger and more semantically unified encoders for both depth and broader geometric understanding, leveraging unlabeled and multimodal data.
• Efficient Cross-Domain Transfer: Systematic adaptation for verticals such as medical imaging, scientific measurement, and earth observation with minimal compute and carbon footprint.
• Flexible Inference Pipelines: Modular conditioned refinement models capable of fusing arbitrary priors, facilitating plug-and-play deployment as sensors and underlying predictors evolve.
• Amodal and Generative Geometry: Broader application of generative modeling and amodal reasoning for ambiguity-resolving scene understanding.
• Theoretical Unification: Integration of algorithmic depth, computational depth, foundation depth models, and statistical data depth as complementary tools for measuring and utilizing structure in data and systems.

DepthAny-thing thus represents not only a convergence of historical notions but a rapidly evolving frontier in which foundational, scalable, and generalizable depth modeling is a central enabler for both scientific discovery and real-world artificial intelligence deployments.