Entropy Modeling

Progressive image compression reconstructs images from prefixes of a single bitstream and therefore involves sequential latent inference under partial observations. Existing trit-plane codecs mainly treat decoded trits as discrete symbols for probability prediction, without fully exploiting their role as interval constraints on the underlying continuous latents. We propose bound-conditioned latent inference (BLI), a framework that reformulates progressive trit-plane coding as sequential latent inference under shrinking interval constraints. At each coding step, the trits decoded so far define lower and upper bounds for each latent element. BLI uses the center and width of these bounds, together with hyperprior information shared by the encoder and decoder, to jointly refine the Gaussian parameters $(μ,σ)$ for entropy modeling and the latent estimate $\hat{y}$ for reconstruction. As additional trits are decoded, the feasible intervals shrink monotonically, yielding tighter constraints for the next inference step and enabling flexible in-plane refinement schedules. Experiments on Kodak, CLIC, and JPEG-AI show that BLI achieves state-of-the-art rate-distortion performance among progressive image codecs, reducing BD-rate by $17.78\%$ over DPICT and by $7.96\%$ over CTC on Kodak, while replacing CTC's plane-specific predictors with a shared model that uses about $5.3\times$ fewer parameters and achieves lower latency than CTC.

Random composition trees model the hierarchical semantic structures of meaningful texts as a statistical ensemble generated by recursively splitting texts into coherent chunks. A recent study shows that different text types can be described by random composition trees with distinct branching factors, and that the entropy rates of these trees exhibit a similar dependence across text types as the entropy rates of the actual texts. Motivated by this correspondence, we analyze information-theoretic properties of this stochastic splitting model. First, we demonstrate that the entropy of random composition trees grows linearly with the root size, and we derive an analytical expression for the entropy rate as a function of the branching factor. Second, we prove that large trees exhibit typicality: with high probability, they belong to a typical set in which each tree is approximately equally likely. Our analysis reveals that the random composition tree model shares two key information-theoretic properties of language when modeled as a stationary and ergodic stochastic process: linear entropy growth and typicality.

Lossless image compression fundamentally relies on accurate entropy modeling, driving a recent shift toward large language model (LLM)-based methods. However, existing approaches mainly emphasize the modeling capacity of LLMs while directly serializing images into pixel or residual sequences, leaving heterogeneous signal patterns entangled within a single prediction stream. In this paper, we argue that effective entropy modeling depends not only on model capacity, but also on the predictability of the input representation. Motivated by this insight, we propose WaveLLM, a wavelet-domain LLM-based entropy model for lossless image compression. By decomposing an image into multi-level wavelet subbands, WaveLLM converts a heterogeneous prediction problem into a set of more structured and statistically homogeneous subproblems that are more amenable to autoregressive probability estimation. To bridge numerical coding symbols with the LLM's native text token space, we introduce a dedicated numerical interface featuring value embeddings and a coefficient prediction head. Furthermore, a task-adaptive prompting strategy is designed to enable a single shared model to adapt to diverse subbands within a unified framework. Built on Llama-3.2-3B with LoRA adaptation, WaveLLM achieves 2.51, 2.24, and 2.27 bpsp on Kodak, CLIC Pro, and DIV2K, respectively, outperforming the PixelCNN-based codec using the same wavelet transform by 0.43, 0.36, and 0.46 bpsp, and surpassing recent LLM-based lossless compressors on all three benchmarks.

Learned image codecs (LICs) achieve high reconstruction quality, but their decoding speed is often insufficient for immersive virtual reality (VR). Gaussian splatting (GS) codecs render much faster, yet still lag in reconstruction quality and typically decide primitive allocation without considering the coding cost of each primitive. We introduce OIC-GS, an omnidirectional GS codec with a new hierarchical HEALPix primitive grid representation. Gaussian primitives are anchored at predefined spherical locations, eliminating explicit coordinate coding. Finer levels refine their coarser ancestors, naturally supporting coarse-to-fine reconstruction and layered transmission. The predefined grid also enables efficient viewport decoding by selecting only view-relevant primitives. We further introduce a lightweight entropy model for quantized primitives and optimize the codec under a spherical rate-distortion objective. Primitives with insufficient rate-distortion benefit are automatically removed when their quantized opacity becomes zero, allowing OIC-GS to adapt both primitive density and level of detail without a fixed primitive budget. A single bitstream supports full-sphere, viewport-dependent, and progressive decoding. The first viewport reaches final quality after decoding only 52% of the bitstream, and is then rendered at 1,270 FPS. On a 100-image omnidirectional benchmark, OIC-GS outperforms all evaluated GS codecs, reducing WS-PSNR BD-rate by 49.6% over GaussianImage++ and 68.6% over SGI, which uses a learned entropy model.

Distributed deployment of large vision foundation models often partitions a ViT backbone and exchanges intermediate token features between computing nodes, making efficient feature compression critical under bandwidth and computation constraints. Existing ViT feature codecs typically flatten heterogeneous global and patch tokens into an L x C pseudo image, causing entropy models to mainly capture sequence-axis dependencies while overlooking the native two-dimensional patch-grid structure. In this paper, we show that ViT patch tokens retain strong local spatial correlations on the original grid. To exploit this structural prior, we propose the Visual Token Codec (VTC), a dual-path learned codec that separates global and patch tokens into dedicated coding paths. Global tokens are compressed with a lightweight factorized prior, whereas patch tokens are encoded on the patch-token grid using a spatial-channel context entropy model. To support intermediate-layer compression and practical rate adaptation, VTC further incorporates feature-matching supervision after subsequent ViT blocks and variable-rate modules within a single codec. Experiments on DINOv2 and SAM3 show that VTC consistently outperforms representative ViT feature coding baselines on classification, segmentation, and detection tasks. At 90% of uncompressed-feature performance, VTC reduces bitrate by 15.7x-37.4x across these tasks. We further provide intermediate-layer rate-utility analyses for practical transmission- and storage-oriented deployment scenarios.

LiDAR point clouds provide explicit, deterministic physical boundaries critical for collaborative safety-critical perception. However, wireless channels inherently impair and corrupt transmitted signals. Existing robust frameworks (such as deep JSCC or MDC) attempt to counter these channel impairments through statistical or parametric estimation, turning exact physical measurements into unverified algorithmic estimates. To address this, we propose Proteus, a learned LiDAR codec operating on 2D range images. By decoupling the frame representation into independent coders for the significant range bit-planes (SIG) and the insignificant range bit-planes and attributes (INS), Proteus achieves overall stream-level truncation robustness. The non-truncatable SIG block encodes the most significant range bit-planes to establish a necessary, self-contained perceptual lower bound, below which the reconstructed point cloud is severely degraded. Meanwhile, INS employs bit-plane slicing representation and coding, ensuring that range truncation mathematically maps to a deterministic spatial precision degradation. Subordinate attributes are reconstructed via a hybrid lossless-predictive method, leveraging the decoded geometry as a strong structural prior for fine-grained approximation. Furthermore, strategic ordering within INS prioritizes geometry over attributes under bandwidth drops. Experimental results on the Waymo Open Dataset and SemanticKITTI demonstrate that Proteus tolerates up to approximately 70% bitstream truncation, while outperforming established standards (G-PCC, Draco, and JPEG XL) and the representative learned compressor Unicorn under ideal channel conditions.

Implicit neural representations (INRs) have recently emerged as a promising approach to video compression, delivering competitive rate-distortion performance alongside rapid decoding. However, existing neural video codecs struggle to balance complexity and scalability. Lightweight models often suffer from degraded compression performance when scaled to different bitrate/quality levels, whereas high-performance models exhibit limited scalability, as their model complexity typically increases with quality. This lack of a unified architecture capable of maintaining consistent complexity across a wide range of bitrates severely limits their diverse real-world deployment. To address these challenges, we introduce NVRC++, a novel INR-based video codec that utilizes a lightweight INR with multiple high-resolution feature grids, providing high scalability at any given complexity level. This is paired with an optimization framework that enables efficient overfitting on high-resolution grids for long video sequences, thereby exploiting spatio-temporal redundancies without prohibitive computational or memory overhead. Additionally, an advanced entropy model is designed for efficiently compressing the high-dimensional grid parameters. As a result, NVRC++ provides four complexity levels (from 7kMACs/pixel to 360kMACs/pixel), each spanning wide bitrate and quality ranges while supporting real-time decoding. The experimental results show that NVRC++ offers a much faster decoding speed (up to 7.6x) compared to the SOTA INR-based video codec, NVRC, while delivering comparable performance.

Fine-tuning is often believed to reduce uncertainty and diversity in large language models, but existing analyses overlook output length, a key confounder, and therefore fail to capture how uncertainty is distributed across an entire generation rollout. To address this, we propose Canopy Entropy ($\mathrm{CE}^\star$), a measure that views language generation from a tree perspective, where ``canopy'' represents the space of all possible rollouts, making $\mathrm{CE}^\star$ naturally quantify the effective size of the generation space. $\mathrm{CE}^\star$ jointly captures uncertainty in both the output length $N$ and the generated sequence $Y_{1:N}$ -- indeed, we show that it equals to total Shannon entropy $H(N, Y_{1:N}\mid X)$, where $X$ denotes the prompt. This formulation yields interpretable metrics, including a length-entropy correlation term $ρ(N, r_N)$, where $r_N$ is the entropy rate, quantifying information conveyance efficiency by indicating whether longer outputs are more or less informative per token. Empirically, across tasks and model families, we find that fine-tuned models consistently exhibit stronger positive correlation $ρ(N, r_N)$, even when total entropy decreases. Furthermore, after controlling for model family, task, prompt, and output-length effects, we find that fine-tuning nearly triples the correlation strength between entropy rate and semantic diversity, suggesting that aligned models convert token uncertainty into semantic diversity more efficiently. Overall, these results demonstrate that fine-tuning does not simply reduce uncertainty, but fundamentally reorganizes it into more informative and semantically meaningful generations. Our code is available at https://github.com/WeiyiTian/canopy-entropy.

State-of-the-art learned image compression (LIC) schemes are increasingly based on hybrid CNN-transformer architectures. To further improve rate-distortion performance, we introduce channel-wise wavelet transforms into both the transformer and entropy-coding components. First, we propose a channel-wise wavelet-domain transformer attention (ChWDTA) mechanism. ChWDTA keeps the efficient windowed spatial self-attention used in modern LIC backbones, but computes the Q/K/V projections on channel-wise wavelet-transformed features before mapping the attention output back with the inverse transform. The resulting Channel-wise Wavelet-Domain Transformer Block (ChWDTB) therefore preserves the spatial tokenization pattern of windowed attention while sparsifying the channel covariance seen by the attention projections. Second, in the entropy-coding stage, we introduce a channel-wise wavelet packet (ChWP) decomposition that produces four equal-sized subbands, which better fit channel-wise slice-based autoregressive entropy modeling. When each channel-wise subband is divided into two slices, we use eight slices for entropy coding. With this configuration, the proposed scheme obtains BD-rate reductions of -17.82%, -19.15%, and -22.56% on the Kodak, CLIC Professional Validation, and Tecnick test sets, respectively. Even when each channel-wise subband is coded as a single slice, the scheme still retains most of the coding gains with lower complexity. The results confirm the advantage of introducing wavelet transform in CNN-transformer-based LIC schemes.

In natural language processing, the entropy of a language is a measure of its unpredictability and complexity. The first study on this subject was conducted by Claude Shannon in 1951. By having participants predict the next character in a sentence, he was able to approximate the entropy of the English language. Several follow-up studies by other authors have since been conducted for English, and one for Hebrew. However, to date, Shannon's experiment has never been conducted for Ukrainian. In this paper, we perform this experiment for Ukrainian by recruiting 184 volunteers using social media channels. We rely on techniques used for English to approximate the entropy value of Ukrainian. The final result is an upper bound of $H_{upper}\approx1.201$ bits per character. We compare this to the performance of current Large Language Models. The methods and code used are also documented and published, along with a discussion of the main challenges encountered.

Parameter-efficient fine-tuning of pre-trained codecs is a promising direction in image compression for human and machine vision. While most existing works have primarily focused on tuning the feature structure within the encoder-decoder backbones, the adaptation of the statistical semantics within the entropy model has received limited attention despite its function of predicting the probability distribution of latent features. Our analysis reveals that naive adapter insertion into the entropy model can lead to suboptimal outcomes, underscoring that the effectiveness of adapter-based tuning depends critically on the coordination between adapter type and placement across the compression pipeline. Therefore, we introduce Structure-Semantics Co-Tuning (S2-CoT), a novel framework that achieves this coordination via two specialized, synergistic adapters: the Structural Fidelity Adapter (SFA) and the Semantic Context Adapter (SCA). SFA is integrated into the encoder-decoder to preserve high-fidelity representations by dynamically fusing spatial and frequency information; meanwhile, the SCA adapts the entropy model to align with SFA-tuned features by refining the channel context for more efficient statistical coding. Through joint optimization, S2-CoT turns potential performance degradation into synergistic gains, achieving state-of-the-art results across four diverse base codecs with only a small fraction of trainable parameters, closely matching full fine-tuning performance. Code is available at https://github.com/Brock-bit4/S2-CoT.