Transformer HRTF Upsampling

• The paper demonstrates that transformer models utilizing self-attention in the spherical harmonic domain can effectively upsample sparse HRTF data.
• The architecture integrates positional encoding via RoPE and a neighbor dissimilarity loss to maintain local spectral smoothness and global spatial coherence.
• Comparative evaluations reveal that the transformer approach achieves lower log-spectral distortion and improved localization accuracy over traditional methods.

A transformer-based architecture for Head-Related Transfer Function (HRTF) upsampling refers to the use of attention-driven neural models, operating predominantly in the spherical harmonic (SH) domain, to reconstruct dense, high-resolution HRTFs from sparse measurements. HRTFs encode the acoustic filtering effect of a listener's anatomy for every spatial direction and are essential for spatial audio rendering, localization, and immersive audio applications. Transformer-based approaches introduce self-attention mechanisms that model long-range spatial dependencies across directions and frequency bins, overcoming limitations of convolution-driven or interpolation-based methods which often struggle with global spatial consistency and generalization under extreme data sparsity. The design further integrates positional encoding and custom losses to enhance both spectral fidelity and local spatial smoothness.

1. Spherical Harmonics Representation of HRTFs

HRTFs are naturally defined as functions on the sphere, parameterized by azimuth $\theta$ and elevation $\phi$. They are commonly projected onto a basis of spherical harmonics $Y_l^m(\theta, \phi)$, yielding SH coefficients $F_l^m$ that encode spatial information compactly:

$$F_l^m = \int_0^{2\pi}\int_0^{\pi} f(\theta, \phi) \cdot Y_l^m(\theta, \phi) \sin\phi \, d\phi \, d\theta$$

$$f(\theta, \phi) = \sum_{l=0}^{\infty}\sum_{m=-l}^{l} F_l^m Y_l^m(\theta, \phi)$$

This representation allows efficient manipulation and interpolation of HRTF data and is particularly well suited for transformer architectures, which treat SH coefficients as sequential inputs with explicit spatial ordering. The SH domain also ensures that upsampling and reconstruction are operated on physically meaningful spatial features.

2. Transformer Encoder–Decoder Architectures for HRTF Upsampling

Modern transformer-based HRTF upsampling frameworks, such as HRTFformer (Hu et al., 2 Oct 2025), employ encoder–decoder structures. The encoder ingests sparse SH coefficients and passes them through a stack of transformer blocks—each equipped with self-attention—to capture global correlations. Downsampling layers may be interleaved to compress sequence length. The decoder iteratively reconstructs high-resolution SH coefficients using transformer modules, up- and downsampling layers, and residual connections for stability.

A critical aspect is the use of rotary position embeddings (RoPE) to infuse spatial location information within the SH sequence. For a $d$-dimensional input $x$ at position $p$, RoPE applies sinusoidal rotations:

$$Q' = \text{RoPE}(Q, p), \quad K' = \text{RoPE}(K, p)$$

$$\text{Attention}(Q', K', V) = \text{softmax} \left( \frac{Q' K'^T}{\sqrt{d_k}} \right) \cdot V$$

The resulting architecture models both local and global dependencies among spatial coefficients, learning relationships across the full HRTF sphere.

3. Spatial Coherence: Neighbor Dissimilarity Loss

Objective spectral losses such as log-spectral distortion (LSD) and interaural level difference (ILD) provide frequency-domain accuracy but do not guarantee smooth magnitude variation between neighboring spatial positions. To address this, transformer-based models incorporate a neighbor dissimilarity loss (NDL):

$$\mathcal{L}_{nd} = \frac{1}{N}\sum_{n} \left[ \left(H_{HR}^{(n)} - \frac{1}{|\mathcal{K}(n)|} \sum_{k \in \mathcal{K}(n)} H_{HR}^{(k)}\right) - \left(H_G^{(n)} - \frac{1}{|\mathcal{K}(n)|} \sum_{k \in \mathcal{K}(n)} H_G^{(k)}\right) \right]^2$$

where $\mathcal{K}(n)$ is the set of spatial neighbors for position $n$. By minimizing this loss, the upsampled HRTFs exhibit spatially coherent magnitude transitions, crucial for perceptual realism and localization.

4. Comparative Performance and Generalization

Transformer architectures, leveraging self-attention, consistently outperform convolution-based and interpolation approaches in both objective and perceptual evaluations (Hu et al., 2 Oct 2025). Metrics include:

• Log-spectral distortion (LSD):

$$LSD = \frac{1}{N} \sum_n \sqrt{ \frac{1}{W} \sum_{w=1}^W \left[ 20\log_{10}\left(\frac{|H_{HR}(f_w, x_n)|}{|H_G(f_w, x_n)|}\right) \right]^2 }$$

• Interaural level difference (ILD) and interaural time difference (ITD) errors.
• Perceptual localization models: polar accuracy error, quadrant error, polar RMS error.

Transformers yield lower LSD and perceptual localization errors across sparsity levels (e.g., 3, 5, 19, 100 measurements), showing improved interpolation precision and preservation of binaural cues.

Model	Domain	LSD (Sparse Inputs)	ILD/ITD Error	Localization Error
Transformer	SH + RoPE	lowest	lowest	lowest (polar/quadrant)
SH Interp	SH	mid/high	mid/high	mid/high
ConvNet	Spatial	higher	higher	higher

The transformer’s capacity for long-range spatial modeling enables robust generalization under both regular and irregular measurement configurations.

5. Integration with Related Upsampling Methodologies

While transformer-based systems emphasize global spatial consistency, other ML architectures have explored magnitude correction (Arend et al., 2023), spherical CNNs (Chen et al., 2023), or retrieval augmentation (Masuyama et al., 22 Jan 2025). These methods may provide local spectral enhancement or efficient parameterization, but attention-based transformers integrated in the SH domain uniquely combine both global and local spatial fidelity.

For instance, spherical CNNs leverage SH representations for rotational equivariance but primarily capture local features. Magnitude correction post-processing mitigates spectral errors in interpolated HRTFs but lacks explicit modeling of global spatial dependencies. Retrieval-augmented neural fields reduce errors in sparse regimes by leveraging similar subject data, but transformer models offer built-in self-attention to learn spatial relations directly from the HRTF geometry.

6. Challenges and Future Directions

Implementing transformer-based HRTF upsampling involves computational and data requirements. Model training on high-dimensional SH sequences, especially at higher orders, can incur significant overhead. Positional encoding and proper loss balancing (between LSD, ILD/ITD, NDL) are crucial for convergence. Handling individual variability and anatomical diversity remains an open area—integrating retrieval augmentation or efficient adaptation modules may synergize with transformers.

A plausible implication is that continued refinement in transformer design, domain-informed embeddings, and loss formulations will further decrease the required number of measurement points for perceptually accurate HRTFs, democratizing personalized spatial audio rendering for consumer-scale applications.

7. Summary and Significance

Transformer-based architectures represent a maturation of data-driven HRTF upsampling, combining SH domain physical fidelity and self-attention’s long-range modeling. The inclusion of neighbor-aware losses enforces local smoothness, yielding upsampled HRTFs that align both with objective spectral cues and perceptual localization benchmarks. Comparative studies demonstrate substantial gains over prior methods, confirming the approach’s relevance for personalized immersive audio and efficient individualized HRTF acquisition.