OPE-Upscale Module: Orthogonal Image SR

• The OPE-Upscale module is a parameter-free upsampling mechanism that uses orthogonal position encoding to reconstruct high-resolution images for arbitrary scales.
• It replaces conventional INR-based modules with a deterministic process employing fixed trigonometric evaluations and matrix-vector multiplications to ensure efficient, mathematically interpretable inference.
• Empirical evaluations demonstrate competitive fidelity with state-of-the-art methods while significantly reducing computational and memory overhead, enabling faster super-resolution rendering.

The OPE-Upscale module is a parameter-free upsampling mechanism designed for arbitrary-scale image super-resolution (SR). It replaces conventional implicit neural representation (INR)-based upsampling modules by leveraging orthogonal position encoding (OPE). The OPE-Upscale module reconstructs high-resolution images in a mathematically interpretable and efficient manner, achieving competitive fidelity with state-of-the-art approaches while significantly reducing computational and memory requirements (Song et al., 2023).

1. Orthogonal Position Encoding: Mathematical Formulation

The foundation of the OPE-Upscale module is the orthogonal position encoding (OPE), an extension of standard position encoding. OPE defines an explicit, orthonormal basis for mapping 2D coordinates within $[-1,1]^2$ to a high-dimensional embedding. For an input coordinate $(x, y)$ and maximum frequency $n \in \mathbb{N}$:

• The 1D position encoding map is

$$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$

This yields a vector in $\mathbb{R}^{2n+1}$.

• The 2D encoding $P(x, y) \in \mathbb{R}^{(2n+1)^2}$ is constructed by the outer product and flattening:

$$X = \gamma(x),\quad Y = \gamma(y),\quad P(x, y) = \text{flatten}(X^T Y)$$

• The local reconstruction of a continuous image channel $f(x, y)$ is expressed as

$f(x, y) \approx ZP(x, y)^T$

where $Z$ is a learned coefficient (“projection”) vector.

This encoding forms an orthonormal basis under the $(x, y)$0 inner product, with explicit expressions for all basis functions (combining $(x, y)$1 and $(x, y)$2 terms in both directions).

2. OPE-Upscale Module: Architecture and Rendering Procedure

The OPE-Upscale module is structured around a clear separation of learning and deterministic inference:

• Encoder $(x, y)$3: A conventional convolutional network (e.g., EDSR-baseline or RDN) processes the low-resolution image $(x, y)$4 to generate the feature map $(x, y)$5. Each spatial location $(x, y)$6 in $(x, y)$7 contains the concatenated coefficient vectors $(x, y)$8, $(x, y)$9, $n \in \mathbb{N}$0 for the RGB channels.
• Rendering at Arbitrary Grid: For each output pixel coordinate $n \in \mathbb{N}$1:
  1. Locate the nearest feature-map cell center $n \in \mathbb{N}$2.
  2. Compute local relative coordinates:

  $n \in \mathbb{N}$3

1. Form $n \in \mathbb{N}$4 using OPE described above.
2. Reconstruct the SR pixel value for each channel via

   $n \in \mathbb{N}$5

3. To ensure seamless stitching, a weighted patch-ensemble of the four nearest neighbors (using bilinear interpolation weights) is used:

   $n \in \mathbb{N}$6

This process efficiently handles arbitrary-scale and continuous coordinates.

3. Parameter-Free and Analytical Properties

The OPE-Upscale module is characterized by its complete absence of trainable parameters in the upsampling stage:

• All learned parameters are contained within the encoder $n \in \mathbb{N}$7.
• The upsampling pipeline consists solely of fixed trigonometric evaluations, matrix-vector products, and linear combinations without any neural network layers (such as MLPs or convolutions) in the module itself.
• Given a feature map $n \in \mathbb{N}$8, every SR pixel is deterministically computed, establishing the OPE-Upscale module as analytically interpretable and fully parameter-free at inference.

4. Orthonormality and Mathematical Justification

OPE’s encoding functions form a mathematically orthonormal basis within the finite domain $n \in \mathbb{N}$9 under the $$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$0 inner product:

$$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$1

The family of basis functions consists of combinations of

• $$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$2
• $$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$3
• $$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$4
• $$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$5

for $$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$6, with normalization factors to ensure orthonormality. The orthogonality can be demonstrated by direct calculation of the relevant inner products—after incorporating the $$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$7 scaling in $$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$8, it follows that

$$\gamma(x) = [1, \sqrt{2}\cos(\pi x), \sqrt{2}\sin(\pi x), \ldots, \sqrt{2}\cos(n\pi x), \sqrt{2}\sin(n\pi x)]\ .$$9

where $\mathbb{R}^{2n+1}$0 denote the corresponding basis functions. The encoding $\mathbb{R}^{2n+1}$1 thus provides an orthonormal expansion suitable for analytical super-resolution reconstruction.

5. Algorithmic Workflow

The rendering algorithm for a high-resolution image is as follows:

$P(x, y) \in \mathbb{R}^{(2n+1)^2}$8

This procedure leverages only cos/sin evaluations and matrix-vector multiplications per pixel for highly efficient rendering.

6. Empirical Evaluation and Resource Analysis

Extensive experimentation confirms the following properties:

• Fidelity: On DIV2K-val (arbitrary scales $\mathbb{R}^{2n+1}$2–$\mathbb{R}^{2n+1}$3), the OPE-SR method narrows the PSNR gap to LIIF/LTE to less than 0.1 dB in most cases. On standard benchmarks (Set5, Set14, B100, Urban100), the drop is less than 0.15 dB. For extreme super-resolution factors ($\mathbb{R}^{2n+1}$4–$\mathbb{R}^{2n+1}$5), OPE matches or outperforms competitors.
• Efficiency: The computational requirements per SR pixel are approximately $\mathbb{R}^{2n+1}$6 multiply–accumulates and a handful of trigonometric function calls, compared to $\mathbb{R}^{2n+1}$7 in LIIF. Overall FLOPs for a full image are $\mathbb{R}^{2n+1}$8 million versus $\mathbb{R}^{2n+1}$9 billion. OPE-Upscale achieves $P(x, y) \in \mathbb{R}^{(2n+1)^2}$0–$P(x, y) \in \mathbb{R}^{(2n+1)^2}$1 times faster system inference (e.g., EDSR+LIIF $P(x, y) \in \mathbb{R}^{(2n+1)^2}$2 s/image vs. EDSR+OPE $P(x, y) \in \mathbb{R}^{(2n+1)^2}$3 s/image on DIV2K-val); rendering alone is $P(x, y) \in \mathbb{R}^{(2n+1)^2}$4–$P(x, y) \in \mathbb{R}^{(2n+1)^2}$5\% faster on large images.
• Memory Use: The module uses zero additional activations or gradients during training. LIIF/LTE incur $P(x, y) \in \mathbb{R}^{(2n+1)^2}$6–$P(x, y) \in \mathbb{R}^{(2n+1)^2}$7 MB of memory overhead, while OPE-Upscale incurs virtually none.

These results establish that the OPE-Upscale module enables mathematically interpretable, parameter-free, and resource-efficient arbitrary-scale image super-resolution while maintaining competitive output quality (Song et al., 2023).