QuadINR: Efficient Implicit Neural Representations

• QuadINR is an implicit neural representation framework that uses a periodically extended, piecewise quadratic activation to mitigate spectral bias while remaining hardware friendly.
• It replaces complex nonlinear functions with a simple two-segment quadratic design, enabling efficient FPGA/ASIC integration using only multipliers, adders, and lightweight control logic.
• Empirical results demonstrate up to 2.06 dB PSNR improvement and significant reductions in LUT usage, power, and latency compared to prior activation methods.

Searching arXiv for QuadINR and related INR papers to ground the article in the cited literature. QuadINR denotes, in the literature provided here, two related but distinct ideas within implicit neural representations (INRs). In its explicit and primary usage, QuadINR is a hardware-efficient INR architecture that replaces conventional activation functions with a periodically extended piecewise quadratic activation in order to reduce spectral bias while minimizing FPGA and ASIC cost (Zhou et al., 20 Aug 2025). In a separate, conceptual usage, the term is invoked to describe a unified INR philosophy for dynamic and quantitative MRI, instantiated by MoE-dqINR as a scan-specific, state-conditioned mixture-of-experts representation over shared spatial functions (Wu et al., 29 May 2026). The common thread is an attempt to improve INR efficiency without abandoning high-frequency expressivity, but the two usages target different optimization regimes, hardware assumptions, and application domains.

1. Terminological scope and research context

In the 2025 paper "QuadINR: Hardware-Efficient Implicit Neural Representations Through Quadratic Activation" (Zhou et al., 20 Aug 2025), QuadINR is a concrete INR design for images and videos in which the central innovation is the activation function. The stated motivation is that standard MLP-based INRs exhibit spectral bias, fitting low-frequency components first, while many prior remedies rely on sine, Gaussian, Snake, WIRE, or FINER-style nonlinearities that are comparatively hardware-heavy because they require LUTs, CORDIC, Taylor expansions, exponentials, or products of trigonometric functions. QuadINR therefore targets reduced spectral bias and improved high-frequency expressivity using activation functions that are harmonic-rich yet hardware-friendly, requiring only multipliers and adders and enabling a unified pipeline for FPGA/ASIC (Zhou et al., 20 Aug 2025).

The 2026 MRI paper "MoE-dqINR: A Unified Mixture-of-Experts Implicit Neural Representation Framework for Scan-Specific Dynamic and Quantitative MRI Reconstruction" (Wu et al., 29 May 2026) does not use the name QuadINR as its formal method name. However, the provided details state that MoE-dqINR embodies the essence of a unified INR for dynamic and quantitative MRI and can be viewed as a concrete instantiation of a "QuadINR" philosophy. In that usage, QuadINR refers to a scan-specific image-first prior that reconstructs time-resolved cine frames and ordered qMRI contrast states using the same state-conditioned mixture-of-experts over shared spatial functions. This suggests that the term has acquired a broader interpretive role beyond the hardware-focused activation-function design.

A recurrent misconception is to treat these two usages as a single architecture. They are not. The hardware-oriented QuadINR is defined by a quadratic activation and a pipeline-compatible accelerator design (Zhou et al., 20 Aug 2025), whereas the MRI-oriented usage refers to a unified reconstruction framework built around decoupled spatial experts and state-conditioned routing (Wu et al., 29 May 2026).

2. Piecewise quadratic activation and periodic construction

The defining mechanism of QuadINR in the explicit sense is a simple two-segment quadratic activation function, periodically extended with base period $T=4$ (Zhou et al., 20 Aug 2025). On the interval $(-2,2)$ it is defined as

$$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$

and then extended periodically to all $x\in\mathbb{R}$ with period $4$.

Its derivative on the base interval is

$$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$

The coefficient choices ensure continuity and differentiability at $x=0$ and at the periodic endpoints $x=\pm 2$, so that with periodic extension both $\phi$ and $\phi'$ are continuous across period boundaries; hence $(-2,2)$0 is $(-2,2)$1 periodic (Zhou et al., 20 Aug 2025).

The paper also gives a general formulation for piecewise quadratic activations over one period,

$(-2,2)$2

with continuity and differentiability constraints at each boundary $(-2,2)$3 between intervals:

$(-2,2)$4

$(-2,2)$5

For periodic extension with period $(-2,2)$6, one additionally enforces $(-2,2)$7 and $(-2,2)$8 at a reference point to avoid boundary artifacts. QuadINR adopts the two-segment case because it yields what the paper describes as the minimal hardware footprint: two multiplies and one add per activation, plus lightweight segment selection (Zhou et al., 20 Aug 2025).

This design choice is central to the method’s identity. Unlike SIREN or wavelet-like nonlinearities, the activation is evaluated exactly as a polynomial and therefore introduces zero approximation error at the activation level under the FP32 arithmetic used in the pipeline (Zhou et al., 20 Aug 2025).

3. Harmonic content, spectral bias, and NTK characterization

The theoretical justification for QuadINR is that periodic piecewise polynomial functions admit Fourier expansions with nontrivial harmonic content. For the odd periodic extension of $(-2,2)$9 with period $$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$0, the Fourier series is written as

$$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$1

Because $$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$2 is odd, $$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$3, and the sine coefficients satisfy

$$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$4

The paper therefore writes the activation approximately as

$$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$5

with higher odd harmonics decaying as $$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$6 (Zhou et al., 20 Aug 2025).

The significance attributed to this expansion is that the activation injects a controlled harmonic ladder. Compared with a single-frequency sine activation, this is presented as substantially reducing spectral bias while retaining smooth gradients. The cubic decay on $$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$7 is said to encourage low-to-mid frequency components during the early training stage while preserving high-frequency capacity (Zhou et al., 20 Aug 2025).

The paper also analyzes training dynamics through the Neural Tangent Kernel (NTK). For a single-layer coordinate MLP with output $$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$8, the NTK is

$$\phi(x)= \begin{cases} x^2 + 2x, & -2 < x \le 0 \ -x^2 + 2x, & 0 < x < 2 \end{cases}$$9

The comparison given is

$x\in\mathbb{R}$0

and

$x\in\mathbb{R}$1

The stated distinction is that QuadINR’s kernel exhibits polynomial scaling in $x\in\mathbb{R}$2, with a dominant $x\in\mathbb{R}$3 term, whereas SIREN’s scales quadratically but is modulated by oscillatory cosine terms. The paper interprets this as indicating strong coupling to high-frequency components while maintaining smooth gradient propagation (Zhou et al., 20 Aug 2025).

A plausible implication is that QuadINR’s contribution lies not merely in replacing one activation with another, but in aligning spectral expressivity, optimization behavior, and hardware realizability within a single construction.

4. Pipeline architecture and hardware realization

QuadINR is embedded in a hardware/software co-design framework. The accelerator architecture described in the paper consists of a specialized input layer and three linear layers with pipelined processing units for coordinate-to-pixel generation, yielding five layers total with the output layer lacking an activation. The linear layer comprises 256 parallel MAC arrays, a 256-input addition tree, and two RAM blocks, while dedicated memory control orchestrates pipeline timing. Weight and bias RAMs store parameters, intermediate RAM buffers partial results, and activation-function modules are instantiated for the input and three linear layers, giving four AF modules in total (Zhou et al., 20 Aug 2025).

A notable aspect is the unified $x\in\mathbb{R}$4-stage pipeline framework that can accommodate Taylor-expanded AFs as well as the native piecewise quadratic AF. The conceptual stages include power-term multiplication, coefficient multiplication, repeated accumulation stages, and final polynomial addition. For QuadINR itself, the mapping is much simpler: only two pipeline stages are needed, with one stage computing $x\in\mathbb{R}$5 and $x\in\mathbb{R}$6 and a second stage performing the polynomial addition. Segment selection and periodic folding use a lightweight comparator and periodic mapping into $x\in\mathbb{R}$7, with no division, LUT, CORDIC, or transcendental units required (Zhou et al., 20 Aug 2025).

The timing model reported is that latency is approximately the number of pipeline stages times $x\in\mathbb{R}$8. At $x\in\mathbb{R}$9 MHz, QuadINR AF latency is $4$0 ns, compared with $4$1 ns for SIREN, $4$2 ns for WIRE, $4$3 ns for FINER, $4$4 ns for Gaussian, and $4$5 ns for sinc. Throughput after pipeline fill is one activation per cycle, and the end-to-end latency of the full INR accelerator is $4$6 ns at $4$7 MHz (Zhou et al., 20 Aug 2025).

The implementation rationale is correspondingly direct: the activation uses arithmetic only, with multipliers and adders in FP32, simple control via comparators, and trivial coefficient storage. The paper attributes low DSP/LUT usage and minimal switching activity to this design, and therefore connects the activation choice to both area and power savings (Zhou et al., 20 Aug 2025).

5. Empirical performance in software and hardware

The experimental setup uses PyTorch 1.11.0 with Adam and a learning rate that decays to $4$8 of its initial value over training. Datasets include Kodak images and a $4$9-second video clip at $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$0 fps, with PSNR as the metric. Positional encodings are not required and are not used, because the activation functions are taken to provide harmonic content intrinsically (Zhou et al., 20 Aug 2025).

For image reconstruction on Kodak samples, the paper reports the following PSNR values for four representative images:

Sample	Prior AFs	QuadINR
Kodak01	SIREN 30.52, WIRE 30.95, FINER 35.17, Gaussian 29.94	32.58
Kodak02	SIREN 35.21, WIRE 35.61, FINER 37.90, Gaussian 34.20	36.21
Kodak03	SIREN 36.26, WIRE 37.96, FINER 40.87, Gaussian 37.91	38.86
Kodak04	SIREN 34.52, WIRE 35.78, FINER 38.65, Gaussian 34.36	36.32

The aggregate statement is that QuadINR achieves up to $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$1 dB PSNR improvement over prior work while delivering major hardware savings (Zhou et al., 20 Aug 2025). In the video regression study, qualitative comparisons are said to show sharper details such as fur texture, whiskers, and eye regions relative to SIREN and WIRE, while FINER is described as competitive in visual quality but more hardware-demanding (Zhou et al., 20 Aug 2025).

The hardware comparisons are equally central. On FPGA (Xilinx VCU128 at $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$2 MHz), the AF module for QuadINR uses $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$3 LUTs, $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$4 FFs, $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$5 DSPs, $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$6 mW, and $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$7 ns latency; on ASIC (SMIC 28 nm, $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$8 V, $$\phi'(x)= \begin{cases} 2x + 2, & -2 < x \le 0 \ -2x + 2, & 0 < x < 2 . \end{cases}$$9C, $x=0$0 GHz), it occupies $x=0$1, with $x=0$2 mW static power, $x=0$3 mW dynamic power, and $x=0$4 energy for a $x=0$5 image (Zhou et al., 20 Aug 2025). The paper further reports reductions versus WIRE of $x=0$6 LUTs, $x=0$7 power, and $x=0$8 latency on FPGA.

These results frame QuadINR as a co-designed compromise: lower hardware overhead than sine-, Gaussian-, or wavelet-like activations, while retaining competitive reconstruction quality.

6. Limitations, trade-offs, and practical deployment

The paper explicitly notes that more expressive activations can still outperform QuadINR on some inputs. For example, FINER reaches $x=0$9 dB on Kodak03, compared with QuadINR’s $x=\pm 2$0 dB, which the authors interpret as evidence that extremely high-frequency or localized spectral components may benefit from more complex AFs (Zhou et al., 20 Aug 2025). QuadINR’s activation is also strictly periodic with period $x=\pm 2$1, so periodic folding imposes a global spectral prior that may not match signals with strongly non-periodic localized content.

Another trade-off concerns scale sensitivity. The NTK analysis depends on $x=\pm 2$2, and the paper states that excessively large $x=\pm 2$3 can amplify polynomial terms and risk gradient explosion if not paired with an appropriate learning-rate schedule. Scalability is likewise bounded by the broader INR system: even with an efficient activation, extremely deep or wide INRs or 3D signals can remain memory-bound or bandwidth-limited, and the $x=\pm 2$4 harmonic decay means extremely fine textures may require more layers or residual connections (Zhou et al., 20 Aug 2025).

The practical guidance given is correspondingly specific. For deployment, one can replace all activations in coordinate MLPs with QuadINR; for SIREN-like systems, one can swap sine AF modules with QuadINR modules while retaining $x=\pm 2$5 for scale control. The paper recommends moderate $x=\pm 2$6, a two-segment design for the best hardware trade-off, input rescaling into $x=\pm 2$7 for stable behavior, standard Adam with decay to approximately $x=\pm 2$8 of the initial learning rate, and periodic folding implemented with add/subtract and comparators rather than division (Zhou et al., 20 Aug 2025).

A plausible implication is that QuadINR is best understood not as a universally dominant INR nonlinearity, but as a design point optimized for the joint objective of harmonic richness and implementation economy.

7. Relation to the MRI-oriented “QuadINR” interpretation

The MoE-dqINR work extends the semantic scope of QuadINR in a different direction. The paper addresses undersampled multicoil MRI reconstruction over an ordered set of acquisition states, with dynamic cine MRI and quantitative MRI treated in a unified scan-specific INR formulation (Wu et al., 29 May 2026). Its representation factorizes image synthesis into a shared background, shared spatial experts, and a state-conditioned routing pathway:

$x=\pm 2$9

where $\phi$0 is the normalized spatial coordinate and the normalized state index is $\phi$1 (Wu et al., 29 May 2026). Routing weights are produced by an MLP operating on a Fourier-encoded state feature,

$\phi$2

with simplex constraints $\phi$3 and $\phi$4.

In cine MRI, the model optionally augments this mixture with a compact residual branch,

$\phi$5

with $\phi$6 and $\phi$7 (Wu et al., 29 May 2026). The reconstruction is coupled to the multicoil forward model

$\phi$8

and optimized by minimizing multicoil k-space data inconsistency across all states,

$\phi$9

with weight decay on image-generating parameters but not on routing or time-conditioning parameters (Wu et al., 29 May 2026).

The paper reports approximately $\phi'$0 s per scan for cine and approximately $\phi'$1 s for T1-mapping on a single NVIDIA RTX 3090, with inference time below $\phi'$2 s. On CMRxRecon cine, it reports best PSNR/SSIM/LPIPS at acceleration factors $\phi'$3, $\phi'$4, and $\phi'$5, including $\phi'$6 dB / $\phi'$7 / $\phi'$8 at $\phi'$9 and $(-2,2)$00 / $(-2,2)$01 / $(-2,2)$02 at $(-2,2)$03; on CMRxRecon T1-mapping, it reports the best per-state metrics and lowest T1-map NMSE across acceleration factors, including T1 NMSE values of $(-2,2)$04, $(-2,2)$05, and $(-2,2)$06 for $(-2,2)$07, $(-2,2)$08, and $(-2,2)$09 (Wu et al., 29 May 2026).

The provided text explicitly states that MoE-dqINR is a concrete instantiation of a “QuadINR” philosophy: a unified, scan-specific implicit representation that covers both dynamic MRI and qMRI within one decoupled image-first prior. This usage differs fundamentally from the activation-centric QuadINR of (Zhou et al., 20 Aug 2025). The former is a domain-specific formulation for multicoil MRI reconstruction; the latter is a hardware-efficient INR primitive for generic continuous signal regression. Their connection is conceptual rather than architectural: both seek to reconcile expressivity and efficiency, but they do so through different abstractions.