Directional Total Variation-Regularized Implicit Neural Representations (DTV-INR) for Continuous Super-Resolution in Degraded Imaging Domains

Mahmoud Saeedi Kelishami

cs.CVSep 21, 2026

Directional Total Variation-Regularized Implicit Neural Representations (DTV-INR) for Continuous Super-Resolution in Degraded Imaging Domains

Authors: Mahmoud Saeedi Kelishami

Organizations: Department of Mathematics, Institute of Biosocial and Quantum Science and Technologies, Rasht Branch, Islamic Azad University, Rasht, Iran

Abstract

In this paper, we introduce the Directional Total Variation-Regularized Implicit Neural Representation (DTV-INR), an advanced variational paradigm that synergistically integrates coordinate-driven implicit neural networks with an anisotropic, structure-tensor-informed total variation regularizer tailored for resolution-agnostic image super-resolution. Casting the continuous-to-discrete acquisition process into an ill-posed inverse problem framework, our formulation equips a SIREN-architected coordinate network with a dynamic Riemannian metric tensor field D(x). By leveraging its spectral decomposition, the proposed regularizer preferentially directs diffusion parallel to dominant structural contours while penalizing cross-edge dissipation, successfully circumventing the classical staircasing artifacts inherent to scalar total variation schemes. We rigorously prove the well-posedness of this formulation in H^1(Omega) by establishing the existence, uniqueness, and metric stability of the variational minimizer, and realize this via an alternating projected optimization algorithm that decouples network parameter tuning from adaptive tensor field updates. Comprehensive experiments conducted on clinical brain magnetic resonance imaging (MRI) and biomedical transmission electron microscopy confirm substantial quantitative and qualitative improvements, yielding PSNR enhancements reaching +5.05 dB over baseline unregularized INRs and +1.71-2.85 dB over isotropic TV-INR across continuous (non-integer) upsampling factors, alongside remarkable noise robustness up to sigma_eta = 0.10 and monotonic preconditioned convergence behavior.

Figures & tables

Method	Type	$\times 2$	$\times 3$	$\times 4$	$\times 6$	$\times 8$
Bicubic Interpolation	Continuous	31.84 / 0.884	28.52 / 0.812	26.21 / 0.745	23.12 / 0.638	21.20 / 0.542
Discrete TV (CP)	Discrete	33.40 / 0.902	29.80 / 0.835	27.42 / 0.778	24.15 / 0.672	22.08 / 0.579
EDSR	Discrete	35.42 / 0.931	31.80 / 0.875	29.20 / 0.821	25.24 / 0.710 †	22.80 / 0.612 †
LIIF (Continuous SR)	Implicit	36.21 / 0.942	32.74 / 0.891	30.52 / 0.852	27.40 / 0.781	25.12 / 0.715
Vanilla SIREN INR	Implicit	33.95 / 0.910	30.12 / 0.846	27.65 / 0.784	24.38 / 0.680	22.25 / 0.584
Isotropic TV-INR	Implicit	35.80 / 0.935	32.15 / 0.880	30.70 / 0.856	26.85 / 0.765	24.45 / 0.692

Table 1: Quantitative reconstruction performance (PSNR [dB] / SSIM) across varying magnification factors

s

on biomedical microscopic and neuroimaging test suites. Best results are highlighted in bold , and second-best are underlined .

Method	$\sigma_{\eta}{=}0.01$	$\sigma_{\eta}{=}0.03$	$\sigma_{\eta}{=}0.05$	$\sigma_{\eta}{=}0.07$	$\sigma_{\eta}{=}0.10$
Bicubic	27.40/0.781/0.384	24.80/0.695/0.452	22.10/0.584/0.540	20.20/0.490/0.621	17.80/0.385/0.710
LIIF	31.85/0.882/0.220	28.60/0.804/0.295	25.40/0.712/0.385	22.90/0.605/0.480	19.80/0.485/0.602
Vanilla SIREN	30.50/0.865/0.252	27.00/0.762/0.334	23.40/0.640/0.442	20.90/0.528/0.560	18.20/0.412/0.685
Isotropic TV-INR	32.20/0.895/0.205	29.30/0.825/0.278	26.50/0.742/0.360	24.10/0.650/0.445	21.50/0.535/0.565
Proposed DTV-INR	34.60 / 0.938 / 0.142	32.10 / 0.886 / 0.198	29.40 / 0.820 / 0.270	27.20 / 0.755 / 0.345	24.80 / 0.670 / 0.440

Table 2: Reconstruction robustness under increasing measurement noise

\sigma_{\eta}\in[0.01,0.10]

under

\times 4

super-resolution. Metrics reported: PSNR [dB] / SSIM / LPIPS (

\downarrow

Configuration	SIREN	Tikhonov	Isotropic	Anisotropic	Dynamic	PSNR [dB]
	Backbone	$L^{2}$ ( $\mu$ )	TV	$D(x)$	Tensor	/ SSIM
(1) Vanilla INR	✓	–	–	–	–	27.00 / 0.762
(2) Tikhonov Only	✓	✓	–	–	–	28.15 / 0.795
(3) Isotropic TV	✓	✓	✓	–	–	29.30 / 0.825
(4) Static Anisotropic $D(x)$	✓	✓	–	✓	–	31.42 / 0.856
(5) Complete DTV-INR	✓	✓	–	✓	✓	32.10 / 0.886

Table 3: Ablation study evaluating the functional components of the proposed DTV-INR formulation (

\times 4

super-resolution,

\sigma_{\eta}=0.03

Figure 1: Overall schematic of the Directional Total Variation-Regularized Implicit Neural Representation (DTV-INR) framework. The degraded observation $\mathbf{y}$ is transformed into an anisotropic diffusion tensor field $\mathbf{D}(\mathbf{x})$ that guides the coordinate-based implicit network $f_{\theta}$ to preserve structural boundaries and fine edges.

Figure 2: Quantitative comparison of PSNR across benchmark methods on medical and microstructure test datasets, highlighting the performance margins attained by DTV-INR.

Figure 3: Continuous super-resolution scaling evaluation. Reconstruction quality (PSNR) as a continuous function of magnification factor from $\times 2$ to $\times 8$ .

Figure 4: Robustness analysis against severe additive Gaussian and Rician noise profiles across varying signal-to-noise ratios (SNR).

Figure 5: Quantitative computational profiling: (a) Variational objective convergence over training iterations $k$ , demonstrating accelerated, monotonic descent and lower terminal energy for DTV-INR; (b) Robustness trajectory under severe observational noise $\sigma_{\eta}$ at $\times 4$ resolution; (c) Continuous scale generalization behavior across non-integer magnification factors $s\in[2.0,8.0]$ .

Figure 6: Qualitative super-resolution comparison ( $\times 4$ magnification with additive noise $\sigma_{\eta}=0.05$ ): (a) Ground Truth HR reference; (b) Degraded low-resolution observation; (c) Bicubic interpolation; (d) Vanilla unregularized SIREN; (e) Isotropic TV-INR showing characteristic blocky staircasing; (f) Proposed DTV-INR preserving sharp, smooth curvilinear edges without piecewise-constant segmentation artifacts.

Figure 7: Continuous super-resolution benchmark on the analytical Shepp–Logan phantom ( $4\times$ degradation factor, generated via main2.py ). Comparison highlights: (a) Bicubic baseline ( $17.52\,\mathrm{dB}$ , $\mathrm{SSIM}=0.584$ ) exhibiting severe blur; (b) Vanilla SIREN ( $18.15\,\mathrm{dB}$ , $\mathrm{SSIM}=0.612$ ) with high-frequency ringing; (c) SIREN + IsoTV ( $19.45\,\mathrm{dB}$ , $\mathrm{SSIM}=0.689$ ) afflicted by staircasing faceting; and (d) Proposed DTV-INR ( $20.61\,\mathrm{dB}$ , $\mathrm{SSIM}=0.748$ ) maintaining smooth curvilinear level sets and high geometric fidelity.

Figure 8: Magnified inspection of curvilinear boundary interfaces from the Shepp–Logan phantom reconstructions. While standard isotropic Total Variation induces false piecewise-planar faceting (staircasing) across continuous curves, the directional diffusion tensor in DTV-INR channels smoothing tangentially, thereby recovering pristine elliptical contours with intact boundary contrast.

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Image super-resolution reconstructs high-resolution images from lower-resolution observations, balancing detail, fidelity, and computational cost.

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Resolving complex fiber geometries in brain white matter requires high-resolution diffusion MRI at the cost of long acquisition times. This leads many clinical protocols to opt for low-resolution scans, making downstream microstructure estimation and tractography challenging. Implicit neural representations (INRs) can model the diffusion signal continuously, enabling native single-subject super-resolution by querying the network at arbitrary spatial coordinates, yet existing methods often suffer from long training times and lack a mechanism to incorporate anatomical priors to regularize super-resolution by constraining the space of plausible reconstructions. To address these limitations, we propose a novel transfer-learning framework that pre-trains an INR on a high-resolution template and then adapts it to subject-specific scans via registration and fine-tuning. For

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