Abstract
Coordinate-based neural networks rely on Random Fourier Features (RFF) to overcome spectral bias; however, the sinusoidal bases underlying RFF impose smooth priors that produce Gibbs ringing at sharp discontinuities. In medical imaging, edge fidelity determines diagnostic accuracy, so this smoothness prior is a poor match for anatomical boundaries. We reframe positional encoding as a basis-geometry problem and introduce \emph{Piecewise Positional Encodings} (PPE), which replace the sinusoidal basis functions of RFF with square, triangle, sawtooth, and trapezoid waveforms while preserving the random-projection framework. A single square wave at frequency $f$ implicitly encodes all odd harmonics ($f, 3f, 5f, \dots$) decaying as $1/n$ through its Fourier series, thus natively representing step-like tissue boundaries. PPE requires no architectural change, adds no hyperparameters, and is a drop-in replacement for sinusoidal encodings. We evaluated PPE as an image-representation method: we fit an implicit neural representation(INR) to single 2D slices from five modalities (MRI, CT, X-ray, ultrasound, and histopathology) and measure how faithfully each encoding reproduces the image, rather than solving an inverse problem from raw sensor data. Square-wave encoding achieves $44.84 \pm 2.90 \ \mathrm{dB}$ PSNR, outperforming Fourier features ($39.87 \pm 5.64 \ \mathrm{dB}$, a $+5.0 \ \mathrm{dB}$ gain), with tighter variance. The gain track: they are largest in MRI ($+8.6 \ \mathrm{dB}$) and histopathology ($+8.3 \ \mathrm{dB}$), intermediate in CT ($+5.9\ \mathrm{dB}$) and ultrasound($+4.1 \ \mathrm{dB}$) and slightly negative in X-ray ($-2.1 \ \mathrm{dB}$). Square-wave encodings reduce the edge-to-flat PSNR gap from $9.5 \ \mathrm{dB}$ (Fourier) to $2.2\ \mathrm{dB}$ and lowers the ringing scores by $63\%$, at the same parameter count and training cost as Fourier features. A learned hash encoding (Instant-NGP) reaches higher absolute fidelity, but PPE is the strongest fixed, parameter-free encoding. These results establish the geometry of the basis function as a design variable for coordinate networks in medical image representation.
Links to Paper and Supplementary Materials
Main Paper (Open Access Version): https://papers.miccai.org/miccai-2026-sat/paper/Off_Grid_031.pdf
SharedIt Link: Not yet available
SpringerLink (DOI): Not yet available
Supplementary Material: Not Submitted
Link to Open Review
Open Review Page: https://openreview.net/forum?id=4DcJHzWVRJ
BibTex
@InProceedings{KanAnu_Piecewise_MICCAISAT2026,
author = { Kanagala, Anusha AND Aguilera, Stephanie M. AND Panjwani, Dhiral D. AND Pudhota, Abhiraj AND Bodduluri, Sandeep},
title = { { Piecewise Positional Encodings (PPE) for Medical Image Representation } },
booktitle = {Medical Image Computing and Computer Assisted Intervention -- MICCAI 2026 Workshops and Challenges},
year = {2026},
publisher = {Springer Nature Switzerland},
volume = {LNCS 17277},
month = {pending},
page = {pending}
}