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In classical mechanics, motion is often described through relationships between spatial coordinates and time. A car moving along a straight highway with constant acceleration serves as a simple case where velocity is an explicit function of time. This scenario results in a linear equation, enabling straightforward analysis using basic differentiation techniques.In contrast, a satellite in circular orbit follows a path defined by an implicit function. The position of the satellite is constrained...
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Implicit memories, also known as non-declarative memories, are long-term memories that function outside of conscious awareness. These memories influence behavior and skills without explicit knowledge. This type of memory is evident in tasks like playing tennis, snowboarding, and texting. Implicit memory has three subsystems: procedural memory, conditioning, and priming. This type of memory is essential in various activities, from everyday tasks to specialized skills.
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The sign test for matched pairs offers a robust method for comparing two paired samples, often for the effects of an intervention in one of them. This method is very useful in situations where the underlying distribution of the data is unknown. The test compares two related samples—often pre- and post-treatment measurements on the same subjects—to determine if there are significant differences in their median values.
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Curves defined implicitly, where variables cannot be separated algebraically, require specialized techniques for analysis. The conchoid of Nicomedes exemplifies such a case. Its equation links x and y in a way that prevents isolation of one variable, making implicit differentiation essential to determine the slope and behavior at any point on the curve.The implicit form of the conchoid can be expressed as:To differentiate this equation, y is treated as a function of x, and the chain rule is...
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Related Experiment Video

Updated: Feb 5, 2026

Co-analysis of Brain Structure and Function using fMRI and Diffusion-weighted Imaging
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Landmark matching and B-spline implicit neural representations for diffusion-weighted imaging distortion correction.

Yunxiang Li1, Yen-Peng Liao1, Yan Dai1

  • 1Department of Radiation Oncology, University of Texas Southwestern Medical Center, Dallas, TX 75390, United States of America.

Physics in Medicine and Biology
|February 3, 2026
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Summary

Geometric distortions in diffusion-weighted imaging (DWI) hinder radiation therapy planning. This study introduces a novel framework for accurate DWI distortion correction, improving tumor delineation and treatment assessment.

Keywords:
diffusion-weighted imagingdistortion correctionimplicit neural representationlandmark matchingmultimodal registration

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Area of Science:

  • Medical Imaging
  • Radiotherapy
  • Computational Anatomy

Background:

  • Geometric distortions in diffusion-weighted imaging (DWI) impede precise tumor delineation and localization for radiation therapy.
  • Conventional mutual information optimization methods for correcting these distortions often result in non-smooth and physically implausible deformations due to local minima.

Purpose of the Study:

  • To propose and evaluate a novel Landmark Matching B-spline Implicit Neural Representation (LMBS-INR) framework for accurate DWI distortion correction.
  • To overcome the limitations of traditional optimization methods by integrating anatomical landmark correspondences and B-spline deformation fields.

Main Methods:

  • The LMBS-INR framework utilizes a foundation landmark matching model to establish anatomical correspondences.
  • B-spline deformation fields, modeled using Fourier-encoded multi-layer perceptrons, ensure physically plausible transformations for robust multimodal registration between DWI and anatomical references.

Main Results:

  • The proposed method demonstrated superior performance on brain and abdominal datasets, achieving high Dice coefficients (0.919 ± 0.038 for brain, 0.926 ± 0.032 for abdomen).
  • Evaluations on simulated data yielded excellent metrics, including PSNR (25.912 ± 3.148 dB), NCC (0.911 ± 0.137), and SSIM (0.888 ± 0.107), outperforming all baseline methods.

Conclusions:

  • The LMBS-INR framework effectively corrects geometric distortions in DWI by combining B-spline parameterization with foundation model capabilities.
  • This approach offers enhanced precision for intra/post-radiotherapy assessment, improving the utility of DWI in clinical practice.