Intrinsic Neural Fields: Learning Functions on Manifolds

Koestler, Lukas; Grittner, Daniel; Moeller, Michael; Cremers, Daniel; Lähner, Zorah

doi:10.1007/978-3-031-20086-1_36

Cited by 8 publications

(4 citation statements)

References 43 publications

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“…Early methods include sinusoidal functions [VSP*17; XRK*19], while more recent approaches consider learned positional embeddings [LSL*21]. The use of the eigenvectors of the Laplacian to encode locations has recently been explored in the context of graphs [DLL*21] and for texture reconstruction of meshes from multiple views [KGM*22].…”

Section: Related Effortsmentioning

confidence: 99%

Partial Matching of Nonrigid Shapes by Learning Piecewise Smooth Functions

David

Rotstein

Goldenstein

et al. 2023

Computer Graphics Forum

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Learning functions defined on non‐flat domains, such as outer surfaces of non‐rigid shapes, is a central task in computer vision and geometry processing. Recent studies have explored the use of neural fields to represent functions like light reflections in volumetric domains and textures on curved surfaces by operating in the embedding space. Here, we choose a different line of thought and introduce a novel formulation of partial shape matching by learning a piecewise smooth function on a surface. Our method begins with pairing sparse landmarks defined on a full shape and its part, using feature similarity. Next, a neural representation is optimized to fit these landmarks, efficiently interpolating between the matched features that act as anchors. This process results in a function that accurately captures the partiality. Unlike previous methods, the proposed neural model of functions is intrinsically defined on the given curved surface, rather than the classical embedding Euclidean space. This representation is shown to be particularly well‐suited for representing piecewise smooth functions. We further extend the proposed framework to the more challenging part‐to‐part setting, where both shapes exhibit missing parts. Comprehensive experiments highlight that the proposed method effectively addresses partiality in shape matching and significantly outperforms leading state‐of‐the‐art methods in challenging benchmarks. Code is available at https://github.com/davidgip74/Learning-Partiality-with-Implicit-Intrinsic-Functions

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Section: Related Effortsmentioning

confidence: 99%

Partial Matching of Nonrigid Shapes by Learning Piecewise Smooth Functions

David

Rotstein

Goldenstein

et al. 2023

Computer Graphics Forum

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“…CNRs for non-Euclidean data. Recently, researchers have also considered generalizing CNRs to non-Euclidean data (Esteves et al, 2022;Grattarola & Vandergheynst, 2022;Koestler et al, 2022;Schwarz et al, 2023;Huang & Hoefler, 2022). A simple approach is to lift the two-dimensional sphere into three-dimensional Euclidean space so that the Cartesian coordinates of the sphere can be used as inputs to the existing three-dimensional CNRs (Schwarz et al, 2023;Huang & Hoefler, 2023).…”

Section: Related Workmentioning

confidence: 99%

“…A simple approach is to lift the two-dimensional sphere into three-dimensional Euclidean space so that the Cartesian coordinates of the sphere can be used as inputs to the existing three-dimensional CNRs (Schwarz et al, 2023;Huang & Hoefler, 2023). To incorporate the geometry of the data, researchers considered using the eigenfunctions of the Laplace-Beltrami operator for general manifolds (Grattarola & Vandergheynst, 2022;Koestler et al, 2022). For spherical data, this corresponds to using spherical harmonics to featurize the spherical coordinates (Esteves et al, 2022).…”

Section: Related Workmentioning

confidence: 99%

“…To this end, recent works (Esteves et al, 2022;Grattarola & Vandergheynst, 2022;Koestler et al, 2022;Schwarz et al, 2023;Huang & Hoefler, 2023) have investigated CNRs based on coordinate-based multilayer perceptrons (MLPs). To featurize the spherical coordinates, they use the sinusoids from the Cartesian coordinate system (Schwarz et al, 2023;Huang & Hoefler, 2023), spectral embeddings of the discretized manifold (Grattarola & Vandergheynst, 2022;Koestler et al, 2022), or the spherical harmonics with fixed frequency components (Esteves et al, 2022). However, these methods suffer from the inherent limitation of coordinate-based MLPs: lack of expressive power to approximate highly nonlinear signals to their finest details.…”

Section: Introductionmentioning

confidence: 99%

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Relationship of Neural Correlates of Gait Characteristics and Cognitive Dysfunction in Patients With Mild Cognitive Impairment

Kim¹,

Park²,

Choi³

et al. 2023

Preprint

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Background Some patients with mild cognitive impairment (MCI) experience gait disturbances. However, there are few reports on the relationship between gait disturbance and cognitive function in patients with MCI. Therefore, we investigated the neural correlates of gait characteristics related to cognitive dysfunction. Methods Eighty patients diagnosed with MCI from three dementia centres in Gangwon-do, Korea were recruited for this study. We defined MCI as a Clinical Dementia Rating global score of 0.5 or higher, with a memory domain score of 0.5 or greater. The patients were classified as having either early or late MCI based on their Mini Mental Status Examination z-scores. Multiple logistic regression analysis was performed to examine the association between the gait characteristics and cognitive impairment. Analyses included variables such as age, sex, years of education, number of comorbidities, body mass index and height. Results Gait velocity, step count, step length, heel-to-heel base support, swing and stance phase duration, and support time were associated with cognitive function. A decrease in gray matter volume in the right pericalcarine area was associated with gait characteristics related to cognitive dysfunction. An increase in the curvature of gray matter in the right entorhinal, right lateral orbitofrontal, right cuneus, and right and left pars opercularis areas was also associated with gait characteristics related to cognitive dysfunction. Conclusion Since gait impairment is an important factor in determining activities of daily living in patients with mild cognitive impairment, the evaluation of gait and cognitive functions in patients with mild cognitive impairment is important.

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