Shape Learning in Biomedical Imaging

Nina Miolane

**Concept Graph & Resume using Claude 3 Opus | Chat GPT4o | Llama 3:**

graph LR
classDef shape fill:#f9d4d4, font-weight:bold, font-size:14px
classDef learning fill:#d4f9d4, font-weight:bold, font-size:14px
classDef manifolds fill:#d4d4f9, font-weight:bold, font-size:14px
classDef applications fill:#f9f9d4, font-weight:bold, font-size:14px
classDef geomstats fill:#f9d4f9, font-weight:bold, font-size:14px
A[Shape Learning in

Biomedical Imaging] --> B[Shape learning:

machine learning on shapes 1] A --> C[Biomedical imaging:

biological structures, various scales 2] C --> D[Shape-biology link:

structure, function, health 3] C --> E[Imaging techniques:

MRI, microscopy, cryo-EM 4] C --> F[Biomedical questions:

shape changes, disease 5] C --> G[Shape reconstruction:

extraction from images 6] B --> H[Shape modeling/representation:

numerical representation for analysis 7] A --> I[Manifolds: curved

generalizations of vector spaces 8] I --> J[Shapes as manifolds:

object surface 9] I --> K[Shape spaces as manifolds:

all possible shapes 10] I --> L[Shape transformation spaces

as manifolds: motions, deformations 11] B --> M[Statistics generalized

to manifolds 12] M --> N[Manifold geometry:

inductive bias for learning 13] B --> O[ML problems on manifolds:

supervised, unsupervised, RL, optimization 14] O --> P[Manifold learning building blocks:

points, tangents, geodesics, distances 15] O --> Q[Vector space to

manifold ML conversion 16] A --> R[Geomstats: Python package

for manifold computations 17] R --> S[Learning on Riemannian manifolds 18] R --> T[Pose/transformation manifolds: SE3 19] R --> U[Manifold taxonomy:

abstract to concrete 20] B --> V[Manifold-adapted ML overview 21] V --> W[Dimension reduction on manifolds:

PCA, autoencoders 22] W --> X[Geometric view:

principal subspaces 23] V --> Y[Variational autoencoders VAEs 24] Y --> Z[Generalizing VAEs to manifolds 25] Z --> AA[mVAE components:

exponential map, geodesic distance 26] Z --> AB[mVAE vs alternatives:

faster than MCMC 27] Z --> AC[mVAE explains

VAE latent space curvature 28] A --> AD[Shape analysis pipeline:

extraction, modeling, manifolds, insights 29] A --> AE[Ongoing research:

generalize ML, biomedical applications 30] class A,H,M,N,V,W,X,Y,Z,AA,AB,AC learning class B,D,F,G,J shape class C,E applications class I,K,L,O,P,Q,S,T,U manifolds class R geomstats

Biomedical Imaging] --> B[Shape learning:

machine learning on shapes 1] A --> C[Biomedical imaging:

biological structures, various scales 2] C --> D[Shape-biology link:

structure, function, health 3] C --> E[Imaging techniques:

MRI, microscopy, cryo-EM 4] C --> F[Biomedical questions:

shape changes, disease 5] C --> G[Shape reconstruction:

extraction from images 6] B --> H[Shape modeling/representation:

numerical representation for analysis 7] A --> I[Manifolds: curved

generalizations of vector spaces 8] I --> J[Shapes as manifolds:

object surface 9] I --> K[Shape spaces as manifolds:

all possible shapes 10] I --> L[Shape transformation spaces

as manifolds: motions, deformations 11] B --> M[Statistics generalized

to manifolds 12] M --> N[Manifold geometry:

inductive bias for learning 13] B --> O[ML problems on manifolds:

supervised, unsupervised, RL, optimization 14] O --> P[Manifold learning building blocks:

points, tangents, geodesics, distances 15] O --> Q[Vector space to

manifold ML conversion 16] A --> R[Geomstats: Python package

for manifold computations 17] R --> S[Learning on Riemannian manifolds 18] R --> T[Pose/transformation manifolds: SE3 19] R --> U[Manifold taxonomy:

abstract to concrete 20] B --> V[Manifold-adapted ML overview 21] V --> W[Dimension reduction on manifolds:

PCA, autoencoders 22] W --> X[Geometric view:

principal subspaces 23] V --> Y[Variational autoencoders VAEs 24] Y --> Z[Generalizing VAEs to manifolds 25] Z --> AA[mVAE components:

exponential map, geodesic distance 26] Z --> AB[mVAE vs alternatives:

faster than MCMC 27] Z --> AC[mVAE explains

VAE latent space curvature 28] A --> AD[Shape analysis pipeline:

extraction, modeling, manifolds, insights 29] A --> AE[Ongoing research:

generalize ML, biomedical applications 30] class A,H,M,N,V,W,X,Y,Z,AA,AB,AC learning class B,D,F,G,J shape class C,E applications class I,K,L,O,P,Q,S,T,U manifolds class R geomstats

**Resume: **

**1.-** Shape learning: Machine learning on data that are shapes, with each data point being one shape.

**2.-** Biomedical imaging: Studying biological structures through imaging techniques at various scales (organs to molecules).

**3.-** Shape and biology link: A biological structure's shape is linked to its function and health/disease state.

**4.-** Imaging techniques: MRI, microscopy, cryo-electron microscopy enable observing biological shapes at different scales.

**5.-** Biomedical questions: Research aims to answer questions about biological shape changes, e.g. brain atrophy in Alzheimer's.

**6.-** Shape reconstruction: Extracting shapes from images through algorithms like segmentation before analysis.

**7.-** Shape modeling/representation: Representing extracted shapes numerically in a computer for analysis.

**8.-** Manifolds: Generalizations of vector spaces that can be curved. Many shape representations give rise to manifolds.

**9.-** Shapes as manifolds: A shape itself can be a manifold (e.g. surface of an object).

**10.-** Shape spaces as manifolds: The space of all possible shapes of a certain type forms a manifold.

**11.-** Spaces of shape transformations as manifolds: Sets of shape motions or deformations can be modeled as manifolds.

**12.-** Generalizing statistics to manifolds: Traditional statistics don't apply on manifolds. Statistics must be generalized.

**13.-** Manifold geometry as inductive bias: Incorporating knowledge of shape space geometry can improve machine learning algorithms.

**14.-** Types of machine learning problems: Supervised, unsupervised, reinforcement learning, optimization - all can be generalized to manifolds.

**15.-** Manifold learning building blocks: Points, tangent vectors, geodesics, distances, exponential map - used to translate ML to manifolds.

**16.-** Vector space to manifold conversion: Translating ML building blocks allows converting many algorithms to manifolds.

**17.-** Geomstats package: Open-source Python package implementing manifold computations for manifold/geometric learning.

**18.-** Learning on Riemannian manifolds: Package supports various algorithms on Riemannian (metrizable) manifolds.

**19.-** Pose/transformation manifolds: Example manifold is SE(3) - the space of 3D rotations and translations.

**20.-** Numerical taxonomy of manifolds: Hierarchy/taxonomy of manifolds from abstract to concrete, implemented in geomstats.

**21.-** Manifold-adapted ML overview: Research aims to generalize ML algorithms across different manifolds.

**22.-** Dimension reduction on manifolds: Generalizing linear (PCA) and nonlinear (autoencoders) dim reduction techniques to manifolds.

**23.-** Geometric view of dim reduction: Finding principal subspaces within linear spaces or manifolds.

**24.-** Variational autoencoders (VAEs): Learn nonlinear latent subspace of data space assuming generative model.

**25.-** Generalizing VAEs to manifolds: Adapting VAE components (generative model, loss function) to manifold versions.

**26.-** Manifold VAE (mVAE) components: Exponential map replaces addition. Geodesic distance replaces Euclidean. Loss functions adapted.

**27.-** mVAE vs alternatives: mVAE faster than MCMC-based manifold dim reduction method.

**28.-** mVAE explains VAE latent space curvature: Manifold perspective shows why VAEs tend to learn flatter latent spaces.

**29.-** Overall shape analysis pipeline: Shape extraction, modeling, computing on manifolds, generating insights.

**30.-** Ongoing research: Further generalizing ML to manifolds, applying to biomedical shape data.

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