Group Effects in Shape and Deformation Data

Definitions and mathematical statement

Let $N\in{\mathbb{N}}$ be the number of subjects, and let $G_1,\ldots,G_K$ be a partition of $\{1,\ldots,N\}$ into $K\geq2$ nonempty groups. Write $g_i=k$ when $i\in G_k$. These symbols record group membership only; no sampling or assignment model is supplied.

1. For each subject $i$, let $X_i$ denote the observed image or shape. The source asks whether the shapes depend on group membership and, if they do, where they are most affected.
2. In the paired setting, let $B_i$ and $F_i$ denote the baseline and follow-up images of subject $i$, respectively. The source asks whether the deformation from $B_i$ to $F_i$ depends on group membership and, if it does, where it is affected. Here "deformation" is an unspecified relation or transformation between the paired images; existence, uniqueness, regularity, and a method for selecting such a transformation are not assumed. The original wording refers to deformation "between" the images, whereas the cleaned statement specifies the baseline-to-follow-up direction.

Unresolved formulation. These are requests to formulate and investigate group effects, rather than fully specified mathematical propositions. The source does not specify an image or shape space, a spatial domain or correspondence between subjects, a deformation model, probability laws, a sampling scheme, or a definition of group dependence. It also does not specify a local effect measure, the meaning of "most affected," or an inferential criterion. Consequently, neither a precise hypothesis about group dependence nor a precise spatial localization target is determined by the record. Such choices require additional assumptions and cannot be inferred from the finite observations alone.

Short English statement

For subjects divided into at least two groups, investigate whether images or shapes vary with group membership and identify where any effects occur. For paired baseline and follow-up images, ask the corresponding questions about deformation. The source leaves the statistical model, deformation representation, and spatial effect criterion unspecified, so this remains a research-question draft requiring further formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001689$, title: Group Effects in Shape and Deformation Data; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. No individual proposer is identified in the supplied record; the recorded proposed year is 2011.
• Supplied report citation: Trends in Mathematical Imaging and Surface Processing (2011), Oberwolfach Reports, pp. 293--348, DOI: https://doi.org/10.4171/owr/2011/07. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; it supplies no more specific passage location or speaker attribution.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2011/07.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and reports that no theorem resolving these broad questions about group effects and paired-image populations was located. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. This is a supplied editorial qualification, not independent verification of a precise mathematical formulation or of open status.