The TPS Direct Transport: A New Method for Transporting Deformations in the Size-and-Shape Space

Varano, Valerio; Gabriele, Stefano; Teresi, Luciano; Dryden, Ian L.; Puddu, Paolo; Torromeo, Concetta; Piras, Paolo

doi:10.1007/s11263-017-1031-9

Cited by 14 publications

(33 citation statements)

References 34 publications

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“…By following the notation of Varano et al and Dryden and Mardia, we summarize the construction of the TPS and we refer to previous studies for further details. In the Euclidean space

E^{m}

, the m ‐tuple of interpolating TPS is a function Φ represented by the pair ( A X , W X ), where: A X is a linear transformation of

E^{m}

, represented by a ( m × m ) matrix and W X is a ( k × m ) matrix.…”

Section: Geometry Of the Tps Spacementioning

confidence: 99%

“…As it vanishes on linear deformations, its eigenvectors associated to nonvanishing eigenvalues are only ( k −1− m ). They are called principal warps and represent the principal modes of deformation …”

Section: Geometry Of the Tps Spacementioning

confidence: 99%

“…It is possible to prove (see Kent and Mardia for the proof) that fixing the landmarks and changing the integration domain, in such a way that

normalΩ \to R^{2}

, then B Ω →16 πB . This can be qualitatively understood by considering that in case the boundary term C ∂ Ω vanishes, and for the bulk term C Ω , there are no landmarks belonging to ∂ Ω and then α i =2 π for each i , and by using the notable property Γ 11 X S X Γ 11 X =Γ 11 X . It is important to stress that for

normalΩ \subset R^{2}

, the bulk term C Ω has the closed form representation , while the boundary term C ∂ Ω is calculated solving, numerically, the single variable integrals appearing in Equation , which involve the derivatives of the biharmonic function.…”

Section: Geometry Of the Tps Spacementioning

confidence: 99%

“…In Varano et al, a connection named TPS connection was introduced. It has the following properties: It is compatible with the TPS metric. It is compatible with the decomposition given in . It preserves the Γ‐energy. It is independent of the path. …”

Section: Geometry Of the Tps Spacementioning

confidence: 99%

“…We also assume all the configurations to be centred, and represented by the Helmertized landmarks, then, if not otherwise specified, each matrix is a ( k −1)× m matrix. Furthermore, deformation vectors will have a subscript denoting the starting point, that is, the source configuration (for details see Varano).…”

Section: Geometry Of the Tps Spacementioning

confidence: 99%

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In a previous contribution, a new Riemannian shape space, named TPS space, was introduced to perform statistics on shape data. This space was endowed with a Riemannian metric and a flat connection, with torsion, compatible with the given metric. This connection allows the definition of a Parallel Transport of the deformation compatible with the three‐fold decomposition in spherical, deviatoric, and non‐affine components. Such a parallel transport also conserves the Γ‐energy, strictly related to the total elastic strain energy stored by the body in the original deformation. A new approach is here presented in order to calculate the bending energy on the body alone (body bending energy) and to restrict it exclusively within physical boundaries of objects involved in the deformation analysis. The novelty of this new procedure resides in the fact that we propose a new metric to be preserved during the TPS direct transport. This allows transporting the shape change more coherently with the mechanical meaning of the deformation. The geometry of the TPS space is then further discussed in order to better represent the relationship between the Γ‐energy, the strain energy, and the so‐called bending energy densities.

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“…By following the notation of Varano et al and Dryden and Mardia, we summarize the construction of the TPS and we refer to previous studies for further details. In the Euclidean space