Wavelet-Based Multiscale Flow For Realistic Image Deformation in the Large Diffeomorphic Deformation Model Framework

TL;DR

This work tackles the challenge of high-dimensional image registration and atlas estimation within the LDDMM framework by introducing a Haar-wavelet-based multiscale parameterization of the initial velocity fields. The coarse-to-fine reparameterization preserves the RKHS structure of the velocity fields, enabling efficient gradient-based optimization while guiding the search away from unrealistic minima. Empirical results across toy data, digits, artificial characters, and fetal brain MRIs show improved residuals, more realistic and stable templates, and robustness to initialization, with modest overhead from the wavelet transforms. The approach maintains model simplicity, is adaptable to other deformation frameworks, and offers perspectives for further multiscale extensions and smoother wavelet choices.

Abstract

Estimating accurate high-dimensional transformations remains very challenging, especially in a clinical setting. In this paper, we introduce a multiscale parameterization of deformations to enhance registration and atlas estimation in the Large Deformation Diffeomorphic Metric Mapping framework. Using the Haar wavelet transform, a multiscale representation of the initial velocity fields is computed to optimize transformations in a coarse-to-fine fashion. This additional layer of spatial regularization does not modify the underlying model of deformations. As such, it preserves the original kernel Hilbert space structure of the velocity fields, enabling the algorithm to perform efficient gradient descent. Numerical experiments on several datasets, including abnormal fetal brain images, show that compared to the original algorithm, the coarse-to-fine strategy reaches higher performance and yields template images that preserve important details while avoiding unrealistic features. This highly versatile strategy can easily be applied to other mathematical frameworks for almost no additional computational cost.

Figures (12)

• Figure 1: Computing shape deformations from an initial image $I(0)$ and a set of initial momentum vectors $\alpha(0)$. 1- Given $\alpha(0)$ and the initial control points $c_0$, integration of the Hamiltonian (\ref{['eq:HamSyst']}) gives the trajectory of the momentum vectors $\alpha(t)$. 2- The velocity field $v_t$ is computed by interpolating the momentum vectors with \ref{['eq:velocity']}. 3- Integration of the flow equation (\ref{['eq:flow_equation']}) gives a flow of diffeomorphisms $(\Phi_t)_{t \in [0,1]}$. 4- Finally, $\Phi_t$ is applied to the object $I(0)$, giving the deformed image $I(t)$ at any time $t$.
• Figure 2: Overview of the original LDDMM algorithm durrleman2012 and the coarse-to-fine optimization strategy in dimension 2. For the sake of clarity, only a single gradient descent iteration is presented in each panel. The original algorithm iterates between two classical steps: computing the gradients (1) and updating the parameters (2). In the multiscale strategy, step (1) is preserved and step (2) is replaced by another procedure involving updating the representation of the momentum vectors in the wavelet basis. Red arrows denote the coordinates of the momentum vectors either in the RKHS basis (for $\alpha_0(j)$) either in the wavelet basis (for $\beta_0(j)$). Purple arrows denote the gradients of the cost function $\nabla E_{\alpha_0}$ and $\nabla E_{\beta_0}$ with regard to $\alpha_0$ and $\beta_0$ (respectively).
• Figure 3: The 1D Haar wavelet. At scale 0, $V_0$ is the space of piecewise constant functions of size $1$. A basis for $V_0$, i.e. $\left\{\Phi_{0,k} \right\}_k$, is obtained by translating the scaling function $\Phi_{0,0}=\psi^L$ by factors $k \in \mathbb{Z}$. A basis for $W_0$, i.e. $\left\{\psi_{0,k} \right\}_k$, is obtained by translating the mother wavelet function $\psi_{0,0}=\psi^H$ by factors $k \in \mathbb{Z}$. At scale $s$, a basis for the space $V_s$, i.e. $\left\{\Phi_{s,k} \right\}_k$, is obtained by dilating $\Phi_{0,0}$ by $2^s$, translating it by $2^sk$ and normalizing it by $2^{-s/2}$. A basis for $W_s$, i.e. $\left\{\psi_{s,k} \right\}_k$, is obtained by performing the same operations on $\psi_{0,0}$.
• Figure 4: Decomposition of a 4-by-4 grid in two bases. The letters a and d refer to approximation and detail coefficients, respectively. Subscripts indicate the scale of the coefficient and the x-y position of the related wavelet function. Superscripts indicate the orientation of the wavelet function. In the grids, empty cells denote null values.
• Figure 5: Original LDDMM and LDDMM combined with multiscale optimization applied to a registration example. The source and target images (panel (a)) present feature differences at a large scale (character translation) and at a finer scale (orientation of the arms and legs). Registration was performed with $\sigma_g=4$ ($k_g=49$ control points). For each algorithm, we display the source-to-target vector fields (orange arrows, scaling factor $=5$), the corresponding deformation field $L_2$ norm in RGB, the deformation grid and the transformed source image every 20 iterations until convergence. $S_j$ is the scale constraining the momentum vectors at iteration $j$. Notice how the multiscale strategy first estimates coarse displacements to move the character to the top right, then finer transformations to adjust the position of the arms. The original LDDMM algorithm applies fine-scale deformations to the entire image (except the borders), while the multiscale algorithm estimates fine deformations around the character's right leg, and smoother deformations elsewhere.