Let $\mu$ be the maximum posterior curve, given an index set, $\mathbf{x}$, arranged as a single column in "xyzxyz" format. Let $\mu^{(3)}$ be the 3 by N/3 matrix obtained by rearranging the points of $\mu$ into column vectors. That is, the i-th column $\mu^{(3)}_i$ is the i-th reconstructed point.

Let $\eta$ be the vector of absolute distances between adjacent points in $mu^{3}$. Formally,

$$\eta_i = \| \mu^{(3)}_i - \mu^{(3)}_{i-1} \|.$$ Note that $\eta^\top \eta = \|\eta\|^2 = \mu^\top D^\top D \mu$, where $D$ is the adjacent differences matrix, adapted to operate on column vectors in the "xyzxyz" format.

$$\begin{align} E &= \frac{1}{2} \left ( Dx - \eta \right ) ^2 \\ &= \frac{1}{2} x^\top D^\top D x + \frac 12 \eta^\top \eta - x^\top D^\top \eta \\ &= \frac{1}{2} x^\top D^\top D x + \frac 12 \mu^\top D^\top D \mu - x^\top D^\top \eta \\ \end{align}$$

$$\frac{\partial E}{\partial x} = x^\top D^\top D + \mu^\top D^\top D J_\mu - \eta^\top D - x^\top D^\top J_\eta$$

Note the identity $\eta_i^2 = \| \mu^{(3)}_i - \mu^{(3)}_{i-1} \|^2$ can be rewritten in terms of the full vector $\eta$ as $$\eta \odot \eta = \operatorname{sum_{3x1}}(D \mu \odot D \mu)$$ where $\operatorname{sum_{kx1}}$ implements k-way blockwise summation over columns of a matrix. Formally, it is the function $f : \mathbb{R}^{NxM} \rightarrow R^{(N/k)xM}$ (for any N divisible by k), such that $$(f(A))_{ij} = \sum_{s = (i-1)\times k+1}^{i \times k} A_{sj}$$

$$\begin{align} \frac{\partial \eta}{\partial x_i} &= \frac{\partial}{\partial x_i} \left ( \eta \odot \eta \right)^{\circ\frac12} \\ &= \frac{\partial}{\partial x_i} \left ( \operatorname{sum_{3x1}}(D \mu \odot D \mu) \right)^{\circ\frac12} \\ &= \frac12 \left ( \eta \odot \eta \right)^{\circ \frac{-1}{2}} \odot \frac{\partial}{\partial x_i} \left ( \operatorname{sum_{3x1}}(D \mu \odot D \mu )\right) \\ &= \frac12 \eta^{\circ (-1)} \odot \left ( \operatorname{sum_{3x1}}(\frac{\partial}{\partial x_i} D \mu \odot D \mu )\right) \\ J_\mu &= \frac12 \eta^{\circ (-1)} \odot \left ( \operatorname{sum_{3x1}}( 2 D \mu \odot D J_\mu )\right) \\ &= \eta^{\circ (-1)} \odot \left ( \operatorname{sum_{3x1}}( D \mu \odot D J_\mu )\right) \\ \end{align}$$ where $x^{\circ(-1)} = \left( \frac{1}{x_{ij}} \right)$ is the Hadamard (i.e. element-wise) inverse, and $x^{\circ \frac12}$ is the Hadamard root.