Bayesian hierarchical modeling of simply connected 2D shapes

Our random shape generation process starts with a closed curve and performs a sequence of multiscale deformations to generate a final shape. In §2.2, we introduce the Roth curve developed by ?, which is used to represent the shape boundary. Then, in §2.3, we demonstrate how to deform a Roth curve at multiple scales to produce any simply-connected shape. Using the mechanisms developed in §2.2 and §2.3, we present the full random shape process in §2.5.

A Roth curve is a closed parametric curve, $C:[-\pi,\pi]\rightarrow\mathbb{R}^{2}$, defined by a set of $2n+1$ control points $\{c_{j},j=1,\ldots,2n+1\}$, where $n$ is the degree of the curve and we may choose it to be any positive integer, depending on how many control points are desired. For convenience, we will refer to the total number of control points as $J$, where $J(n)=2n+1$. For notational simplicity, we will drop the dependence of $n$ in $J(n)$. As a function of $t$, the curve can be viewed as the trajectory of a particle over time. At every time $t$, the particle’s location is defined as some convex combination of all control points. The weight accorded to each control point in this convex combination varies with time according to a set of basis functions, $\{B^{n}_{j}(t),j=1,\ldots,J\}$, where $B^{n}_{j}(t)>0$ and $\sum_{j=1}^{J}B^{n}_{j}(t)=1$ for all $t$.

	$\displaystyle C(t)$	$\displaystyle=$	$\displaystyle\sum_{j=1}^{J}c_{j}B_{j}^{n}(t),\enspace t\in[-\pi,\pi]\enspace,$		(1)
	$\displaystyle B_{j}^{n}(t)$	$\displaystyle=$	$\displaystyle\frac{h_{n}}{2^{n}}\left\{1+\cos\bigg{(}t+\frac{2\pi(j-1)}{2n+1}\bigg{)}\right\}^{n},\enspace h_{n}=\frac{(2^{n}n!)^{2}}{(2n+1)!}\enspace,$		(2)

where $c_{j}=[c_{j,x}\enspace c_{j,y}]^{\prime}$ specifies the location of the $j^{th}$ control point and $B^{n}_{j}:[-\pi,\pi]\rightarrow[0,1]$ is the $j^{th}$ basis function. For simplicity, we omit the superscript $n$ denoting a basis function’s degree, unless it requires special attention. This representation is a type of Bezier curve. Refer to Figure 1 for an illustration of the Roth basis functions.

A Roth curve can be deformed simply by translating some of its control points. We now formally define deformation and illustrate it in Figure 2.

• Definition

  Suppose we are given two Roth curves,

  $\displaystyle C(t)=\sum_{j=1}^{J}c_{j}B_{j}(t),\quad\widetilde{C}(t)=\sum_{j=1}^{J}\widetilde{c}_{j}B_{j}(t),$

  (3)

  where for each $j$, $\widetilde{c}_{j}=c_{j}+R_{j}d_{j}$, $d_{j}\in\mathbb{R}^{2}$ and $R_{j}$ is a rotation matrix. Then, we say that $C(t)$ is deformed into $\widetilde{C}(t)$ by the deformation vectors $\{d_{j},j=1,\ldots,J\}$.

Each $R_{j}$ orients the deformation vector $d_{j}$ relative to the original curve’s surface. As a result, positive values for the y-component of $d_{j}$ always correspond to outward deformation, negative values always correspond to inward deformation, and $d_{j}$’s x-component corresponds to deformation parallel to the surface. We will call $R_{j}$ a deformation-orienting matrix. In precise terms,

$\displaystyle R_{j}$

$\displaystyle=$

$\displaystyle\begin{bmatrix}\cos(\theta_{j})&-\sin(\theta_{j})\\ \sin(\theta_{j})&\cos(\theta_{j})\end{bmatrix}$

(4)

where $\theta_{j}$ is the angle of the curve’s tangent line at $q_{j}=\frac{-2\pi(j-1)}{2n+1}$, the point where the control point $c_{j}$ has the strongest influence: $q_{j}=\arg\displaystyle\max_{t\in[a,b]}B_{j}(t)$. $\theta_{j}$ can be obtained by computing the first-derivative of the Roth curve, also known as its hodograph.

• Definition

  The hodograph of a Roth curve is given by:

  	$\displaystyle H(t)$	$\displaystyle=$	$\displaystyle\sum_{j=1}^{J}c_{j}\frac{d}{dt}B_{j}(t),$		(5)
  	$\displaystyle\frac{d}{dt}B_{j}(t)$	$\displaystyle=$	$\displaystyle-\frac{2}{(2n+1){2n\choose n}}\sum_{j=1}^{J}c_{j}\sum_{k=0}^{n-1}{2n\choose k}(n-k)\sin\bigg{(}(n-k)t+\frac{2(n-k)(j-1)\pi}{2n+1}\bigg{)},$		(6)

  where $t\in[-\pi,\pi]$. If we view $C(t)$ as the trajectory of a particle, $H(t)$ intuitively gives the velocity of the particle at point $t$.

We can now use simple trigonometry to determine that

$\displaystyle\theta_{j}=\arctan\left(\frac{H_{y}(q_{j})}{H_{x}(q_{j})}\right).$

(7)

Note that $R_{j}$ is ultimately just a function of $\{c_{j}\in\mathbb{R}^{2},j=1,\ldots,J\}$.

Next, we show how to alter the scale of deformation, using an important concept called degree elevation.

• Definition

  Given any Roth curve, we can use degree elevation to re-express the same curve using a larger number of control points (a higher degree). More precisely, if we are given a curve of degree $n$, $C(t)=\sum_{j=1}^{2n+1}c_{j}B_{j}^{n}(t)$, we can elevate its degree by any positive integer $v$, to obtain a new degree elevated curve: $\widehat{C}(t)=\sum_{j=1}^{2(n+v)+1}\widehat{c}_{j}B_{j}^{n+v}(t)$ such that $C(t)=\widehat{C}(t)$ for all $t\in[-\pi,\pi]$. In $\widehat{C}(t)$, each new degree-elevated control point, $\widehat{c}_{j}$, can be defined in terms of the original control points, $\{c_{i},i=1,\ldots,2n+1\}$:

  $\displaystyle\widehat{c}_{j}:=\frac{1}{2n+1}\sum_{i=1}^{2n+1}c_{i}+\frac{{2(n+v)\choose n+v}h_{n}}{2^{2n-1}}\sum_{k=0}^{n-1}\frac{{2n\choose k}}{{2(n+v)\choose v+k}}\sum_{i=1}^{2n+1}\cos\left((n-k)\left(\frac{-2(j-1)\pi}{2(n+v)+1}\right)+\frac{2(n-k)(i-1)\pi}{2n+1}\right)c_{i}\enspace.$

It is crucial to note that the ‘influence’ of a single control point shrinks after degree elevation. We quantify this intuition in §3.3. This is because the curve is now shared by a greater total number of control points. This implies that after degree-elevation, the translation of any single control point will cause a smaller, finer-scale deformation to the curve’s shape. Thus, degree elevation can be used to adjust the scale of deformation. We exploit this strategy in the random shape process proposed in §2.5.

To that end, we first rewrite all of the concepts described above in more compact vector notation. Note that the formulas for degree elevation, deformation, the hodograph and the curve itself all simply involve linear operations on the control points.

First, we rewrite the control points in a ‘stacked’ vector of length $2J$.

$\displaystyle c=(c_{1,x},c_{1,y},c_{2,x},c_{2,y},\ldots,c_{J,x},c_{J,y})^{\prime}.$

(8)

The formula for a Roth curve given in (1) can be rewritten as:

	$\displaystyle C(t)$	$\displaystyle=$	$\displaystyle X(t)c$		(9)
	$\displaystyle X(t)$	$\displaystyle=$	$\displaystyle\begin{bmatrix}B_{1}(t)&0&B_{2}(t)&0&\cdots&B_{J}(t)&0\\ 0&B_{1}(t)&0&B_{2}(t)&\cdots&0&B_{J}(t)\end{bmatrix}$		(10)

The formula for the hodograph given in (5) is rewritten as:

$\displaystyle H(t)=\dot{X(t)}c,\quad\dot{X}(t)=\frac{d}{dt}X(t)$

(11)

Deformation can be written as:

$\displaystyle\widetilde{c}=c+T(c)d,\quad d=(d_{1,x},d_{1,y},d_{2,x},d_{2,y},\ldots,d_{J,x},d_{J,y})^{\prime},\quad T(c)=\mbox{block}(R_{1},R_{2},\ldots,R_{J})$

(12)

where $\mbox{block}(A_{1},\ldots,A_{q})$ is a $pq\times pq$ block diagonal matrix using $p\times p$ matrices $A_{i},i=1,\ldots,q$. We call $T$ the stacked deformation-orientating matrix. Note that $T$ is a function of $c$, because each $R_{j}$ depends on $c$. Degree elevation can be written as the linear operator, $E$:

$\displaystyle\widehat{c}=Ec,\quad E=(E_{i,j})_{i=1,j=1}^{n+v,n}.$

where

$\displaystyle E_{i,j}$

$\displaystyle=$

$\displaystyle\frac{1}{2n+1}+\frac{{2(n+v)\choose n+v}h_{n}}{2^{2n-1}}\sum_{k=0}^{n-1}\frac{{2n\choose k}}{{2(n+v)\choose v+k}}\cos\left((n-k)\left(\frac{-2(i-1)\pi}{2(n+v)+1}\right)+\frac{2(n-k)(j-1)\pi}{2n+1}\right).$

We will maintain this vector notation throughout the rest of the paper.

The random shape process starts with some initial Roth curve, specified by an initial set of control points, $c^{(0)}$. From here on, we will refer to all curves by the stacked vector of their control points, $c$. Then, drawing on the deformation and degree-elevation operations defined earlier, we repeatedly apply the following recursive operation $R$ times:

$\displaystyle\widehat{c}^{(r-1)}=E_{r}c^{(r-1)},\quad d^{(r)}\sim\mbox{N}(\mu_{r},\Sigma_{r}),\quad c^{(r)}=\widehat{c}^{(r-1)}+T_{r}(c^{(r-1)})d^{(r)}$

(13)

resulting in a final curve $c^{(R)}$. In other words, the process simply has two steps: (i) degree elevate the current curve, (ii) randomly deform it, and repeat a total of $R$ times. Note that this random process specifies a probability distribution over $c^{(R)}$.

We now elaborate on the details of this recursive process. The parameters of the process are:

1. 1.

   $R\in\mathbb{Z}$, the number of steps in the process

2. 2.

   $n_{r}\in\mathbb{Z}$, the degree of the curve $c^{(r)}$, for each $r=1,\ldots,R$. The sequence of $\{n_{r}\}_{1}^{R}$ must be strictly monotonically increasing. For convenience, we will denote the number of control points at a certain step $r$ to be $J_{r}=2n_{r}+1$.

3. 3.

   $\mu_{r}\in\mathbb{R}^{2J_{r}}$, the average set of deformations applied at step $r=1,\ldots,R$. Note that this vector contains a stack of deformations, not just one.

4. 4.

   $\Sigma_{r}\in\mathbb{R}^{2J_{r}\times 2J_{r}}$, the covariance in the set of deformations applied at step $r=1,\ldots,R$.

According to these parameters, $E_{r}$ is then the degree-elevation matrix going from degree $n_{r-1}$to $n_{r}$, $\mbox{N}(\cdot,\cdot)$ is a $2J_{r}$-variate normal distribution and $T_{r}$ is the stacked deformation orienting matrix.

We take special care in defining the initial curve, $c^{(0)}$. We choose $c^{(0)}$ to be degree $n_{0}=1$, which guarantees that it is an ellipse. For $j=1,2,3$, we define each control point as:

	$\displaystyle c_{j}^{(0)}$	$\displaystyle=$	$\displaystyle(0,0)^{\prime}+R_{\theta_{j}}d_{j}^{(0)},$
	$\displaystyle R_{\theta_{j}}$	$\displaystyle=$	$\displaystyle\mbox{rotation matrix where }\theta_{j}=\frac{2\pi j}{3},$

and where each $d_{j}^{(0)}\in\mathbb{R}^{2}$ is a random deformation vector. In words: we start with a curve that is just a point at the origin, $C(t)\equiv(0,0)$, and apply three random deformations which are rotated by a radially symmetric amount: $0^{\circ},120^{\circ}$ and $240^{\circ}$ (note that the final deformations are not radially symmetric, since each $d_{j}$ is randomly drawn). We will write this in vector notation as:

	$\displaystyle d^{(0)}$	$\displaystyle\sim$	$\displaystyle\mbox{N}(\mu_{0},\Sigma_{0})$
	$\displaystyle c^{(0)}$	$\displaystyle=$	$\displaystyle{\bf 0}+T_{0}d^{(0)}$

The deformations essentially ‘inflate’ the curve into some ellipse. This completes our definition of the random shape process.

We now give some intuition about the process and each of its parameters, and define several additional concepts which make the process easier to interpret. The random shape process gives a multiscale representation of shapes, because each step in the process produces increasingly fine-scale deformations, through degree-elevation.

$R$ is then the number of scales or ‘resolutions’ captured by the process. Each $n_{r}$ specifies the number of control points at resolution $r$. We will use $\mathbb{S}_{r}$ to denote the class of shapes that can be exactly represented by a degree $n_{r}$ Roth curve. See the definition of $\mathbb{H}^{n_{r}}$ in §3 for a formal characterization of a special case of $\mathbb{S}_{r}$ . If $\{n_{r}\}_{1}^{R}$ is monotonically increasing, then $\mathbb{S}_{1}\subset\mathbb{S}_{2}\subset\ldots\subset\mathbb{S}_{R}$. Thus, the deformations $d^{(r)}$ roughly describe the additional details gained going from $\mathbb{S}_{r-1}$ to $\mathbb{S}_{r}$.

$\mu_{r}$ is the mean deformation at level $r$. Based on $\{\mu_{r},r=0,\ldots,R\}$, we define the ‘central shape’ of the random shape process, $c^{*}$ as:

	$\displaystyle c^{*}$	$\displaystyle:=$	$\displaystyle c^{*(R)}$
	$\displaystyle c^{*(r)}$	$\displaystyle=$	$\displaystyle E_{r}c^{*(r-1)}+T_{r}(c^{*(r-1)})\mu_{r}$

Note that $c^{*}$ is simply the deterministic result of the random shape process when each $d^{(r)}=\mu_{r}$, rather than being drawn from a distribution centered on $\mu_{r}$. Thus, all shapes generated by the process tend to be deformed versions of the central shape. We illustrate this in Figure 4. If the random shape process is used to describe a population of shapes, the central shape provides a good summary.

$\Sigma_{r}$ determines the covariance of the deformations at level $r$. This naturally controls the variability among shapes generated by the process. If the variance is very small, all shapes will be very similar to the central shape. $\Sigma_{r}$ can also be chosen to induce correlation between deformation vectors at the same resolution, in the typical way that correlation is induced between dimensions of a multivariate normal distribution. This allows us to incorporate higher-level assumptions about shape, such as reflected or radial symmetry. For example, if $R=2,n_{1}=1$ and $n_{2}=2$, we can specify perfect correlation in $\Sigma_{2}$, such that $d_{1}^{(2)}=d_{4}^{(2)}$ and $d_{2}^{(2)}=d_{3}^{(2)}$. The resulting shapes are guaranteed to be symmetrical along an axis of reflection.

In the subsequent sections 4 and 5, we show how to use our random shape process to guide curve-fitting for various types of data. When doing so, we would like each resolution $r$ to describe the shape as best as possible, within its class $\mathbb{S}_{r}$. This can be achieved by setting $\mu_{r}={\bf 0}$ for $r=1,\ldots,R$ (not including $\mu_{0}$).

The supremum and $\mbox{L}_{1}$-norm are denoted by $||\cdot||_{\infty}$ and $||\cdot||_{1}$, respectively. We let $||\cdot||_{p,\nu}$ denote the norm of $L_{p}(\nu)$, the space of measurable functions with $\nu$-integrable $p$th absolute power. The notation $C(\mathcal{X})$ is used for the space of continuous functions $f:\mathcal{X}\rightarrow\mathbb{R}$ endowed with the uniform norm. For $\alpha>0$ , we let $C^{\alpha}(\mathcal{X})$ denote the Hölder space of order $\alpha$, consisting of the functions $f\in C(\mathcal{X})$ that have $\lfloor\alpha\rfloor$ continuous derivatives with the $\lfloor\alpha\rfloor$th derivative $f^{\lfloor\alpha\rfloor}$ being Lipshitz continuous of order $\alpha-\lfloor\alpha\rfloor$. We write “$\precsim$” for inequality up to a constant multiple and $\{a_{(1)},a_{(2)},\ldots,a_{(n)}\}$ to denote the order statistics of the set $\{a_{i}:a_{i}\in\mathbb{R},i=1,\ldots,n\}$.

Let the Hölder class of periodic functions on $[-\pi,\pi]$ of order $\alpha$ be denoted by $C^{\alpha}([-\pi,\pi])$. Define the class of closed parametric curves $\mathcal{S}_{\mathcal{C}}(\alpha_{1},\alpha_{2})$ having different smoothness along different coordinates as

$\displaystyle\mathcal{S}_{\mathcal{C}}(\alpha_{1},\alpha_{2}):=\{S=(S^{1},S^{2}):[-\pi,\pi]\to\mathbb{R}^{2},S^{i}\in C^{\alpha_{i}}([-\pi,\pi]),i=1,2\}.$

(14)

Consider for simplicity a single resolution Roth curve with control points $\{c_{j},j=0,\ldots,2n\}$. Assume we have independent Gaussian priors on each of the two coordinates of $c_{j}$ for $j=0,\ldots,2n$, i.e., $C(t)=\sum_{j=0}^{2n}c_{j}B_{j}^{n}(t),c_{j}\sim\mbox{N}_{2}(0,\sigma_{j}^{2}I_{2}),j=0,\ldots,2n$. Denote the prior for $C$ by $\Pi_{C^{n}}$. $\Pi_{C^{n}}$ defines an independent Gaussian process for each of the components of $C$. Technically speaking, the support of a prior is defined as the smallest closed set with probability one. Intuitively, the support characterizes the variety of prior realizations along with those which are in their limit. We construct a prior distribution to have large support so that the prior realizations are flexible enough to approximate the true underlying target object. As reviewed in ?, the support of a Gaussian process (in our case $\Pi_{C^{n}}$) is the closure of the corresponding reproducing kernel Hilbert space (RKHS). The following Lemma 3.1 describes the RKHS of $\Pi_{C^{n}}$, which is a special case of Lemma 2 in ?.

The following theorem describes how well an arbitrary closed parametric surface $S_{0}\in\mathcal{S}_{\mathcal{C}}(\alpha_{1},\alpha_{2})$ can be approximated by the elements of $\mathbb{H}^{n}$ for each $n$. Refer to Appendix A for a proof.

This shows that the Roth basis expansion is sufficiently flexible to approximate any closed curve arbitrarily well. Although we have only shown large support of the prior under independent Gaussian priors on the control points, the multiscale structure should be even more flexible and hence rich enough to characterize any closed curve. We can also expect minimax optimal posterior contraction rates using the prior $\Pi_{C^{n}}$ similar to Theorem 2 in ? for suitable choices of prior distributions on $n$.

The unique maximum of basis function $B_{j}^{n}(t)$ defined in (1) is at $t=-2\pi(j-1)/J$, therefore the control point $c_{j}$ has the most significant effect on the shape of the curve in the neighborhood of the point $C(-2\pi(j-1)/J)$. Note that $B_{j}^{n}(t)$ vanishes at $t=\pi-2\pi(j-1)/J$, thus $c_{j}$ has no effect on the corresponding point i.e., the point of the curve is invariant under the modi cation of $c_{j}$. The control point $c_{j}$ affects all other points of the curve, i.e. the curve is globally controlled. These properties are illustrated in Figure 5.

However, we emphasize following Proposition 5 in ? that while control points have a global effect on the shape, this in uence tends to be local and dramatically decreases on further parts of the curve, especially for higher values of $n$.

Denote by $v_{i}=(H_{x}(t_{i}),H_{y}(t_{i}))\in\mathbb{R}^{2}$ the velocity vector of the curve $C(t)$ at the parameterization location $t_{i},i=1,\ldots,N$. Note that $v_{i}$ is always tangent to the curve. Since each $\omega_{i}$ points roughly normal to the curve, we can rotate all of them by 90 degrees, $\theta_{i}=\omega_{i}+\frac{\pi}{2}$, and treat each $\theta_{i}$ as a noisy estimate of $v_{i}$’s orientation. Note that we cannot rotate the vector $g_{i}$ by 90 degrees and directly treat it as a noisy observation of $v_{i}$. In particular, $g_{i}$ ’s magnitude bears no relationship to the magnitude of $v_{i}$: $||g_{i}||$ is the rate of change in image brightness when crossing the edge of the shape, while $||v_{i}||$ describes the speed at which the curve passes through $p_{i}$.

Suppose we did have some noisy observation of $v_{i}$, denoted $u_{i}$. Then, we could have specified the following linear model relating the curve $\{c_{j},j=1,\ldots,J\}$ to the $u_{i}$’s:

	$\displaystyle u_{i}$	$\displaystyle=$	$\displaystyle v_{i}+\delta_{i}$		(22)
		$\displaystyle=$	$\displaystyle\sum_{j=1}^{J}c_{j}\frac{d}{dt}B_{j}(t_{i})+\delta_{i}$		(23)

for $i=1,\ldots,N$ where $\delta_{i}\sim N_{2}(0,\tau^{2}I_{2})$. Instead, we only know the angle of $u_{i}$, $\theta_{i}$. In §7, we show that using this model, we can still write the likelihood for $\theta_{i}$, by marginalizing out the unknown magnitude of $u_{i}$. The resulting likelihood still results in conditional conjugacy of the control points.

Observe that since $T_{r}(c^{(r),k})$ may not be linear in $c^{(r),k}$, due to the $\arctan$ in (7), the full conditional distribution of $c^{(r),k}$ is not conditionally conjugate. Below, we develop a novel approximation to the rotation matrix $\tilde{T}_{r}$ of $T(c^{(r-1),k})$. The approximation ensures that $T_{r}(c^{(r-1),k})$ is linear in the level $r-1$ control points $c^{(r-1),k}$ which results in conditional conjugacy of $c^{(r-1),k}$. We resort to a Metropolis Hastings algorithm with an independent proposal suggested by the approximation to correct for the approximation error. For $j=1,\ldots,J_{r}$, let $q_{j,r}=-2\pi(j-1)/J_{r}$. Recall that $\dot{X}(q_{j,r})c^{(r),k}$ is the hodograph evaluated at $q_{j,r}$. \prop $R_{j}$ in (4) can be approximated by $\tilde{R}_{j},j=1,\ldots,J_{r}$, where $\tilde{R}_{j}$ are approximate rotation matrices given by

$\displaystyle\frac{2\pi}{L_{A}}\times\begin{bmatrix}\dot{X}_{x}(q_{j,r})c^{(r-1),k}&-\dot{X}_{y}(q_{j,r})c^{(r-1),k}\\ \dot{X}_{y}(q_{j,r})c^{(r-1),k}&\dot{X}_{x}(q_{j,r})c^{(r-1),k}\end{bmatrix}$

(25)

and $L_{A}$ is an approximation for the length of the curve $A(\pi;c^{(r-1),k})$ formed by the control points $c^{(r-1),k}$.

• Proof

  Refer to the definition of $R_{j}$ in (4). First we derive an approximation for $\cos(\theta_{j}^{*})$ and $\sin(\theta_{j}^{*})$ where $\theta_{j}^{*}$ is the angle at the point $q_{j,r}$ of the curve specified by $c^{(r-1),k}$. We write $\theta_{j}^{*}$ in vector notation

  $\displaystyle\theta_{j}^{*}=\arctan\bigg{(}\frac{\dot{X}_{y}(q_{j,r})c^{(r-1),k}}{\dot{X}_{x}(q_{j,r})c^{(r-1),k}}\bigg{)}.$

  (26)

  Using the identities

  $\displaystyle\cos\arctan(x/y)=\frac{x}{\sqrt{x^{2}+y^{2}}},\quad\sin\arctan(x/y)=\frac{y}{\sqrt{x^{2}+y^{2}}},$

  (27)

  we obtain,

  	$\displaystyle\cos(\theta_{j}^{*})$	$\displaystyle=$	$\displaystyle\frac{\dot{X}_{x}(q_{j,r})c^{(r-1),k}}{\sqrt{(\dot{X}_{x}(q_{j,r})c^{(r-1),k})^{2}+(\dot{X}_{y}(q_{j,r})c^{(r-1),k})^{2}}}$
  	$\displaystyle\sin(\theta_{j}^{*})$	$\displaystyle=$	$\displaystyle\frac{\dot{X}_{y}(q_{j,r})c^{(r-1),k}}{\sqrt{(\dot{X}_{x}(q_{j,r})c^{(r-1),k})^{2}+(\dot{X}_{y}(q_{j,r})c^{(r-1),k})^{2}}}$

  The magnitude $\sqrt{(\dot{X}_{x}(q_{j,r})c^{(r-1),k})^{2}+(\dot{X}_{y}(q_{j,r})c^{(r-1),k})^{2}}$ of the velocity vector at the point $q_{j,r}$ can be well approximated by the quantity $\frac{A(\pi;c^{(r-1),k})}{2\pi}$ in view of the uniform arc-length parameterization discussed in §LABEL:sec:para. Hence

  	$\displaystyle\cos(\theta_{j}^{*})\approx\frac{2\pi}{A(\pi;c^{(r-1),k})}\dot{X}_{x}(q_{j,r})c^{(r-1),k}$
  	$\displaystyle\sin(\theta_{j}^{*})\approx\frac{2\pi}{A(\pi;c^{(r-1),k})}\dot{X}_{y}(q_{j,r})c^{(r-1),k}$

  Plugging in in a fixed approximation $L_{A}$ for the length of the curve $A(\pi;c^{(1),k})$, we obtain the required result. $\Box$