Homotopic Affine Transformations in the 2D Cartesian Grid

Abstract

Topology preservation is a property of affine transformations in ${{{\mathbb {R}}}^2}$, but not in ${{\mathbb {Z}}}^2$. In this article, given a binary object ${\mathsf {X}} \subset {{\mathbb {Z}}}^2$ and an affine transformation ${{\mathcal {A}}}$, we propose a method for building a binary object $\widehat{{\mathsf {X}}} \subset {{\mathbb {Z}}}^2$ resulting from the application of ${{\mathcal {A}}}$ on ${\mathsf {X}}$. Our purpose is, in particular, to preserve the homotopy type between ${\mathsf {X}}$ and $\widehat{{\mathsf {X}}}$. To this end, we formulate the construction of $\widehat{{\mathsf {X}}}$ from ${{\mathsf {X}}}$ as an optimization problem in the space of cellular complexes, and we solve this problem under topological constraints. More precisely, we define a cellular space ${{\mathbb {H}}}$ by superimposition of two cellular spaces ${{\mathbb {F}}}$ and ${{\mathbb {G}}}$ corresponding to the canonical Cartesian grid of ${{\mathbb {Z}}}^2$ where ${{\mathsf {X}}}$ is defined, and a regular grid induced by the affine transformation ${{{\mathcal {A}}}}$, respectively. The object $\widehat{{\mathsf {X}}}$ is then computed by building a homotopic transformation within the space ${{\mathbb {H}}}$, starting from the complex in ${{\mathbb {G}}}$ resulting from the transformation of ${\mathsf {X}}$ with respect to ${{\mathcal {A}}}$ and ending at a complex fitting $\widehat{{\mathsf {X}}}$ in ${{\mathbb {F}}}$ that can be embedded back into ${{\mathbb {Z}}}^2$.

A Construction of the Cellular Space \({{\mathbb {H}}}\)

We describe hereafter the way we build the cellular space ${\mathbb {H}}$ that refines the two cubical spaces ${{\mathbb {F}}}$ and ${{\mathbb {G}}}$.

1.1 A.1 Input

Although ${{\mathbb {F}}}$, ${{\mathbb {G}}}$ and ${{\mathbb {H}}}$ are infinite spaces, our purpose is to handle finite digital objects. As a consequence, our first input is a finite subset ${\mathsf {S}}$ of ${{{\mathbb {Z}}}}^2$ that will include these digital objects. Without loss of generality, we assume that

$$\begin{aligned} {{\mathsf {S}}} = [\![-s,s]\!]^2 \subset {{{\mathbb {Z}}}}^2 \end{aligned}$$

(90)

is a square with $s \in \mathbb N^\star$. The continuous analogue of this digital set ${\mathsf {S}}$ is the Euclidean square subset of ${{{\mathbb {R}}}}^2$ defined as

$$\begin{aligned} S = \square {\mathsf {S}} = [-s - \frac{1}{2},s+\frac{1}{2}]^2 \subset {{{\mathbb {R}}}}^2 \end{aligned}$$

(91)

The parameters that define the affine transformation ${{\mathcal {A}}}$ are also required, namely the values $a_{11}, a_{12}, a_{21}, a_{22}, t_x, t_y \in {\mathbb Q}$ (see Eq. 2). Consequently, the information required as input is a 7-uple $(s,a_{11}, a_{12}, a_{21},$ $a_{22},t_x,t_y) \in {\mathbb N}^\star \times {\mathbb Q}^6$.

1.2 A.2 Output

The output of the algorithm is the finite subspace (namely a complex) of the cellular space ${{\mathbb {H}}}$ that intersects a Euclidean square

$$\begin{aligned} Q = [q_x,q_x +w] \times [q_y,q_y+w] \subset {{{\mathbb {R}}}}^2 \end{aligned}$$

(92)

with $\mathbf{q}\in ({{{\mathbb {Z}}}}+ \frac{1}{2})^2$ and .

The output cellular space

$$\begin{aligned} {{\mathbb {H}}}(Q) = \{{\mathfrak {h}} \in {{\mathbb {H}}} \mid {\mathfrak {h}} \subset Q\} \end{aligned}$$

(93)

is defined as a finite set of faces of ${{\mathbb {H}}}$, and is partitioned into three subsets ${{\mathbb {H}}}_0(Q)$, ${\mathbb {H}}_1(Q)$ and ${{\mathbb {H}}}_2(Q)$ that contain the 0-, 1- and 2-faces of ${{\mathbb {H}}}(Q)$, respectively. In particular, ${{\mathbb {H}}}_0(Q) \cup {{\mathbb {H}}}_1(Q) \cup {{\mathbb {H}}}_2(Q) = {{\mathbb {H}}}(Q)$ is a partition of Q.

For each face ${\mathfrak {h}}$, we also compute the closure $C({\mathfrak {h}})$ and/or the star $S({\mathfrak {h}})$ within the subspace ${{\mathbb {H}}}(Q)$. If ${\mathfrak {h}}$ is a 0- (resp. 1-, resp. 2-) face, we compute $S_1({\mathfrak {h}})$ and $S_2({\mathfrak {h}})$ (resp. $C_0({\mathfrak {h}})$ and $S_2({\mathfrak {h}})$, resp. $C_0({\mathfrak {h}})$ and $C_1({\mathfrak {h}})$) where $S_d({\mathfrak {h}})$ (resp. $C_d({\mathfrak {h}})$) is the part of $S({\mathfrak {h}})$ (resp. $C({\mathfrak {h}})$) composed by the d-faces ($0 \le d \le 2$).

For each 2-face ${\mathfrak {h}}_2$, we also compute the functions $\phi$ and $\gamma$. More precisely, we compute the functions ${{\widetilde{\phi }}}, {{\widetilde{\gamma }}} : {{\mathbb {H}}}_2 \rightarrow {{{{\mathbb {Z}}}}^{2}}$ such that

$$\begin{aligned} {{\widetilde{\phi }}}({\mathfrak {h}}_2)&= \boxdot (\phi ({\mathfrak {h}}_2))\end{aligned}$$

(94)

$$\begin{aligned} {{\widetilde{\gamma }}}({\mathfrak {h}}_2)&=\boxdot ( {{\mathcal {A}}}^{-1}(\gamma ({\mathfrak {h}}_2))) \end{aligned}$$

(95)

This is indeed relevant since $\boxdot$ is a bijection between ${{\mathbb {F}}}_2$ and ${{{{\mathbb {Z}}}}^{2}}$ while ${\boxdot \circ {{\mathcal {A}}}^{-1}}$ is a bijection between ${{\mathbb {G}}}_2$ and ${{{{\mathbb {Z}}}}^{2}}$. In particular, these functions allow us to define the two functions ${{\widetilde{\varPhi }} : \boxdot Q \subset {{{{\mathbb {Z}}}}^{2}} \rightarrow 2^{{{\mathbb {H}}}_2}}$ and ${{{\widetilde{\varGamma }}} : {\mathsf {S}} \subset {{{{\mathbb {Z}}}}^{2}} \rightarrow 2^{{\mathbb {H}}_2}}$ such that for any $\mathbf{p}\in \phi ({{\mathbb {H}}}_2)$ (resp. $\mathbf{p}\in {\mathsf {S}}$), we have ${{{\widetilde{\varPhi }}}(\mathbf{p}) = \widetilde{\phi }^{-1}(\{\mathbf{p}\})}$ (resp. ${{{\widetilde{\varGamma }}}(\mathbf{p}) = \widetilde{\gamma }^{-1}(\{\mathbf{p}\})})$.

1.3 A.3 Definition of the Square Q

The first task is to define the square Q, i.e. to define $\mathbf{q}$ and w so that Q includes the image of the square S by the affine transformation ${{\mathcal {A}}}$. Since S is convex, we can simply compute the images of the four vertices of S by ${{\mathcal {A}}}$ to reach that goal. In particular, for $0 \le i, j \le 1$, we set

$$\begin{aligned} \mathbf{c}^{i,j} = ((-1)^i (s + \frac{1}{2}),(-1)^j (s + \frac{1}{2})) \end{aligned}$$

(96)

The four points $\mathbf{c}^{i,j}$ are the vertices of S. For $0 \le i, j \le 1$, we compute $\mathbf{r}^{i,j} = {{\mathcal {A}}}(\mathbf{c}^{i,j})$. We set

$$\begin{aligned} r^-_x&= \min _{i,j}\{\lfloor r^{i,j}_x + \frac{1}{2}\rfloor \} - \frac{1}{2} \end{aligned}$$

(97)

$$\begin{aligned} r^-_y&= \min _{i,j}\{\lfloor r^{i,j}_y + \frac{1}{2}\rfloor \} - \frac{1}{2} \end{aligned}$$

(98)

$$\begin{aligned} r_x^+&= \max _{i,j}\{\lceil r^{i,j}_x - \frac{1}{2}\rceil \} + \frac{1}{2} \end{aligned}$$

(99)

$$\begin{aligned} r_y^+&= \max _{i,j}\{\lceil r^{i,j}_y- \frac{1}{2}\rceil \} + \frac{1}{2} \end{aligned}$$

(100)

Finally, we define

$$\begin{aligned} q_x&= r_x^- \end{aligned}$$

(101)

$$\begin{aligned} q_y&= r_y^- \end{aligned}$$

(102)

$$\begin{aligned} w&= \max \{r_x^+ - r_x^-, r_y^+ - r_y^-\} \end{aligned}$$

(103)

The square Q is then defined by its four vertices $\mathbf{q}^{0,0} = \mathbf{q}$, $\mathbf{q}^{w,0} = \mathbf{q}+ w\mathbf{e}_x$, $\mathbf{q}^{0,w} = \mathbf{q}+ w\mathbf{e}_y$ and $\mathbf{q}^{w,w} = \mathbf{q}+ w\mathbf{e}_x + w\mathbf{e}_y$.

1.4 A.4 Definition of the Generator Lines of ${{\mathbb {H}}}(Q)$

The cellular subspace ${{\mathbb {H}}}(Q)$ is induced by the subdivision of Q by the lines of ${\mathscr {V}}_\varDelta$, $\mathscr {H}_\varDelta$, ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )$ and ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )$. These four sets are infinite, but for each of them, the subset of lines that contribute to the subdivision of Q is finite and corresponds to the lines that intersect the square Q. The subsets ${\mathscr {V}}_\varDelta (Q)$ of ${\mathscr {V}}_\varDelta$ and $\mathscr {H}_\varDelta (Q)$ of ${\mathscr {H}}_\varDelta$ are defined by

$$\begin{aligned} {\mathscr {V}}_\varDelta (Q)&= \{V_\delta \mid \delta \in \varDelta \cap [q_x,q_x + w]\} \end{aligned}$$

(104)

$$\begin{aligned} {\mathscr {H}}_\varDelta (Q)&= \{H_\delta \mid \delta \in \varDelta \cap [q_y,q_y + w]\} \end{aligned}$$

(105)

while the subsets ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)$ of ${{\mathcal {A}}}(\mathscr {V}_\varDelta )$ and ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)$ of ${{\mathcal {A}}}(\mathscr {H}_\varDelta )$ can be determined as follows. For $0 \le i, j \le 1$, we compute

$$\begin{aligned} \mathbf{u}^{i,j} = {{\mathcal {A}}}^{-1}(\mathbf{q}^{iw,jw}) \end{aligned}$$

(106)

These four points $\mathbf{u}^{i,j}$ are the vertices of the parallelogram ${{\mathcal {A}}}^{-1}(Q)$. We set

$$\begin{aligned} \delta _x^-&= \min _{i,j}(\lceil u_x^{i,j} +\frac{1}{2} \rceil ) - \frac{1}{2} \end{aligned}$$

(107)

$$\begin{aligned} \delta _x^+&= \max _{i,j}(\lfloor u_x^{i,j} -\frac{1}{2} \rfloor ) + \frac{1}{2} \end{aligned}$$

(108)

$$\begin{aligned} \delta _y^-&= \min _{i,j}(\lceil u_y^{i,j} +\frac{1}{2} \rceil ) - \frac{1}{2} \end{aligned}$$

(109)

$$\begin{aligned} \delta _y^+&= \max _{i,j}(\lfloor u_y^{i,j} -\frac{1}{2} \rfloor ) + \frac{1}{2} \end{aligned}$$

(110)

The only lines of ${\mathscr {V}}_\varDelta$ (resp. ${\mathscr {H}}_\varDelta$) that intersect ${{\mathcal {A}}}^{-1}(Q)$ are the lines $V_\delta$ (resp. $H_\delta$) for $\delta _x^- \le \delta \le \delta _x^+$ (resp. $\delta _y^- \le \delta \le \delta _y^+$). This leads us to define the subsets ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)$ of ${{\mathcal {A}}}(\mathscr {V}_\varDelta )$ and ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)$ of ${{\mathcal {A}}}(\mathscr {H}_\varDelta )$ as follows

$$\begin{aligned} {{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)&= \{ {{\mathcal {A}}}(V_\delta ) \mid \delta _x^- \le \delta \le \delta _x^+ \} \end{aligned}$$

(111)

$$\begin{aligned} {{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)&= \{ {{\mathcal {A}}}(H_\delta ) \mid \delta _y^- \le \delta \le \delta _y^+ \} \end{aligned}$$

(112)

1.5 A.5 Definition of ${{\mathbb {H}}}_0(Q)$

Any 0-face ${\mathfrak {f}}_0$ of ${{\mathbb {H}}}(Q)$ corresponds to the intersection of (at least) two lines of ${\mathscr {V}}_\varDelta (Q)$, ${\mathscr {H}}_\varDelta (Q)$, ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)$ and ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)$ inside the square Q. (Of course, such two lines cannot belong to a same subset.) Reversely, two lines from two of these subsets can induce at most one such 0-face. In order to ensure that the intersection between two lines is indeed inside the square Q, we define for each line L the segment, noted Q(L), which corresponds the intersection between L and Q. In particular, two lines $L_1$ and $L_2$ will intersect inside Q iff $Q(L_1)$ and $Q(L_2)$ intersect.

Let L be a line of ${\mathscr {V}}_\varDelta (Q)$, ${\mathscr {H}}_\varDelta (Q)$, ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)$ or ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)$. We compute the putative intersections

$$\begin{aligned} L \cap V_{q_x}&= \{\mathbf{m}^{x-}\} \end{aligned}$$

(113)

$$\begin{aligned} L \cap V_{q_x+w}&= \{\mathbf{m}^{x+}\} \end{aligned}$$

(114)

$$\begin{aligned} L \cap H_{q_y}&= \{\mathbf{m}^{y-}\} \end{aligned}$$

(115)

$$\begin{aligned} L \cap H_{q_y+w}&= \{\mathbf{m}^{y+}\} \end{aligned}$$

(116)

with the convention that $\mathbf{m}^{x-} = (m^{y-}_x,-\infty )$ and $\mathbf{m}^{x+} = (m^{y+}_x,+\infty )$ if L is colinear to $V_{q_x}$ and $V_{q_x+w}$ and $\mathbf{m}^{y-} = (-\infty ,m^{x-}_y)$ and $\mathbf{m}^{y+} = (+\infty ,m^{x+}_y)$ if L is colinear to $H_{q_y}$ and $H_{q_y+w}$. The segment associated to L is then

$$\begin{aligned}&Q(L) = [(\max \{m_x^{x-},m_x^{y-}\},\max \{m_y^{x-},m_y^{y-}\}),\nonumber \\&(\min \{m_x^{x+}, m_x^{y+}\}, \min \{m_y^{x+},m_y^{y+}\})] \end{aligned}$$

(117)

For the sake of concision, a 0-face ${\mathfrak {f}}_0$ will be also noted as $\langle \mathbf{h}\rangle$ where $\mathbf{h}$ is the point that defines this face, i.e. the intersection point of these two lines.

Let $L_1, L_2$ be two lines of two distinct subsets of $\mathscr {V}_\varDelta (Q)$, ${\mathscr {H}}_\varDelta (Q)$, ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)$ or ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)$. If $L_1 \cap L_2 = \{\mathbf{h}\}$ (i.e. if $L_1 \cap L_2 \ne \emptyset$ and $L_1 \ne L_2$, i.e. $L_1, L_2$ are non-colinear), then $\langle \mathbf{h}\rangle$ is a 0-face of ${{\mathbb {H}}}_0(Q)$ iff $\mathbf{h}\in Q(L_1)$ and $\mathbf{h}\in Q(L_2)$. The exhaustive scanning of all the couples of lines within $\mathscr {V}_\varDelta (Q)$, ${\mathscr {H}}_\varDelta (Q)$, ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)$ or ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)$ then allows us to build ${\mathbb {H}}_0(Q)$.

1.6 A.6 Definition of ${{\mathbb {H}}}_1(Q)$

For each line L of ${\mathscr {V}}_\varDelta (Q)$, ${\mathscr {H}}_\varDelta (Q)$, ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)$ or ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)$, we keep track of all the 0-faces of ${{\mathbb {H}}}_0(Q)$ induced by the intersection of L with another line. In particular, we note I(L) the sets of all the points corresponding to these 0-faces.

For the sake of concision, a 1-face ${\mathfrak {f}}_1$ will be also noted as $\langle \mathbf{h}_1,\mathbf{h}_2 \rangle$ where ${\mathfrak {f}}_1 = \ ]\mathbf{h}_1, \mathbf{h}_2[$.

Let $I(L) = \{\mathbf{h}_0,\ldots ,\mathbf{h}_i\ldots ,\mathbf{h}_t\}$ ($t \ge 0$). Without loss of generality, we assume that the points $\mathbf{h}_i$ are sorted in the lexicographic order in ${\mathbb Q}^2$. (Note that we have $\mathbf{h}_0$ and $\mathbf{h}_t$ equal to the bounds of Q(L).) Then, for any $0 \le i \le t-1$, $\langle \mathbf{h}_i,\mathbf{h}_{i+1} \rangle$ is a 1-face of ${\mathbb {H}}_1(Q)$. Reversely, each 1-face of ${{\mathbb {H}}}_1(Q)$ satisfies this property for one (or two) line(s) L of $\mathscr {V}_\varDelta (Q)$, ${\mathscr {H}}_\varDelta (Q)$, ${{\mathcal {A}}}({\mathscr {V}}_\varDelta )(Q)$ or ${{\mathcal {A}}}({\mathscr {H}}_\varDelta )(Q)$.

For any 1-face ${\mathfrak {f}}_1 = \langle \mathbf{h}_i,\mathbf{h}_{i+1} \rangle$, the set $C_0({\mathfrak {f}}_1)$ is defined as $\{\langle \mathbf{h}_i \rangle , \langle \mathbf{h}_{i+1} \rangle \}$. Reversely, for any 0-face ${\mathfrak {f}}_0 \in {{\mathbb {H}}}_0$, the set $S_1({\mathfrak {f}}_0)$ is defined as $\{{\mathfrak {f}}_1 \in {{\mathbb {H}}}_1 \mid {\mathfrak {f}}_0 \in C_0({\mathfrak {f}}_1)\}$.

1.7 A.7 Definition of ${{\mathbb {H}}}_2(Q)$

Let ${\mathfrak {f}}_0 = \langle \mathbf{h}\rangle$ be a 0-face of ${{\mathbb {H}}}_0$. The set $S_1({\mathfrak {f}}_0)$ contains many 1-faces $\langle \mathbf{h}, \mathbf{h}' \rangle$ that can be easily sorted in the clockwise order with respect to the orientation of the vectors $\mathbf{h}' - \mathbf{h}$. For each 1-face ${\mathfrak {f}}_1 = \langle \mathbf{h}, \mathbf{h}' \rangle$ of ${{\mathbb {H}}}_1$, we can then define the successor of ${\mathfrak {f}}_1$ in $S(\langle \mathbf{h}\rangle )$ (resp. in $S(\langle \mathbf{h}'\rangle )$) with respect to this ordering. This successor will be noted $\sigma (\langle \mathbf{h}', \mathbf{h}\rangle )$ (resp. $\sigma (\langle \mathbf{h}, \mathbf{h}' \rangle )$); note in particular that the order of $\mathbf{h}, \mathbf{h}'$ in the notation of $\sigma$ will then be important in that case.

For the sake of concision, a 2-face ${\mathfrak {f}}_2$ will be also noted as $\langle \mathbf{h}_1,\ldots ,$ $\mathbf{h}_i,\ldots \mathbf{h}_t \rangle$ ($t \ge 3$) where the $\mathbf{h}_i$ are the vertices of the corresponding convex polygon, clockwise ordered. Each 2-face⁶${\mathfrak {f}}_2$ is defined (up to circular permutations) by $\langle \mathbf{h}_1,\ldots ,\mathbf{h}_i,\ldots \mathbf{h}_t \rangle$ such that for any $1 \le i,j \le t$, we have $\mathbf{h}_i \ne \mathbf{h}_j$, for any $1 \le i \le t-2$, we have $\sigma (\langle \mathbf{h}_i, \mathbf{h}_{i+1} \rangle ) = \langle \mathbf{h}_{i+1}, \mathbf{h}_{i+2} \rangle$ and $\sigma (\langle \mathbf{h}_{t-1}, \mathbf{h}_{t} \rangle ) = \langle \mathbf{h}_{t}, \mathbf{h}_{1} \rangle$.

For any 2-face ${\mathfrak {f}}_2 = \langle \mathbf{h}_1,\ldots ,\mathbf{h}_i,\ldots \mathbf{h}_t \rangle$, the sets $C_0({\mathfrak {f}}_2)$ and $C_1({\mathfrak {f}}_2)$ are defined as

$$\begin{aligned} C_0({\mathfrak {f}}_2)&= \{\langle \mathbf{h}_1\rangle ,\ldots ,\langle \mathbf{h}_i\rangle ,\ldots \langle \mathbf{h}_t \rangle \} \end{aligned}$$

(118)

$$\begin{aligned} C_1({\mathfrak {f}}_2)&= \{\langle \mathbf{h}_1,\mathbf{h}_2\rangle ,\ldots ,\langle \mathbf{h}_i,\mathbf{h}_{i+1}\rangle ,\ldots , \langle \mathbf{h}_{t-1},\mathbf{h}_t \rangle , \langle \mathbf{h}_t,\mathbf{h}_1 \rangle \} \end{aligned}$$

(119)

Reversely, for any 0-face ${\mathfrak {f}}_0 \in {{\mathbb {H}}}_0$, the sets $S_1({\mathfrak {f}}_0)$ and $S_2({\mathfrak {f}}_0)$ are defined as

$$\begin{aligned} S_1({\mathfrak {f}}_0)&= \{{\mathfrak {f}}_2 \in {{\mathbb {H}}}_2 \mid {\mathfrak {f}}_0 \in C_0({\mathfrak {f}}_2)\} \end{aligned}$$

(120)

$$\begin{aligned} S_2({\mathfrak {f}}_0)&= \{{\mathfrak {f}}_2 \in {{\mathbb {H}}}_2 \mid {\mathfrak {f}}_1 \in C_1({\mathfrak {f}}_2)\} \end{aligned}$$

(121)

1.8 A.8 Definition of the Functions ${{{\widetilde{\phi }}}}, {{{\widetilde{\gamma }}}}, {{{\widetilde{\varPhi }}}}, {{{\widetilde{\varGamma }}}}$

Let ${\mathfrak {f}}_2 = \langle \mathbf{h}_1,\ldots ,\mathbf{h}_i,\ldots \mathbf{h}_t \rangle$ be a 2-face of ${{\mathbb {H}}}(Q)$. We set⁷

$$\begin{aligned} \mathbf{b}({\mathfrak {f}}_2) = \frac{1}{t} \sum ^t_{i=1} \mathbf{h}_i \end{aligned}$$

(122)

a point inside ${\mathfrak {f}}_2$, and . We set

(123)

(124)

where $[\,\cdot \,]$ is the rounding operator. From and , we then define and .

In particular, these functions allow us to define the two functions and as follows

(125)

(126)