Geometry of the Hough Transforms with Applications to Synthetic Data

Abstract

In the framework of the Hough transform technique to detect curves in images, we provide a bound for the number of Hough transforms to be considered for a successful optimization of the accumulator function in the recognition algorithm. Such a bound is consequence of geometrical arguments. We also show the robustness of the results when applied to synthetic datasets strongly perturbed by noise. An algebraic approach, discussed in the appendix, leads to a better bound of theoretical interest in the exact case.

Appendix A. An Algebraic Bound

We keep the notation and assumptions as in the previous sections. A better understanding of the behavior of equations defining the Hough transforms in the parameter space leads to a refinement of Proposition 3.4 (see Proposition A.4).

To begin with, let’s add some comments on the degree and dimension of the Hough transform of points in ${{\mathbb {A}}}_{{{\varvec{x}}}}^n(K)$, $K={{\mathbb {R}}},{{\mathbb {C}}}$.

Clearly, there exists a Zariski open set ${{\mathcal {U}}}_1\subseteq {{\mathbb {A}}}_{{{\varvec{x}}}}^n(K)$ such that, for each point $p\in {{\mathcal {U}}}_1$, the Hough transform $\Gamma _p({{\mathcal {F}}}): f_p({{\varvec{\Lambda }}})=0$ of p is a zero locus of a polynomial of degree h (not depending on p) in the parameter space. Since the Euclidean topology is finer than the Zariski topology, this holds true on a Euclidean open set ${{\mathcal {U}}}_1$ as well. If $K={{\mathbb {C}}}$, the Hough transform $\Gamma _p({{\mathcal {F}}})$ is a hypersurface. If $K={{\mathbb {R}}}$, then $\Gamma _p({\mathcal {F}})$ is $(t-1)$-dimensional if and only if the polynomial $f_p=f_p({{\varvec{\Lambda }}})\in {{\mathbb {R}}}[\Lambda _1,\ldots ,\Lambda _t]$ has a non-singular zero in ${{\varvec{\lambda }}}\in {{\mathbb {R}}}^t$, that is, the gradient $\Big (\frac{\partial f_p}{\partial \Lambda _1}({{\varvec{\lambda }}}), \ldots , \frac{\partial f_p}{\partial \Lambda _t}({{\varvec{\lambda }}}) \Big )\ne 0$ (see again [5, Theorem 4.5.1] for details and equivalent conditions). A standard argument then shows that there exists a Euclidean open set ${{\mathcal {U}}}_2\subseteq {{\mathbb {A}}}_{{{\varvec{x}}}}^n({{\mathbb {R}}})$ such that for each point $p\in {{\mathcal {U}}}_2$ the Hough transform $\Gamma _p({{\mathcal {F}}})$ is a hypersurface in ${{\mathbb {A}}}_{{{\varvec{\Lambda }}}}^t({{\mathbb {R}}})$ (for instance, see [20] for details). Indeed, as a special case of a more general result (see [16, Proposition 2.25]), it holds true that the Hough transform $\Gamma _p({\mathcal {F}})$ is $(t-1)$-dimensional for a generic point $p\in {{\mathbb {A}}}_{(x,y)}^2(K)$, if K is a field. The above comments amount to conclude that, for each point p varying in the Euclidean open set ${{\mathcal {U}}}_1\cap {{\mathcal {U}}}_2\subseteq {{\mathbb {A}}}_{{{\varvec{x}}}}^n(K)$, the Hough transform $\Gamma _p({{\mathcal {F}}})$ is a hypersurface of given degree h not depending on p. Following [20, Section 4] we then define the Hough transforms invariance degree open set as ${{\mathcal {U}}}_1$ if $K={{\mathbb {C}}}$ and ${{\mathcal {U}}}_1\cap {{\mathcal {U}}}_2$ if $K={{\mathbb {R}}}$.

From now on, we assume $n=2$. First, we note a fact we subsume in the sequel. Let ${{\mathcal {B}}}_{\mathrm{aff}}$ be the base locus associated to a family ${{\mathcal {F}}}=\{{{\mathcal {C}}}_{{\varvec{\lambda }}}\}$ of curves (see Definition 3.1). Since clearly ${{\mathcal {U}}}_1\cap {{\mathcal {B}}}_{\mathrm{aff}}=\emptyset$, one has

$$\begin{aligned} {{\mathcal {C}}}_{{\varvec{\lambda }}}\cap {{\mathcal {U}}}_1\subseteq {{\mathcal {C}}}_{{\varvec{\lambda }}}\setminus {{\mathcal {B}}}_{\mathrm{aff}} \end{aligned}$$

for each curve ${{\mathcal {C}}}_{{\varvec{\lambda }}}$ from the family.

Given a point $p=(x_p,y_p)$ in the image space, belonging to the invariance degree open set ${{\mathcal {U}}}_1\subset {{\mathbb {A}}}_{(x,y)}^2(K)$, write the polynomial $f_p({{\varvec{\Lambda }}})$, defining the Hough transform $\Gamma _p({\mathcal {F}})$ of p, as

$$\begin{aligned} f_p({{\varvec{\Lambda }}}) = \sum _{i+j=0}^dx_p^iy_p^j~g_{ij}({{\varvec{\Lambda }}})=\sum _{m_1,\ldots ,m_t} f_{m_1,\ldots ,m_t}(x_p,y_p) \Lambda _1^{m_1} \ldots \Lambda _t^{m_t} \in K[{{\varvec{\Lambda }}}], \end{aligned}$$

(9)

with $0 \le m_1+\cdots +m_t \le h$, where h is the degree of $f_p(\Lambda )$. Let $f_{p}({{\varvec{\Lambda }}})=f_h+\cdots +f_0$ be the decomposition of $f_{p}({{\varvec{\Lambda }}})$ into homogeneous components, where $f_\alpha \in K[{{\varvec{\Lambda }}}]$ is homogeneous of degree $\alpha$, for $\alpha =0,\ldots ,h$. Let $\Lambda _0$ be the new homogenizing coordinate. The homogenization of $f_{p}({{\varvec{\Lambda }}})$ with respect to $\Lambda _0$ is the polynomial $f_{p}({{\varvec{\Lambda }}})^{\hom }=f_h+f_{h-1}\Lambda _0+\cdots +f_0\Lambda _0^h \in K[\Lambda _0,{{\varvec{\Lambda }}}]$.

We order the monomials of the polynomial ring $K[\Lambda _0,\Lambda _1,\ldots ,\Lambda _t]$; for instance, according to the degree-lexicographic order with $\Lambda _t<\cdots <\Lambda _0$ (see [9, p. 48]). Let’s give some definitions.

Definition A.1

We say that the set ${{\mathscr {S}}}=\bigcup _{p\in {{\mathcal {U}}}_1}\big (\mathrm{Supp}(f_p({{\varvec{\Lambda }}}))^{\mathrm{hom}}\big )$ is the generic ordered support according to the fixed ordering. We also write $s:=\#{{\mathscr {S}}}$.

Definition A.2

Take a finite set of points $p_1,\ldots ,p_\nu$ in the image space, belonging to the invariance degree open set ${{\mathcal {U}}}_1$, and let $M(p_1,\ldots ,p_\nu ;{{\mathcal {F}}})\in \mathrm{Mat}_{\nu \times s}(K)$ be the matrix whose j-th row consists of the coefficients of the polynomial $f_{p_j}({{\varvec{\Lambda }}})^{\hom }$ ordered according to Definition A.1. We say that $M(p_1,\ldots ,p_\nu ;{{\mathcal {F}}})$ is the HT-matrix associated to the points $p_1,\ldots ,p_\nu$ with respect to the family ${{\mathcal {F}}}$. We denote by $\varrho (M(p_1,\ldots ,p_\nu ;{{\mathcal {F}}}))$ its rank.

We are interested to find a minimal set of generators of the ideal $\big (f_{p_1}({{\varvec{\Lambda }}}), \ldots ,f_{p_\nu }({{\varvec{\Lambda }}})\big )$ in $K({{\varvec{\Lambda }}}]$. To this purpose, just for technical reasons we pass to the homogenization, then working in $K[\Lambda _0,{{\varvec{\Lambda }}}]$. The following general fact (not involving specific curves from the family) achieves our goal.

Proposition–Definition A.3

Notation as above. Let ${{\mathcal {F}}}$ be a family of curves in ${{\mathbb {A}}}_{(x,y)}^2(K)$. Let ${{\mathscr {I}}}=\{p_1,\ldots ,p_\nu \}$ be a set of distinct points belonging to the invariance degree open set ${{\mathcal {U}}}_1\subset {{\mathbb {A}}}_{(x,y)}^2(K)$. Let ${{\mathscr {T}}}_\nu :=\bigcap _{i=1, \ldots , \nu } \Gamma _{p_j}({\mathcal {F}})$. Consider the smallest positive integer $\nu _{\mathrm{best}}:=\nu _{\mathrm{best}}(p_1,\ldots ,p_\nu )\le \nu$ defined by the condition that there exist indices $1\le j_1 \le \cdots \le j_{\nu _{\mathrm{best}}}\le \nu$ such that

$$\begin{aligned} \big (f_{p_1}({{\varvec{\Lambda }}})^{\hom },\ldots , f_{p_\nu }({{\varvec{\Lambda }}})^{\hom }\big )= \big (f_{p_{j_1}}({{\varvec{\Lambda }}})^{\hom },\ldots , f_{p_{j_{\nu _{\mathrm{best}}}}}({{\varvec{\Lambda }}})^{\hom }\big ), \end{aligned}$$

and set ${{\mathscr {T}}}_{\mathrm{best}}:= \Gamma _{p_{j_1}}({\mathcal {F}})\cap \ldots \cap \Gamma _{p_{j_{\nu _{\mathrm{best}} }}}({\mathcal {F}})$. Then,

• 1. $\nu _{\mathrm{best}}=\varrho \big (M(p_1,\ldots ,p_\nu ;{{\mathcal {F}}})\big )$;

• 2. ${{\mathscr {T}}}_\nu ={{\mathscr {T}}}_{\mathrm{best}}$.

Proof

To prove statement 1), set $M:=M(p_1,\ldots ,p_\nu ;{{\mathcal {F}}})$ and let us first show $\nu _{\mathrm{best}}\le \varrho (M)$. If $\varrho (M)=\nu$, then obviously $\nu _{\mathrm{best}}\le \varrho (M)$, so we assume that $\varrho (M)<\nu$. We know that there exist $\varrho (M)$ rows of M which are linearly independent and span the vectors space generated by all the rows of M. Up to renaming, we can assume that these are the first $\varrho (M)$ rows of M. Pick the j-th row $R_j$ of M with $j>\varrho (M)$. Then $R_j$ can be written as a linear combination of the rows $R_1,\ldots ,R_{\varrho (M) }$. That is (denoting for simplicity $f_{p_j}:=f_{p_j}({{\varvec{\Lambda }}})$, $j=1,\ldots ,\nu$), there exist $\alpha _1^j,\ldots , \alpha _{\varrho (M) }^j\in K$ such that $R_j=\alpha _1^jR_1+\cdots +\alpha _{\varrho (M) }^jR_{\varrho (M)}$. This implies that $f_{p_j}^{\hom }=\alpha _1^jf_{p_1}^{\hom }+\cdots +\alpha _{\varrho (M) }^jf_{p_{\varrho (M)}}^{\hom }$, so that $f_{p_j}^{\hom }\in \big (f_{p_1}^{\hom },\ldots ,f_{p_{\varrho (M)}}^{\hom }\big )$. Since this holds true for each $j>\varrho (M)$, we have

$$\begin{aligned} \big (f_{p_1}^{\hom },\ldots ,f_{p_{\nu }}^{\hom }\big ) \subseteq \big (f_{p_1}^{\hom },\ldots ,f_{p_{\varrho (M)}}^{\hom }\big ), \end{aligned}$$

which implies $\big (f_{p_1}^{\hom },\ldots ,f_{p_{\nu }}^{\hom }\big )= \big (f_{p_1}^{\hom },\ldots ,f_{p_{\varrho (M)}}^{\hom }\big )$. We then conclude that $\nu _{\mathrm{best}}\le \varrho (M)$ by the minimality of $\nu _{\mathrm{best}}$.

To show the converse, and up to renaming, let $f_{p_1}^{\hom },\ldots ,f_{p_{\nu _{\mathrm{best}}}}^{\hom }$ be the generators of $\big (f_{p_1}^{\hom },\ldots ,f_{p_{\nu }}^{\hom }\big )$. If $\nu _{\mathrm{best}}=\nu$ there is nothing to prove, so we assume that $\nu _{\mathrm{best}}<\nu$. For each $f_{p_j}^{\hom }$ with $j>\nu _{\mathrm{best}}$ we have $f_{p_j}^{\hom }\in \big (f_{p_1}^{\hom },\ldots ,f_{p_{\nu _{\mathrm{best}}}}^{\hom } \big )$, that is, there exist polynomials $h_i^j$, $j=1,\ldots ,\nu$, $i=1,\ldots ,\nu _{\mathrm{best}}$, such that

$$\begin{aligned} f_{p_j}^{\hom }=h_1^jf_{p_1}^{\hom }+\cdots +h_{\nu _{\mathrm{best}}}^jf_{p_{\nu _{\mathrm{best}}}}^{\hom }. \end{aligned}$$

(10)

Since $f_{p_j}^{\hom }$ and $f_{p_1}^{\hom },\ldots ,f_{p_{\nu _{\mathrm{best}}}}^{\hom }$ are homogeneous polynomials of the same degree it follows that the $h_i^j$’s are homogeneous of degree zero, that is, $h_i^j\in K$. Thus, equality (10) is equivalent to say that each row $R_j$ of M is a linear combination of $R_1,\ldots ,R_{\nu _{\mathrm{best}}}$. The conclusion $\varrho (M)\le \nu _{\mathrm{best}}$ then immediately follows.

As to statement 2), consider the ideal $(f_{p_1}({{\varvec{\Lambda }}}),\ldots , f_{p_\nu }({{\varvec{\Lambda }}}))$ in $K[{{\varvec{\Lambda }}}]$. We want to prove that there exist indices $j_1,\ldots ,j_{\mathrm{\nu _\mathrm{best}}}$, $1\le j_1< \cdots <j_{\mathrm{\nu _\mathrm{best}}}\le \nu$, such that

$$\begin{aligned} (f_{p_1}({{\varvec{\Lambda }}}),\ldots , f_{p_\nu }({{\varvec{\Lambda }}}))=\big (f_{p_{j_1}}({{\varvec{\Lambda }}}), \ldots ,f_{p_{j_{\mathrm{\nu _\mathrm{best}} }}}({{\varvec{\Lambda }}})\big ). \end{aligned}$$

The inclusion “$\supseteq$” is obvious, so we only have to prove the converse inclusion “$\subseteq$”. By definition of $\nu _{\mathrm{best}}$, we know that there are indices $1\le j_1< \cdots <j_{\mathrm{\nu _\mathrm{best}}}\le \nu$, such that

$$\begin{aligned} (f_{p_1}^{\mathrm{hom}}({{\varvec{\Lambda }}}),\ldots , f_{p_\nu }^{\mathrm{hom}}({{\varvec{\Lambda }}}))= & {} \big (f_{p_{j_1}}^{\mathrm{hom}}({{\varvec{\Lambda }}}), \ldots ,f_{p_{j_{\mathrm{\nu _\mathrm{best}} }}}^{\mathrm{hom}}({{\varvec{\Lambda }}})\big ) \\\subseteq & {} \big (f_{p_{j_1}}({{\varvec{\Lambda }}}), \ldots ,f_{p_{j_{\mathrm{\nu _\mathrm{best}} }}}({{\varvec{\Lambda }}})\big )^{\mathrm{hom}}, \end{aligned}$$

where the last inclusion follows by definition of ideal homogenization (see [12, Definition 4.3.4]). Passing to the dehomogenization, we get (see [12, Proposition 4.3.12])

$$\begin{aligned} (f_{p_1}^{\mathrm{hom}}({{\varvec{\Lambda }}}),\ldots , f_{p_\nu }^{\mathrm{hom}}({{\varvec{\Lambda }}}))^{\mathrm{deh}}= & {} \big (f_{p_{j_1}}^{\mathrm{hom}}({{\varvec{\Lambda }}}), \ldots ,f_{p_{j_{\mathrm{\nu _\mathrm{best}} }}}^{\mathrm{hom}}({{\varvec{\Lambda }}})\big )^{\mathrm{deh}} \\\subseteq & {} \Big (\big (f_{p_{j_1}}({{\varvec{\Lambda }}}), \ldots ,f_{p_{j_{\mathrm{\nu _\mathrm{best}} }}}({{\varvec{\Lambda }}})\big )^{\mathrm{hom}}\Big )^{\mathrm{deh}}\\= & {} \big (f_{p_{j_1}}({{\varvec{\Lambda }}}), \ldots ,f_{p_{j_{\mathrm{\nu _\mathrm{best}} }}}({{\varvec{\Lambda }}})\big ), \end{aligned}$$

where the last equality is a consequence of [12, Proposition 4.3.5]. Since

$$\begin{aligned} (f_{p_1}^{\mathrm{hom}}({{\varvec{\Lambda }}}),\ldots , f_{p_\nu }^{\mathrm{hom}}({{\varvec{\Lambda }}}))^{\mathrm{deh}} = \big (f_{p_1}({{\varvec{\Lambda }}}), \ldots ,f_{p_{\nu }}({{\varvec{\Lambda }}})\big ), \end{aligned}$$

(see [12, Corollary 4.3.8]), the claimed inclusion follows. Thus, we can conclude that

$$\begin{aligned} {{\mathscr {T}}}_\nu =\Gamma _{p_1}({\mathcal {F}})\cap \ldots \cap \Gamma _{p_\nu }({\mathcal {F}})= \Gamma _{p_{j_1}}({\mathcal {F}})\cap \ldots \cap \Gamma _{p_{j_{\nu _{\mathrm{best}} }}}({\mathcal {F}}). \end{aligned}$$

$\square$

Proposition A.4

Notation as above. Let ${{\mathcal {F}}}=\{{{\mathcal {C}}}_{{\varvec{\lambda }}}\}$ be a family of curves in ${{\mathbb {A}}}_{(x,y)}^2(K)$. Fix a curve ${{\mathcal {C}}}_{{\varvec{\lambda }}}$ from the family, and take $\nu _{\mathrm{opt}}=d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1$ distinct points $p_1,\ldots ,p_{\nu _{\mathrm{opt}}}$ on ${{\mathcal {C}}}_{{\varvec{\lambda }}}\cap {{\mathcal {U}}}_1$. Let ${{\mathscr {T}}}=\cap _{p \in C_{{{\varvec{\lambda }}}}} \Gamma _p({\mathcal {F}})$ and let ${{\mathscr {T}}}_{\mathrm{best}} = \bigcap _{j=1, \ldots , \nu _{\mathrm{best}}} \Gamma _{p_j}({\mathcal {F}})$. Then we have:

• 1. ${{\mathcal {C}}}_{{{{\varvec{\lambda }}}}'}={{\mathcal {C}}}_{{{{\varvec{\lambda }}}} }$ for each ${{{\varvec{\lambda }}}}' \in {{\mathscr {T}}}_{\mathrm{best}}$.

• 2. ${{\mathscr {T}}}_{\mathrm{best}}={{\mathscr {T}}}$.

• 3. If the family ${{\mathcal {F}}}$ is Hough regular, then ${{\mathscr {T}}}_{\mathrm{best}}=\{{{{\varvec{\lambda }}}}\}$.

Proof

If $\nu _{\mathrm{best}}=d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1$, the result simply follows from Proposition 3.4. Then we can assume that $\nu _{\mathrm{best}}=\varrho (M(p_1,\dots ,p_{\nu _{\mathrm{opt}}};{{\mathcal {F}}}))< d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1$. Therefore, Proposition-Definition A.3(2) yields

$$\begin{aligned} {{\mathscr {T}}}_{\mathrm{opt}}= \bigcap _{j=1,\ldots ,\nu _{\mathrm{opt}} } \Gamma _{p_j}({\mathcal {F}})= \bigcap _{j=1, \ldots , \nu _{\mathrm{best}} } \Gamma _{p_j}({\mathcal {F}})= {{\mathscr {T}}}_{\mathrm{best}}. \end{aligned}$$

Thus, Proposition 3.4 applies again to conclude the proof. $\square$

The following remark clarifies the relations between the bounds $\nu _{\mathrm{opt}}$ (see Proposition 3.4) and $\nu _{\mathrm{best}}$, as well as, for families ${\mathcal {F}}$ which are Hough regular, between them and the number of parameters t.

Remark A.5

Assumptions and notation as in Proposition A.4. In fact, instead of $\nu _{\mathrm{best}}$, it is possible to use the easier computable bound

$$\begin{aligned} \nu _{\mathrm{best}}':= \min \{s-1,d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1\}, \end{aligned}$$

with $s=\#{{\mathscr {S}}}$ as in Definition A.1. This follows from the fact that the points $p_j$, $j=1,\ldots ,\nu _{\mathrm{opt}}$, lie on a given curve ${{\mathcal {C}}}_{{\varvec{\lambda }}}$, ${{\varvec{\lambda }}}=(\lambda _1,\ldots ,\lambda _t)$, from the family ${{\mathcal {F}}}$, and consequently the s columns of M are linearly dependent. Precisely, recalling expression (9), one has

$$\begin{aligned} f_{p_j}({{\varvec{\Lambda }}}) = \sum _{m_1,\ldots ,m_t} f_{m_1,\ldots ,m_t}(x_{p_j},y_{p_j}) \Lambda _1^{m_1} \ldots \Lambda _t^{m_t} =0, \end{aligned}$$

and the j-th row of M is made up of the coefficients (ordered according to the fixed ordering) $f_{m_1,\ldots ,m_t}(x_{p_j},y_{p_j})$, for $j=1,\ldots ,\nu _{\mathrm{opt}}$. In conclusion, we have the inequalities:

$$\begin{aligned} \nu _{\mathrm{best}}=\varrho (M)\le \nu _{\mathrm{best}}'= \min \{s-1,d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1\}\le \nu _{\mathrm{opt}}=d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1. \end{aligned}$$

(11)

Now, assume that the family ${\mathcal {F}}$ is Hough regular. Coming back to Sect. 2.1, consider the ideal

$$\begin{aligned} I=\big (f_{p_1}({{\varvec{\Lambda }}}), \ldots ,f_{p_h}({{\varvec{\Lambda }}})\big ) \subset K[\Lambda ] \end{aligned}$$

generated by the polynomials $f_p({{\varvec{\Lambda }}})$ defining the Hough transforms $\Gamma _p({{\mathcal {F}}})$, $p\in {{\mathcal {C}}}_\lambda$. Let m be the minimal number of generators of I in $K[\Lambda ]$, so that, as we noted, $h\ge m$. Furthermore, such a number m has to satisfy the lower bound $m \ge \text {codim}(I)$ (see [11] and also [9, Chapter 10]). By definition, $\text {codim}(I) := \text {dim}(K[\Lambda ]) - \text {dim}(K[\Lambda ]/I)$. Since $\text {dim}(K[\Lambda ])=t$ and $\text {dim}(K[\Lambda ]/I)=0$ (this derives from the assumption that the family ${\mathcal {F}}$ is Hough regular, which implies that the ideal I is zero-dimensional), it then follows $m\ge t$, whence $h\ge t$. Thus, in particular, relations (11) yield

$$\begin{aligned} \nu _{\mathrm{opt}} \ge \nu _{\mathrm{best}}\ge t. \end{aligned}$$

$\square$

We provide here some illustrative examples in the real case.

Example A.6

(Curve of Lamet) Consider in ${\mathbb {A}}^2_{(x,y)}({\mathbb {R}})$ the family ${{\mathcal {F}}}=\{{{\mathcal {C}}}_{a,b}\}$ of curves of degree m of equation $\left( \frac{x}{a}\right) ^m+\frac{y^m}{b}=1$, that is,

$$\begin{aligned} {\mathcal {C}}_{a,b}: bx^m+a^my^m=a^mb, \end{aligned}$$

(12)

for positive real numbers a, b. The curve of Lamet is clearly non-singular (even in the complex projective plane ${{\mathbb {P}}}^2({{\mathbb {C}}})$), and then of genus $\frac{(m-1)(m-2)}{2}$.

We further assume that the degree m is even. As noted below, this assures the boundedness of the curve. (If m is odd the curve is unbounded: think, for example, to the Fermat elliptic cubic of equation $x^3+y^3=1$.) Indeed, the knowledge of some basic facts about p-norms on ${\mathbb {R}}^n$ allows us to show that the curve of Lamet is contained in the rectangular region

$$\begin{aligned} \left\{ (x,y) \in {\mathbb {A}}^2_{(x,y)}({\mathbb {R}}) \, \big |\, -a \le x \le a, \, -b^{1/m} \le y \le b^{1/m} \right\} . \end{aligned}$$

Passing to homogeneous coordinates we have

$$\begin{aligned} \overline{{{\mathcal {C}}}_{a,b}}: bx_0^m+a^mx_1^m-a^mbx_2^m=0, \end{aligned}$$

whence ${{\mathcal {B}}}({{\mathbb {C}}})=\{\emptyset \}$. In order to compute $\nu _{\mathrm{best}}$, note that for any point $p=(x_p,y_p)$ in the invariance degree open set ${{\mathcal {U}}}_1\subset {{\mathbb {A}}}_{(x,y)}^2({{\mathbb {R}}})$, the Hough transform is the $(m+1)$-degree curve $\Gamma _p({{\mathcal {F}}})$ in the parameter plane $\langle A, B \rangle$ of equation

$$\begin{aligned} f_p(A,B)= A^mB -y_p^mA^m-x_p^mB=0. \end{aligned}$$

Therefore the bound from Proposition 3.4 becomes

$$\begin{aligned} \nu _{\mathrm{opt}}=d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1=m^2+1. \end{aligned}$$

For instance, in the case $m=4$, $\nu _{\mathrm{opt}}=17$. Thus, Proposition  A.4 yields

$$\begin{aligned} \nu _{\mathrm{best}}\le \min \{s-1,d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1\} =\min \{2,17\}=2, \end{aligned}$$

where $s:=\#{{\mathscr {S}}}$. To see that $\nu _{\mathrm{best}}=2$, take 17 points $p_\ell =(x_{p_\ell }, y_{p_\ell })$, $\ell =1,\ldots ,17$, on the Lamet curve

$$\begin{aligned} {\mathcal {C}}_{a,b}: bx^4+a^4y^4-a^4b=0, \end{aligned}$$

with a, b fixed and the points belonging to the open set ${{\mathcal {U}}}_1$.

The coefficient of the maximum degree term of $f_p(A,B)$ equals 1, so that it generates the whole ring ${{\mathbb {R}}}[A,B]$, that is, ${{\mathcal {U}}}_1={{\mathbb {R}}}^2$. Keeping the notation as in the proof of Proposition  A.4, consider the (transpose of) HT-matrix M associated to the set of points $\{p_\ell \}_{\ell =1,\ldots , 17}$, that is,

Compute, for $i, j,k\in \{1,\ldots ,17\}$, $i\ne j\ne k$,

to conclude that $\varrho (M)=\nu _{\mathrm{best}}=2$. $\square$

Example A.7

Consider in ${\mathbb {A}}^2_{(x,y)}({\mathbb {R}})$ the family ${{\mathcal {F}}}=\{{{\mathcal {C}}}_{a,b}\}$ of conics of equation

$$\begin{aligned} {{\mathcal {C}}}_{a,b}:a^2x^2+by+x=0, \end{aligned}$$

for real parameters ${{{\varvec{\lambda }}}}=(a,b)$. Passing to homogeneous coordinates we see that $\#{{\mathcal {B}}}({{\mathbb {C}}})=2$. Whence $\nu _{\mathrm{opt}}=d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1=3$. For a general point $p=(x_p,y_p)$, the Hough transform is the conic of equation

$$\begin{aligned} f_p(A,B)=x_p^2A ^2+y_pB+x_p=0, \end{aligned}$$

so that $s=\#\mathrm{Supp}(f_p(A,B))=3=\nu _{\mathrm{opt}}$. Consider the three points $p_1=(1,-2)$, $p_2=(-1,0)$, $p_3=(-2,-2)$ on ${{\mathcal {C}}}_{1,1}$ and the polynomials $f_{p_1}(A,B)=A^2-2B+1$, $f_{p_2}(A,B)=A^2-1$, $f_{p_3}(A,B)=4A^2-2B-2$. The HT-matrix $M \in \mathrm {Mat}_{3 \times 3}({{\mathbb {R}}})$ is

$$\begin{aligned} M=\left( \begin{matrix} 1 &{}\quad -2 &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad -1 \\ 4 &{}\quad -2 &{}\quad -2 \end{matrix} \right) , \end{aligned}$$

whose rank is $\nu _{\mathrm{{best}}}=\varrho (M)=2$. We have ${{\mathscr {T}}}_{\mathrm{best}}=\Gamma _{p_1}({{\mathcal {F}}})\cap \Gamma _{p_2}({{\mathcal {F}}})=\{(1,1), (-1,1)\}$. (Note that $\Gamma _{p_1}({{\mathcal {F}}})$, $\Gamma _{p_2}({{\mathcal {F}}})$ have two more coinciding common points at infinity). The family ${{\mathcal {F}}}$ is not Hough regular unless $a>0$, in which case ${{\mathscr {T}}}_{\mathrm{best}}=\{(1,1)\}$. $\square$

Although most of the times $\varrho (M)<d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1$, so that $\nu _{\mathrm{best}}=\rho (M)$ there are also cases where the equality $\varrho (M)=d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1$ holds true, as the following simple example shows.

Example A.8

Consider in ${\mathbb {A}}^2_{(x,y)}({\mathbb {R}})$ the family ${{\mathcal {F}}}=\{{{\mathcal {C}}}_{a,b,c}\}$ of lines of equation

$$\begin{aligned} {{\mathcal {C}}}_{a,b,c}:ax+by+c=0, \end{aligned}$$

for real parameters ${{{\varvec{\lambda }}}}=(a,b,c)$. The polynomial defining the Hough transform of a general point (x, y) is

$$\begin{aligned} f_{(x,y)}(A,B,C) = xA+yB+C \in {{\mathbb {R}}}[x,y][A,B,C], \end{aligned}$$

having support ${{\mathscr {S}}}=\{A,B,C\}$. Then $s=3$, $d^2-\#{{\mathcal {B}}}({{\mathbb {C}}})+1=s-1=2$, whence $\nu _{\mathrm{best}}\le 2$. Consider the two points $p_1=(0,-1)$, $p_2=(-1,0)$ on ${{\mathcal {C}}}_{1,1,1}$ , and the polynomials $f_{p_1}(A,B,C)=-B+C$, $f_{p_2}(A,B,C)=-A+C$. The HT-matrix $M \in \mathrm {Mat}_{2 \times 3}({{\mathbb {R}}})$ is

$$\begin{aligned} M = \left( \begin{matrix} 0 &{}\quad -1 &{}\quad 1\\ -1 &{}\quad 0 &{}\quad 1 \end{matrix}\right) , \end{aligned}$$

whose rank is $\nu _{\mathrm{best }}=\varrho (M)=2=\nu _{\mathrm{opt}}$. Then the set ${{\mathscr {T}}}_{\mathrm{best}}=\Gamma _{p_1}({{\mathcal {F}}})\cap \Gamma _{p_2}({{\mathcal {F}}})$ coincides with the line $\{(t,t,t)\,|\,t \in {{\mathbb {R}}}\}$ in the parameter space ${{\mathbb {R}}}^3=\langle A, B, C\rangle$. Clearly, ${{\mathcal {C}}}_{t,t,t}={{\mathcal {C}}}_{1,1,1}$ for each $t \in {{\mathbb {R}}}$, according to Proposition A.4(1).

The family ${{\mathcal {F}}}$ is not Hough regular, meeting the regularity property as soon as one of the parameters is fixed. For instance, letting $c=1$, we get the family of lines ${{\mathcal {F}}}'=\{{{\mathcal {C}}}_{a,b}: ax+by+1=0\}$, and now ${{\mathscr {T}}}_{\mathrm{best}}=\Gamma _{p_1}({{\mathcal {F}}}')\cap \Gamma _{p_2}({{\mathcal {F}}}')=\{(1,1)\}$ with $p_1=(0,-1)$, $p_2=(-1,0)$ on ${{\mathcal {C}}}_{1,1}$.

Since the parameters are linear, the same conclusions follow from Lemma  3.2. $\square$