# The mathematical model

#### Origin of the model:

• Hoffman (1989): structure of a contact manifold
• Petitot (1999): structure of a sub-Riemannian manifold (Heisenberg group)

#### then refined by:

• Citti, Sarti (2006): structure of the rototranslations of the plane $$SE(2)$$
• Agrachev, Boscain, Charlot, Gauthier, Rossi, D.P. (2010—)

#### Also studied by:

• Sachkov (2010-)
• Duits (2009–)

# Neurophysiological fact

### Structure of the primary visual cortex

Hubel and Wiesel (Nobel prize 1981) observed that in the primary visual cortex V1, groups of neurons are sensitive to both positions and directions.

# Citti-Petitot-Sarti model

V1 is modeled as the projective tangent bundle: $PT\mathbb R^2 = \mathbb R^2\times P^1$

We need to discuss

#### Receptive fields:

How a greyscale visual stimulus $$f:\mathbb R^2\to [0,1]$$ is lifted to an state on $$V1$$.

#### Spontaneous evolution:

How a state on $$V1$$ evolves via neuronal connections.

# Receptive fields

Receptive fields are aimed to describe a family of neurons $$\xi=(x,y,\theta)$$ in $$V1\cong PT\mathbb R^2$$ with a family of functions $$\{\Psi_\xi\}_\xi$$ on the image plane.

• It is widely accepted that a greyscale visual stimulus $$f:\mathbb R^2\to [0,1]$$ feeds a V1 neuron $$\xi$$ with an extracellular voltage $$f\mapsto Lf(\xi)$$ given by $Lf(\xi) = \langle f, \Psi_\xi \rangle_{L^2(\mathbb R^2)}.$

• A good fit for $$\Psi_\xi$$ is the Gabor filter (sinusoidal wave multiplied by Gaussian function) centered at $$(x,y)$$ of orientation $$\theta$$

# Group theoretical approach

Consider $$SE(2) = \mathbb R^2\rtimes \mathbb S^1$$ with group operation $(x,\alpha)(y,\beta) = (x+R_\beta y,\alpha+\beta),$ where $$R_\alpha$$ is the rotation of $$\alpha$$.

• $$SE(2)$$ is the double covering of $$PT\mathbb R^2$$, so with appropriate care we can work here
• The left regular (unitary) representation $$\Lambda$$ of $$SE(2)$$ acting on $$\varphi\in L^2(SE(2))$$ is $\Lambda(x,\alpha)\varphi(y,\beta) = \varphi((x,\alpha)^{-1}(y,\beta)) = \varphi(R_{-\alpha}(y-x),\beta-\alpha)$
• The quasi-regular (unitary) representation $$\pi$$ of $$SE(2)$$ acting on $$L^2(\mathbb R^2)$$ is $\pi(x,\alpha)f(y)=f(R_{-\alpha}(y-x))$
Definition: A lift operator $$L:L^2(\mathbb R^2)\to L^2(SE(2))$$ is left-invariant if $\Lambda(a) Lf = L(\pi(a)f) \qquad\forall f\in L^2(\mathbb R^2), a\in SE(2).$
Theorem: (J.-P. Gauthier, D.P.)
Let $$L$$ be a left-invariant lift such that
• $$L$$ is linear,
• $$f\mapsto Lf(0)$$ is densely defined and bounded.
Then, there exists $$\Psi\in L^2(\mathbb R^2)$$ such that $Lf(x,\theta) = \langle f,\pi(x,\theta)\Psi\rangle_{L^2(\mathbb R^2)} (= f\star R_{\theta}(-\bar{\Psi})) (x).$

#### Remarks

• V1 receptive fields through Gabor filters define a left-invariant lift.
• The function $$(x,k)\mapsto \langle f,\pi(x,k)\Psi\rangle_{L^2(\mathbb H)}$$ is known as the wavelet transform of $$f$$ w.r.t. $$\Psi$$.

# Spontaneous evolution in V1

Two types of connections between neurons:

• Lateral connections: Between iso-oriented neurons, in the direction of their orientation. Represented by the integral lines of the vector field $X_1(x,y,\theta) = \cos\theta\partial_x + \sin\theta\partial_y.$
• Local connections: Between neurons in the same hypercolumn. Represented by the integral lines of the vector field $X_2(x,y,\theta) = \partial_\theta.$

# Evolution in the Citti-Petitot-Sarti model

Given two independent Wiener process $$W$$ and $$Y$$ on $$PT\mathbb R^2$$, neuron excitation evolve according to the SDE $dA_t = X_1 dW_t + X_2 dY_t$

Its generator $$\mathcal L = (\cos\theta\partial_x + \sin\theta\partial_y)^2+\partial_\theta^2$$ yields to the hypoelliptic equation for the evolution of a stimulus $$\Psi\in L^2(PT\mathbb R^2)$$: $\frac d {dt} \Psi = \mathcal L \Psi.$

• Highly anisotropic evolution
• Natural sub-Riemannian interpretation
• Invariant under the action of $$SE(2)$$

# Semi-discrete model

Conjecture: The visual cortex can detect a finite (small) number of directions only ($$\approx$$ 30).

This suggests to replace $$SE(2)$$ with $$SE(2,N)=\mathbb R^2\times \mathbb Z_N$$ and $$PT\mathbb R^2$$ with $$\mathbb R^2\times \mathbb Z_{N/2}$$.

#### Local connections

We replace $$\partial_\theta dY_t$$ by the jump Markov process $$\Theta_t$$ on $$\mathbb Z_N$$ defined as follows.

• The time of the first jump is exponentially distributed, with probability $$\frac 1 2$$ on either side.
• We obtain a Poisson process with $P(k\text{ jumps in }[0,t]) = \frac {(\beta t)^k}{k!} e^{-\beta t}$

The infinitesimal generator is $$\Lambda_N$$

#### The evolution operator

The semi-discrete evolution operator, acting on $$\Psi=(\Psi_r)_ r \in L^2(\mathbb R^2\times \mathbb Z_N)$$ is then $\mathcal L_N (z,r) = (A\Psi)_r (z) + (\Lambda _N\Psi(z)) _r,$ where, for $$\theta_r=\pi r/N$$, $(A\Psi)_r (z) = \frac 1 2\left( \cos(\theta _r)\partial _x + \sin(\theta _r)\partial _y \right)\Psi _r(z),$ $(\Lambda_N \Psi(z))_r = \frac \beta 2 \left( \Psi _{r-1}(z)-2\Psi _r(z) + \Psi _{r+1}(z) \right).$

• Already naturally discretized the $$\theta$$ variable.
• Invariant w.r.t. semi-discretized rototranslations $$SE(2,N)$$.

# Illusory contours

Neurophysiological assumption: We reconstruct images through the natural diffusion in V1, which minimizes the energy required to activate unexcitated neurons.

# Anthropomorphic image reconstruction

Let $$f\in L^2(\mathbb R^2)$$ with $$f\equiv 0$$ on $$\Omega\subset \mathbb R^2$$ be an image corrupted on $$\Omega$$.

1. Smooth $$f$$ by a Gaussian filter to get generically a Morse function
2. Lift $$f$$ to $$Lf$$ on $$\mathbb R^2\times \mathbb Z_N$$
3. Evolve $$Lf$$ through the CPS semi-discrete evolution
4. Project the evolved $$Lf$$ back to $$\mathbb R^2$$
##### Remarks
• Reasonable results for small corruptions
• Adding some heuristic procedures yield very good reconstructions

# Smoothing

Even if images are not described by Morse functions, the retina smooths images through a Gaussian filter (Peichl & Wässle (1979), Marr & Hildreth (1980))

Theorem (Boscain, Duplex, Gauthier, Rossi 2012)
The convolution of $$f\in L^2(\mathbb R^2)$$ with a two dimensional Gaussian centered at the origin is generically a Morse function.

# Simple Lift

For simplicity, instead than the lift through Gabor filters we chose to lift $$f(x,y)$$ to $Lf(x,y,r) = \begin{cases} f(x,y) & \text{ if } (\nabla f(x,y), (\cos\theta_r,\sin\theta_r))\cong 0\\ 0 & \text{ otherwise} \end{cases}$

# Example: lifting a curve

In the continuous model a curve $$t\mapsto (x(t),y(t))$$ in $$\mathbb R^2$$ is lifted to $$t\mapsto (x(t),y(t),\theta(t))$$ curve in $$PT\mathbb R^2$$, where $\theta(t) = \arctan\left( \frac{\dot x(t)}{\dot y(t)}\right).$

# A remarkable feature

When $$f$$ is a Morse function, the lift Lf is supported on a 2D manifold. (This is false if the angles are not projectivized!)

# Semi-discrete evolution

The hypoelliptic heat kernel of the semi-discretized operator $$\mathcal L_N$$ has been explicitly computed, but is impractical from the numerical point of view.

We use the following scheme:

1. For any $$r\in\mathbb Z_N$$ compute $$\widehat{Lf_r}(k)$$, the Fourier transform of $$Lf(\cdot,\cdot,r)$$.
2. For any $$k=(\lambda\cos\alpha,\lambda\sin\alpha)$$ we let $$U_k(r)$$ be the solution of the decoupled ODE on $$\mathbb Z_N$$ $\begin{cases} \frac{d}{dt}U_k = \Lambda_N U_k - \text{diag}\left( \lambda^2 \cos(\theta_r-\alpha)\right)U_k \\ U_k(r)| _ {t=0}=\widehat{Lf _ r}(k). \end{cases}$
3. For any $$r\in\mathbb Z_N$$ the solution $$\Psi_r(x,y)$$ is the inverse Fourier transform of $$k\mapsto U_r(k)$$

This scheme can be implemented by discretizing $$\mathbb R^2$$.

# A view through almost-periodic functions

To avoid discretizing $$\mathbb R^2$$ we can proceed as follows: Let $$\Sigma\subset\mathbb R^2$$ be the grid of pixels of the image and $$f_d:\Sigma\to [0,1]$$ be the pixel values.

1. Compute $$\hat f_d:\widehat\Sigma\to \mathbb C$$, the discrete Fourier transform (FFT) of $$f_d$$ on $$\Sigma$$
2. Represent $$f_d$$ as the almost-periodic function $f_{\text{AP}}(x) = \sum_{k\in \widehat\Sigma} \hat f_d(k)e^{i\langle z, k\rangle}.$
3. Evolve $$f$$ splitting the evolution in the ODE’s, for each $$k=(\lambda,\mu)\in\widehat\Sigma$$, $\begin{cases} \frac{d}{dt}U_k = \Lambda_N U_k - 2\pi^2 \text{diag}\left( \lambda \cos\theta_r+\mu \sin\theta_r\right)U_k \\ U_k(r)| _ {t=0}=\hat f_d(k). \end{cases}$

The evolution of $$f_{\text{AP}}$$ is exact!

# Projection

Given the result of the evolution $$(\Psi_r)_r$$, we define the reconstructed image by

$\tilde f(x,y) = \max_{r\in\mathbb Z_N} \Psi_r(x,y).$

# Mumford Elastica Model

With the same techniques it is possible to treat the evolution equation underlying the Mumford Elastica model, which is associated with the operator $\mathcal L_M = X_1+X_2^2 = \cos\theta\partial_x + \sin\theta\partial_y +\partial_\theta^2$

# Highly corrupted images reconstruction

Based upon this diffusion and certain heuristic complements, we get nice results on images with more than 85% of pixels missing.

# Spectral invariants

Given a group $$\mathbb G$$ and a representation $$\Phi$$ of $$\mathbb G$$ on some topological space $$X$$, a complete set of invariants is a map $$f\mapsto I_f$$, from $$X$$ to some functional space $$\mathcal B$$ such that $I_f=I_g \iff \exists a\in\mathbb G \text{ s.t. } f= \Phi(a)g.$ If the above holds only for $$f$$ and $$g$$ in some residual subset of $$X$$, the $$I_f$$’s are said to be a weakly complete set of invariants.

We speak of spectral invariants whenever the $$I_f$$’s depend on the Fourier transform of $$f$$.

#### Important cases

1. Since any group acts on itself, we look for invariants for the action of the left regular representation $$\Lambda$$ of $$\mathbb G$$ on $$X=L^2(\mathbb G)$$
2. Given a semidirect product $$\mathbb G = \mathbb H\rtimes \mathbb K$$, we look for invariants for the quasi-regular representation $$\pi$$ of $$\mathbb G$$ on $$X=L^2(\mathbb H)$$.

# Plan

1. Case of an abelian group (e.g. invariants on $$\mathbb R^n$$ w.r.t. translations)
• Fourier transform and Pontryagin duality
• Bispectral invariants
2. Case of $$SE(2,N)$$ (non-compact, non-abelian semi-direct product)
• Generalized Fourier transform and Chu duality
• Bispectral invariants for $$L^2(SE(2,N))$$
• Invariants for lifts of functions in $$L^2(\mathbb R^2)$$ w.r.t. semidiscretized roto-translations.

# Fourier transform on abelian groups

Let $$\mathbb G$$ be a locally compact abelian group with Haar measure $$d\mu$$.

The dual $$\widehat{\mathbb G}$$ of $$\mathbb G$$ is the set of of characters of $$\mathbb G$$, i.e., of continuous group homomorphism $$\lambda:\mathbb G\to \mathbb C$$, $$|\lambda(\cdot)|=1$$.

The Fourier transform of $$f\in L^1(\mathbb G)\cap L^2(\mathbb G)$$ is the map on $$\widehat{\mathbb G}$$ defined by $\hat f(\lambda)= \int_{\mathbb G} f(a) \, \bar\lambda(a)\,d\mu(a).$

Since $$\widehat{\mathbb G}$$ is abelian and locally compact, the Fourier transform extends to an isometry $$\mathcal F:L^2(\mathbb G) \to L^2(\widehat{\mathbb G})$$ w.r.t. the Haar measure on $$\widehat{\mathbb G}$$.

Fundamental property: For any $$a\in\mathbb G$$, $f= \Lambda(a) g \iff \hat f(\lambda) = \lambda(a)\hat g(\lambda) \quad\forall\lambda\in\widehat{\mathbb G}.$

# Example: $$\mathbb R$$

It holds $$\widehat {\mathbb R} \cong \mathbb R$$, where for $$\lambda\in\mathbb R$$ the corresponding element of $$\widehat {\mathbb R}$$ is $$\hat \lambda(x):= e^{i\lambda x}$$. The above defined Fourier transform then reduces to $\hat f(\lambda) = \int_{\mathbb R} f(x) e^{-i \lambda x}\, dx$ Then, $f(y) = g(y-x) \quad\forall y\in\mathbb R \iff \hat f(\lambda) = e^{i\lambda x}\hat g(\lambda) \quad\forall\lambda\in{\mathbb R}$

#### Remark

In this case, it is clear that $$\widehat{\mathbb R}$$ is indeed a group and that $$\widehat {\widehat {\mathbb R}}\cong\mathbb R$$. This is an instance of Pontryagin duality, which works on all abelian groups:

Theorem: (Pontryagin duality) The dual of $$\widehat{\mathbb G}$$ is canonically isomorphic to $$\mathbb G$$. More precisely, $$\Omega:\mathbb G\to \widehat{\widehat{\mathbb G}}$$ defined as $$\Omega(a)(\lambda) = \lambda(a)$$ is a group isomorphism.

# Invariants for abelian groups

Let $$\mathbb G$$ be a locally compact abelian group and consider the action on $$L^2(\mathbb G)$$ of the left-regular representation (e.g. for $$\mathbb G = \mathbb R$$ this corresponds to translations).

The power spectrum invariants for $$f\in L^2(\mathbb G)$$ w.r.t. the action of the left regular representation are the functions $\lambda\in\widehat{\mathbb G} \mapsto I_f(\lambda)=|\hat f(\lambda)|^2.$

These are widely used (e.g. in astronomy), but are not complete:

• Fix any $$\varphi: \widehat{\mathbb G} \to \mathbb C$$ s.t. $$|\varphi(\lambda)|=1$$ which is not a character of $$\widehat{\mathbb G}$$,
• Let $$g = \mathcal F^{-1}( \varphi \hat f )$$, so that $$I_f=I_g$$,
• However, $$f\neq\Lambda(a)g$$ for any $$a\in\mathbb G$$, since $f= \Lambda(a)g \iff \hat f(\lambda) = \lambda(a)\hat g(a).$ Indeed, if it was the case $$\varphi(\lambda) = \lambda(a)$$ which, by Pontryagin duality, is equivalent to $$\varphi$$ being a character of $$\widehat{\mathbb G}$$.

What is missing in the power spectrum invariants is the phase information.

The bispectral invariants for $$f\in L^2(\mathbb G)$$ w.r.t. the action of the left regular representation are the functions $(\lambda_1,\lambda_2)\in\widehat{\mathbb G} \mapsto B_f(\lambda_1,\lambda_2)={\hat f(\lambda_1+\lambda_2)}\,\overline{\hat f(\lambda_1)}\, \overline{\hat f(\lambda_2)}.$

Theorem: The bispectral invariants are weakly complete on $$L^2(K)\subset L^2(\mathbb G)$$, where $$K\subset \mathbb G$$ is compact. In particular, they discriminate on the residual set $$\mathcal G\subset L^2(\mathbb G)$$ of those square-integrable $$f$$’s such that $$\hat f(\lambda)\neq 0$$ on an open-dense subset of $$\widehat{\mathbb G}$$.

#### Remarks

• Note that $$\lambda\mapsto \frac{B(\lambda,0)}{\hat f(0)}$$ allows to recover the power spectral invariants
• The bispectral invariants are used in several areas of signal processing (e.g. to identify music timbre and texture, Dubnov et al. (1997))

# Proof of weak completeness

Let $$f,g\in \mathcal G$$ be compactly supported and such that $$I_f=I_g$$.

• Define $$u(\lambda) = \hat g(\lambda)/\hat f(\lambda)$$, which is a continuous function on an open and dense set of $$\mathbb G$$ satisfying $$|u(\cdot)|\equiv 1$$.
• Since the bispectral invariants coincide it holds $u(\lambda_1+\lambda_2) = u(\lambda_1)u(\lambda_2).$
• Since $$f,g$$ are compactly supported, $$\hat f$$ and $$\hat g$$ are continuous and hence $$u$$ can be extended to a measurable function on $$\widehat{\mathbb G}$$, still satisfying the above.
• Since every measurable character is continuous, this shows that $$u\in\widehat{\widehat{\mathbb G}} \cong \mathbb G$$. That is, there exists $$a\in \mathbb G$$ such that $$u(\lambda)=\lambda(a)$$, and hence $\hat f(\lambda) = \lambda(a)\hat g(\lambda) \iff f = \Lambda(a)g.$

# Generalized Fourier transform

Let $$\mathbb G$$ is a locally compact unimodular group with Haar measure $$d\mu$$ not necessarily abelian.

The dual $$\widehat{\mathbb G}$$ of $$\mathbb G$$ is the set of equivalence classes of unitary irreducible representations of $$\mathbb G$$.

The (generalized) Fourier transform of $$f\in L^1(\mathbb G)\cap L^2(\mathbb G)$$ is the map that to $$R\in \widehat{\mathbb G}$$ acting on the Hilbert space $$\mathcal H_R$$ associates the Hilbert-Schmidt operator on $$\mathcal H_R$$ defined by $\hat f(R)= \int_{\mathbb G} f(a) \, R(a)^{-1}\,d\mu(a) \in HS(\mathcal H_R).$

There exists a measure $$d\hat\mu$$ (the Plancherel measure) on $$\widehat G$$ w.r.t. the Fourier transform can be extended to an isometry $$\mathcal F:L^2(\mathbb G) \to L^2(\widehat{\mathbb G})$$.

Fundamental property: For any $$a\in\mathbb G$$, $f= \Lambda(a) g \iff \hat g(R) = \hat f(R)\circ R(a)$

# Chu duality

Chu duality is an extension of the dualities of Pontryagin (for abelian groups) and Tannaka (for compact groups) to certain (non-compact) MAP groups. Here the difficulty is to find a suitable notion of bidual, carrying a group structure. See Heyer (1973).

Let $$\mathbb G$$ be a topological group

• $$\text{Rep}_n(\mathbb G)$$ is the set of all $$n$$-dimensional continuous unitary representations $$R$$ of $$\mathbb G$$ in $$\mathbb C^n$$. It is endowed with the compact-open topology.
• The Chu dual of $$\mathbb G$$ is the topological sum $\text{Rep}(\mathbb G)=\bigcup_{n\succcurlyeq 1}\text{Rep}_ n(\mathbb G).$
• $$\text{Rep}(\mathbb G)$$ is second countable if $$\mathbb G$$ is so.

# Quasi-representations

• A quasi-representation of $$\mathbb G$$ is a continuous map $$Q$$ from $$\text{Rep}(\mathbb G)$$ to $$\bigcup_{n\succcurlyeq 1}\mathcal U(\mathbb C^n)$$ such that for any $$R,R'\in \text{Rep}(\mathbb G)$$ and $$U\in\mathcal U(\mathbb C^{n(R)})$$

1. $$Q(R)\in\mathcal U(\mathbb C^{n(R)})$$
2. $$Q(R\oplus R')=Q(R)\oplus Q(R')$$
3. $$Q(R\otimes R')=Q(R)\otimes Q(R')$$
4. $$Q(U^{-1}RU) = U^{-1}Q(R)U$$
• The Chu quasi-dual of $$\mathbb G$$ is the union $$\text{Rep} (\mathbb G)^{\vee}$$ of all quasi-representations of $$\mathbb G$$ endowed with the compact-open topology.
• Setting $$E(R)=\text{Id}_{n(R)}$$ and $$Q^{-1}(R)=Q(R^{-1})$$, $$\text{Rep} (\mathbb G)^{\vee}$$ is a Hausdorff topological group with identity $$E$$.
• The mapping $$\Omega:\mathbb G\to \text{Rep} (\mathbb G)^{\vee}$$ defined by $$\Omega(g)(R) = R(g)$$ is a continuous homomorphis, injective if the group is MAP.

Definition: The group $$\mathbb G$$ has the Chu duality property if $$\Omega$$ is a topological isomorphism.

# Chu dual

A Moore group is a group whose irreducible representations are all finite-dimensional.

Theorem (Chu)

The following inclusions hold $\text{Moore groups} \subset \text{Groups with Chu duality} \subset \text{MAP groups}.$
• The group $$SE(2,N)$$ is Moore (i.e. all its irreducible representations are finite-dimensional) and then it has Chu duality.
• The group $$SE(2)$$ is not MAP (i.e. almost-periodic functions do not form a dense subspace of continuous functions) and hence it does not have Cuu duality.
This is why it is more convenient to work in the semi-discretized model

# The case of $$SE(2,N)$$

Let us consider the (non-compact, non-abelian) Moore group $$SE(2,N)=\mathbb R^2\rtimes \mathbb Z_N$$.

The unitary irreducible representations fall into two classes

• Characters: Any $$\hat n\in\mathbb Z_N$$ induces the one-dimensional representation $$K_n(x,k) = \hat n(k)$$.
• $$N$$-dimensional representations: For any $$\lambda\in\widehat{\mathbb R^2}\setminus 0$$ we have the representation that acts on $$\mathbb C^N$$ as $T^\lambda(x,k) = \text{diag }_ n \lambda(R_n x)\, S^k,$ where $$S^k$$ is the shift operator $$S^k v(n) = v(n+k)$$.

Since the Plancherel measure is supported on the $$N$$-dimensional representations, bispectral invariants are generalized to $$\varphi\in L^2(SE(2,N))$$ as the following functions of $$\lambda_1,\lambda_2\neq 0$$: $B_\varphi(\lambda_1,\lambda_2) = \hat\varphi(T^{\lambda_1}\otimes T^{\lambda_2})\circ \hat\varphi(T^{\lambda_1})^* \otimes \hat\varphi(T^{\lambda_2})^*.$

# The left regular representation

Theorem: The bispectral invariants are weakly complete w.r.t. the action of $$\Lambda$$ on $$L^2(K)\subset L^2(SE(2,N))$$, where $$K\subset SE(2,N)$$ is compact. In particular, they discriminate on the residual set $$\mathcal G\subset L^2(K)$$ of those square-integrable $$\varphi$$’s such that $$\hat \varphi(T^\lambda)$$ is invertible for $$\lambda$$ in an open-dense subset of $$\widehat{\mathbb R^2}\setminus\{0\}$$.

The proof is similar to the abelian one: Given $$f,g\in\mathcal G$$ with $$B_{f}=B_g$$:

• Let $$U(T^\lambda) = \hat f(T^\lambda)^{-1}\circ\hat g(T^\lambda)$$ for $$\lambda\neq0$$ s.t. $$\hat f(T^\lambda)$$ is invertible.
• Prove that $$U(T^\lambda)$$ can be extended to a quasi-representation
• Since $$SE(2,N)$$ is Moore, it has Chu duality and hence there exists $$(x,k)\in SE(2,N)$$ s.t. $$U(T) = T(x,k)$$ for any unitary representation $$T$$
• Finally, this implies $\hat f(T) \circ T(x,k) = \hat g(T) \iff \Lambda(x,k) f = g.$

# The quasi-regular representation

Consider now the quasi-regular representation $$\pi$$ acting on $$L^2(\mathbb R^2)$$, which corresponds to rotation and translations.

Fixed a lift $$L:L^2(\mathbb R^2)\to L^2(SE(2,N))$$ we define the bispectral invariants $$B_f$$ as $$B_{Lf}$$.

Corollary: Let $$L:L^2(\mathbb R^2)\to L^2(SE(2,N))$$ be an injective left-invariant lift. Then, for any compact $$K\subset \mathbb R^2$$ the bispectral invariants are complete for the action of $$\pi$$ on the subset $$\mathcal R$$ of $$f\in L^2(K)$$ such that $$\widehat {Lf}$$ is invertible for $$\lambda$$ in an open-dense subset of $$\widehat{\mathbb R^2}\setminus\{0\}$$.

Proof:

Let $$f,g\in L^2(K)$$ be such that $$B_f=B_g$$. Then $B_f=B_g \iff B_{Lf}=B_{Lg} \iff Lf=\Lambda(x,k)Lg \iff f=\pi(x,k)g.$

Unfortunately, the set $$\mathcal R$$ is empty for regular left-invariant lifts:

• Let $$\omega_f(\lambda)\in \mathbb C^N$$ be the vector $$\omega_f(\lambda)(k) = \hat f(R_{-k}\lambda)$$.
• Since $$Lf(x,k) = \langle f,\pi(x,k) \Psi\rangle_{L^2(\mathbb R^2)}$$ we have $\widehat{Lf}(T^\lambda) = \overline{\omega_f(\lambda)}^*\otimes \overline{\omega_\Psi(\lambda)}.$
• Thus $$\widehat{Lf}(T^\lambda)$$ has at most rank $$1$$ and hence $$\mathcal R=\varnothing$$.
Conjecture: The bispectral invariants are weakly complete w.r.t. the action of $$\pi$$ on compactly supported functions of $$L^2(\mathbb R^2)$$.

#### Remark

• There exists non left-invariant lifts for which $$\mathcal R$$ is residual, as the cyclic lift $L_cf(x,k) = f(R_k x + \text{cent } f).$
• The price to pay is that we have to quotient away the translations before the lift. This suggest the following.

# Rotational bispectral invariants

The rotational bispectral invariants for $$f\in L^2(\mathbb R^2)$$ are, for any $$\lambda_1,\lambda_2\neq 0$$ and any $$k\in\mathbb Z_N$$ the quantities $B_f(\lambda_1,\lambda_2,k) = \widehat{Lf}(T^{\lambda_1}\otimes T^{\lambda_2})\circ \widehat{Lf}(T^{\lambda_1})^* \otimes \widehat{Lf}(T^{R_k \lambda_2})^*.$ Observe that $$B_f$$ is invariant under rotations but not under translations.

A function $$f\in L^2(\mathbb R^2)$$ is weakly cyclic if $$\{S^n \omega_f(\lambda)\}_{n\in\mathbb Z _N}$$ is a basis of $$\mathbb C^N$$ for a.e. $$\lambda$$.

Modifying the arguments used in the case of the left-regular representation, we can then prove the following.

Theorem: Consider a regular left-invariant lift with $$\Psi$$ weakly cyclic and such that $$\hat\Psi(\lambda)\neq 0$$ a.e.. Then, the rotational bispectral invariants are weakly complete w.r.t. the action of rotations on $$L^2(K)$$ for any $$K\subset \mathbb R^2$$. More precisely, they discriminate on weakly cyclic functions.

# Experimental results

In Smach et al. (2008), although the theory was not complete, some tests on standard academic databases have been carried out. They yielded results superior to standard strategies.

• ZM denotes the standard Zernike moments
• $$1^{\text{st}}$$ MD are the (non-complete) power spectrum invariants $$I_f(\lambda) = \widehat{Lf}(T^\lambda)\circ \widehat{Lf}(T^\lambda)^*$$
• $$2^{\text{nd}}$$ MD are the bispectral invariants

# Texture recognition

Let $$E\subset \widehat{\mathbb R^2}$$ be countable and invariant under the action of $$\mathbb Z_N$$.

A natural model for texture discrimination are almost periodic functions on $$SE(2,N)$$ in the $$B_2$$ Besicovitch class, i.e., $f(x,k) = \sum_{\substack{\lambda\in E \\ n\in\mathbb Z_N}} a(\lambda,n) e^{i\langle R_n\lambda,x\rangle} \quad\text{s.t.}\quad \sum_{\substack{\lambda\in E \\ n\in\mathbb Z_N}} |a(\lambda,n)|^2<+\infty.$

• $$B_2$$ functions are the pull-back of $$L^2$$ functions on the Bohr compactification.
• The theory above can be adapted to these spaces of functions, and an analog of the weakly completeness of rotational bispectral invariants holds.
• In this space the bispectral invariants are not complete, and thus the (analog of) the above conjecture is false.
• As already mentioned, when $$E$$ is finite this space can be used to exacty solve the hypoelliptic diffusion.

# Remark: Setting

The considerations of this part of the talk work in the general context of a semi-direct product $$\mathbb G = \mathbb H\rtimes \mathbb K$$ where

• $$\mathbb H$$ is an abelian locally compact group
• $$\mathbb K$$ is a finite group
• the Haar measure on $$\mathbb H$$ is invariant under the action $$R:\mathbb K\to \text{Aut }\mathbb H$$,
• the natural action of $$\mathbb K$$ on the dual $$\widehat {\mathbb H}$$ has non-trivial stabilizer only w.r.t. the identity $$\hat o\in\widehat {\mathbb H}$$.

The group law on $$\mathbb G$$ is non-commutative: $(x,k)(y,h) = (x+R_k y, k+h).$

In our case: $$\mathbb H= \mathbb R^2$$, $$\mathbb K=\mathbb Z_N$$ and $$R_k$$ is the rotation of $$2\pi k/N$$.