Spectral Design of Signal-Adapted Tight Frames on Graphs

Abstract

Analysis of signals defined on complex topologies modeled by graphs is a topic of increasing interest. Signal decomposition plays a crucial role in the representation and processing of such information, in particular, to process graph signals based on notions of scale (e.g., coarse to fine). The graph spectrum is more irregular than for conventional domains; i.e., it is influenced by graph topology, and, therefore, assumptions about spectral representations of graph signals are not easy to make. Here, we propose a tight frame design that is adapted to the graph Laplacian spectral content of a given class of graph signals. The design exploits the ensemble energy spectral density, a notion of spectral content of the given signal set that we determine either directly using the graph Fourier transform or indirectly through approximation using a decomposition scheme. The approximation scheme has the benefit that (i) it does not require diagonalization of the Laplacian matrix, and (ii) it leads to a smooth estimate of the spectral content. A prototype system of spectral kernels each capturing an equal amount of energy is defined. The prototype design is then warped using the signal set’s ensemble energy spectral density such that the resulting subbands each capture an equal amount of ensemble energy. This approach accounts at the same time for graph topology and signal features, and it provides a meaningful interpretation of subbands in terms of coarse-to-fine representations.

Appendices

Appendix 1 - Proof of Proposition 1

The sum of squared magnitudes of B-spline based spectral kernels $\{B_{j}(\lambda )\}_{j=1}^{J}$ forms a partition of unity since

$$\begin{aligned} \sum _{j=1}^{J} |B_{j}(\lambda )|^{2}&\,\, {\mathop {=}\limits ^{(38)}} \sum _{i=\varDelta }^{J+\varDelta +1} |\widetilde{B}_{i}(\lambda )|^{2} \\&\,\, {\mathop {=}\limits ^{(39)}} \sum _{i=\varDelta }^{J+\varDelta +1} \beta ^{(n)}\left( \frac{\lambda _{\text {max}}}{J-1} (\lambda - i+1) \right) \\&{\mathop {=}\limits ^{i-1 \rightarrow k}} \sum _{k=\varDelta -1}^{J+\varDelta } \beta ^{(n)}\left( \frac{\lambda _{\text {max}}}{J-1} (\lambda - k) \right) \\&\quad = 1 . \end{aligned}$$

where in the last equality we use the property that integer shifted splines form a partition of unity.

Appendix 2 - Proof of Proposition 2

In order to ensure that the spectral kernels cover the full spectrum, a must be chosen such that

$$\begin{aligned} \lambda _{\text {max}} {\mathop {=}\limits ^{(54c)}} \lambda _{\text {II}}+a {\mathop {=}\limits ^{(j=J)}} \gamma a+(J-2)\varDelta + a, \end{aligned}$$

which using (55a) leads to $a = \frac{\lambda _{\text {max}}}{J\gamma -J -\gamma +3}.$

To prove that the UMT system of spectral kernels form a tight frame, (21) needs to be fulfilled. Since, for all j, the supports of $U_{j-1}(\lambda )$ and $U_{j+1}(\lambda )$ are disjoint, $G(\lambda )$ can be determined as

$$\begin{aligned} G(\lambda )&= \sum _{j=1}^{J} |U_{j}(\lambda )|^{2} \nonumber \\&\, \, {\mathop {=}\limits ^{(54)}} {\left\{ \begin{array}{ll} |U_{1}(\lambda )|^{2} \, {\mathop {=}\limits ^{(54a)}} 1 &{} \forall \lambda \in [0,a]\\ |U_{1}(\lambda )|^{2} + |U_{2}(\lambda )|^{2} &{} \forall \lambda \in ]a,\gamma a]\\ |U_{2}(\lambda )|^{2} + |U_{3}(\lambda )|^{2} &{} \forall \lambda \in ]\gamma a,\gamma a + \varDelta ]\\ \vdots &{} \vdots \\ |U_{J}(\lambda )|^{2} \, {\mathop {=}\limits ^{(54c)}} 1 &{} \forall \lambda \in ]\lambda _{\text {max}}-a,\lambda _{\text {max}}] \end{array}\right. }\nonumber \\&{\mathop {=}\limits ^{(54b)}} {\left\{ \begin{array}{ll} 1 &{} \forall \lambda \in [0,a]\\ \cos ^{2}(x_{\text {I}}) + \sin ^{2}(x_{\text {I}}) &{} \forall \lambda \in ]a,\gamma a]\\ \cos ^{2}(x_{\text {II}}) + \sin ^{2}(x_{\text {II}}) &{} \forall \lambda \in ]\gamma a,\gamma a + \varDelta ]\\ \vdots &{} \vdots \\ 1 &{} \forall \lambda \in ]\lambda _{\text {max}}-a,\lambda _{\text {max}}] \end{array}\right. } \nonumber \\&\, \, = 1 \quad \forall \lambda \in [0,\lambda _{\text {max}}] \end{aligned}$$

(64)

where $x_{\text {I}} = \frac{\pi }{2} \nu (\frac{1}{\gamma -1}(\frac{\lambda }{a}-1))$ and $x_{\text {II}} = \frac{\pi }{2} \nu (\frac{1}{\gamma -1}(\frac{\lambda - \varDelta }{a}-1))$.

For any given $\gamma$, the constructed set of spectral kernels form a tight frame. However, in order for the frame to satisfy the uniformity constraint given in (51), the appropriate $\gamma$ needs to be determined. From (54b), we have $\forall j \in \{2, \ldots ,J-2\}$

$$\begin{aligned} U_{j}(\lambda )&= U_{j+1}(\lambda + \varDelta ) \quad \, \forall \lambda \in ]\lambda _{\text {I}},\lambda _{\text {II}} + \varDelta ]. \end{aligned}$$

(65)

By considering an inverse linear mapping of the spectral support where $U_{1}(\lambda ) \ne 0$, i.e. $[0, \gamma a]$, to the spectral support where $U_{J}(\lambda ) \ne 0$, i.e. $[\lambda _{\text {max}}-\gamma a,\lambda _{\text {max}}]$, we have

$$\begin{aligned} U_{1}(\lambda ) = U_{J}(- \lambda + 2a + J \varDelta ) \quad \, \forall \lambda \in [0,\gamma a]. \end{aligned}$$

(66)

Thus, from (65) and (66) we have

$$\begin{aligned} \int _{0}^{\lambda _{\text {max}}} {U_{j}}(\lambda ) d\lambda&= C_2, \quad j=2,\ldots ,J-1 \end{aligned}$$

(67a)

$$\begin{aligned} \int _{0}^{\lambda _{\text {max}}} U_{1}(\lambda ) d\lambda&= \int _{0}^{\lambda _{\text {max}}} {U_{J}}(\lambda ) d\lambda = C_1, \end{aligned}$$

(67b)

respectively, where $C_1,C_2 \in \mathbb {R}^{+}$. Thus, in order to satisfy (51), $\gamma$ should be chosen such that

$$\begin{aligned} C_1&= C_2 \nonumber \\ \int _{0}^{\lambda _{\text {max}}} U_{1}(\lambda ) d\lambda&= \int _{0}^{\lambda _{\text {max}}} {U_{2}}(\lambda ) d\lambda \nonumber \\ a + \int _{a}^{\gamma a} U_{1}(\lambda ) d\lambda&= \int _{a}^{\gamma a} \sin (\frac{\pi }{2} \nu (\frac{1}{\gamma -1} (\frac{\lambda }{a} -1))) d\lambda \nonumber \\&+ \int _{\gamma a}^{\gamma a+\varDelta } {U_{2}}(\lambda ) d\lambda \nonumber \\ a&{\mathop {=}\limits ^{(65)}} \int _{a}^{\gamma a} \sin (\frac{\pi }{2} \nu (\frac{1}{\gamma -1} (\frac{\lambda }{a} -1))) d\lambda . \end{aligned}$$

(68)

The optimal $\gamma$ that satisfies (68) was obtained numerically by defining

$$\begin{aligned} Q(\gamma ) = \int _{a}^{\gamma a} \sin (\frac{\pi }{2} \nu (\frac{1}{\gamma -1} (\frac{\lambda }{a} -1))) d\lambda - a, \end{aligned}$$

(69)

and discretizing $Q(\gamma )$ within the range $(a,\gamma a]$, with a sampling factor of $1 \times 10^{-4}$. Testing for $\gamma \ge 1$, with a step size of $1 \times 10^{-2}$, the optimal value, which is independent of $\lambda _{\text {max}}$ and J, was found to be $\gamma = 2.73$.