doc: frame operator of the pseudo-inverse

pygsp/filters/filter.py

@@ -517,7 +517,7 @@ def compute_frame(self, **kwargs):
         *analysis operator*
 
         .. math::
-            g(L) = \begin{pmatrix} g_0(L) \\ \vdots \\ g_F(L) \end{pmatrix}
+            g(L) = \begin{pmatrix} g_1(L) \\ \vdots \\ g_F(L) \end{pmatrix}
                    \in \mathbb{R}^{NF \times N}, \quad
             g_i(L) = U g_i(\Lambda) U^\top,
 
@@ -663,14 +663,21 @@ def kernel(x, *args, **kwargs):
     def inverse(self):
         r"""Return the pseudo-inverse filter bank.
 
-        The pseudo-inverse of the filter bank :math:`g` is the filter bank
-        :math:`g^+` such that
+        The pseudo-inverse of the *analysis filter bank* :math:`g` is the
+        *synthesis filter bank* :math:`g^+` such that
 
         .. math:: g(L)^+ g(L) = I,
 
-        where :math:`I` is the identity matrix, and :math:`g(L)^+ = (g(L)\top
-        g(L))^{-1} g(L)^\top` is the left pseudo-inverse of the analysis
-        operator :math:`g(L)`.
+        where :math:`I` is the identity matrix, and the *synthesis operator*
+
+        .. math:: g(L)^+ = (g(L)\top g(L))^{-1} g(L)^\top
+                         = (g_1(L)^+, \dots, g_F(L)^+)
+                           \in \mathbb{R}^{N \times NF}
+
+        is the left pseudo-inverse of the analysis operator :math:`g(L)`. Note
+        that :math:`g_i(L)^+` is the pseudo-inverse of :math:`g_i(L)`,
+        :math:`N` is the number of vertices, and :math:`F` is the number of
+        filters in the bank.
 
         The above relation holds, and the reconstruction is exact, if and only
         if :math:`g(L)` is a frame. To be a frame, the rows of :math:`g(L)`
@@ -680,17 +687,22 @@ def inverse(self):
         will be the closest to :math:`x` in the least square sense.
 
         While there exists infinitely many inverses of the analysis operator of
-        a frame, the pseudo-inverse is unique and corresponds to the canonical
-        dual of the filter kernel.
+        a frame, the pseudo-inverse is unique and corresponds to the *canonical
+        dual* of the filter kernel.
+
+        The *frame operator* of :math:`g^+` is :math:`g(L)^+ (g(L)^+)^\top =
+        (g(L)\top g(L))^{-1}`, the inverse of the frame operator of :math:`g`.
+        Similarly, its *frame bounds* are :math:`A^{-1}` and :math:`B^{-1}`,
+        where :math:`A` and :math:`B` are the frame bounds of :math:`g`.
 
-        If the frame is tight (i.e., :math:`A=B`), the canonical dual filters
-        are given by :math:`h_i = g_i / A`, where :math:`g_i` are the filters
-        composing the filter bank :math:`g`.
+        If :math:`g` is tight (i.e., :math:`A=B`), the canonical dual is given
+        by :math:`g^+ = g / A` (i.e., :math:`g^+_i = g_i / A \ \forall i`).
 
         Returns
         -------
-        inverse: Filter
-            The pseudo-inverse filter bank.
+        inverse : :class:`pygsp.filters.Filter`
+            The pseudo-inverse filter bank, which synthesizes (or reconstructs)
+            a signal from its coefficients using the canonical dual frame.
 
         See also
         --------