Inverting a scalogram

Question

A scalogram is said to discard phase information with modulus; is it still possible to recover the signal to some extent?

Algorithms like Griffin-Lim can invert within an error, but what's the theoretical bound (ideal case) on accuracy? Are there better approaches?

Answer 1

Algorithms aside, a scalogram is proven to be strongly invertible - perfectly for recovering instantaneous frequency and amplitude; see "Invertibility". Besides Griffin-Lim and alike, since CWT is fully differentiable, we can use gradient-based reconstruction - and it should outperform handcrafted algorithms with proper tuning.

Hard part's ensuring every involved operation is differentiable; this is automated with PyTorch and TensorFlow, as long as using their ops. The train loop is trivial:

1. Compute $S(x)$
2. Initialize $x_\text{rec}$ randomly
3. Compute $S(x_\text{rec})$
4. Compute loss, e.g. MSE: $\sum|S(x_\text{rec}) - S(x)|^2$
5. Backpropagate, fetch gradients
6. Update $x_\text{rec}$ with gradients

PyTorch example with Kymatio:

import torch, kymatio
sc = kymatio.Scattering1D(shape=2048, J=6, Q=8, frontend='torch')
x = torch.cos(40*torch.linspace(0, 1, 2048))
Sx = sc(x)
xrec = torch.randn(len(x))
xrec.requires_grad = True
optimizer = torch.optim.SGD([xrec], lr=500, momentum=.9)
loss_fn = torch.nn.MSELoss()
for i in range(100):
    optimizer.zero_grad()
    Sxrec = sc(xrec)
    loss = loss_fn(Sxrec, Sx)
    loss.backward()
    optimizer.step()

Advanced steps can include:

• Learning rate decay
• Swapping L2 to L1 past certain loss, to emphasize small deviations
• Coefficient renormalization (overall Gaussianization and as described in VI. B)

Visualizing

Plot the recovered signal and its scalogram at each gradient iteration; since synchrosqueezed is more informative if we know the ridges, I'll use it instead:

A close approximation is attained in about 20 iterations.

Code available at Github.