What is the importance of the translational invariance of the CWT?

Question

Translational invariance is a property that the continuous wavelet transform (CWT) has but the discrete wavelet transform (DWT) does not have. It says that a shift of the signal, i.e. $x(t)\rightarrow x(t-d)$, leads to a shift of the wavelet coefficients, i.e. $W_x(a,b)\rightarrow W_x(a,b-d)$ without other modification. What is the importance of the translational invariance of the CWT?

Answer 1

First, translational invariance makes signal processing somehow independent on the time origin of the recording. The term "invariance" might be debated, as done here: What is the difference between "equivariant to translation" and "invariant to translation". Related concpets are called rotational invariance, scale invariance, etc.

To explain the time origin stuff: imagine that you record the same signal in different time zones, this allows to yield the "same result", eg same coefficients (translated), or same overall processing result. Sometimes, one only need invariance on some coefficient-based feature (like magnitude), sometimes only approximately so.

You also get shift invariance with undecimated discrete wavelets, cycle-spinning, etc. Since this is a bit oversampled, this was a motivation behind the development of dual-tree or complex discrete wavelets. You can check references in Section 2.3.4 "Translation Invariant Wavelets" from our paper A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity.

For instance if you perform a one-level "Haar" tranform on signals $$[\ldots \, 0 \,2\, 3\, 0\, -1\, 0\, 0\, \ldots]$$ and on $$[\ldots \, 0 \, 0 \,2\, 3\, 0\, -1\, 0\, \ldots]$$ you will get of the detail coefficients (applying the difference $[1 \,-1])$ on every two samples:

$$[\ldots \, -2 \,3\, -1\, 0\, \ldots]$$ and

$$[\ldots \, 0 \,-1\, 1\, 0\, \ldots]$$ which are not shifts of one another. This could lead to different results, for instance if you choose a threshold on 1.

Answer 2

CWT is translation-invariant in feature sense: translating a pattern translates its representation but not modify it. In coefficient sense, it is translation equivariant: shift signal $\Leftrightarrow$ shift representation. CWT is not invariant in coefficient sense. I will stick to latter definition.

Translation equivariance is a basic and powerful prior. It is required for time-frequency localization: instantaneous frequency, amplitude, and phase (if a sample moves and representation doesn't, we've not 'located' it). The Fourier modulus is invariant, but phase isn't, nor is phase equivariant.

Global vs local shifts

Translations(/shifts) can be local and global. Equivariance to

• global shifts, as in changing start/end of a tape recording or trimming an image, is desired to keep features same except at boundaries. The Fourier modulus is invariant to global shifts, but its phase isn't, nor is it equivariant.
• local shifts, as in speaking words in different order or moving an apple in an image, is desired to keep features same but preserving relative locality. These change the Fourier representation entirely, despite only parts of input changing and class labels identical in several regards. Such change is desired neither for classification nor descriptor extraction (e.g. edge detection, measuring transients).

CWT is equivariant to both (to local shifts within time resolution). DWT is equivariant to neither: a shift can change every coefficient without equals to unshifted.

Lastly: among the three, only CWT can be extended to build invariants. This is done with e.g. wavelet scattering. Equivariance is also a core element of stability against warp deformations. Explanation and advantages described in this post.

Power spectrum

Can be discussed in three perspectives:

• along time, sum(abs(Wx)**2, axis=0): determined by LP summation - see "Energy analysis". In the ideal case (tight frame), it's equal to the signal's.
• along frequency, sum(abs(Wx)**2, axis=1): determined by wavelets in the frequency domain, and discretization scheme (number of wavelets per octave, etc). One can define a "transfer function curve" by summarizing energy transfer as a single number, for each wavelet, via Parseval's theorem, similar to LP summation.
• total, sum(abs(Wx)**2): integrate LP summation

Translation equivariance means if we shift signal by $c$, we get same power by shifting temporal integration bounds by $c$:

$$ \int_{t_0}^{t_1} |\text{CWT}_{x(t)}(\lambda, t)|^2 dt = \int_{t_0 + c}^{t_1 + c} |\text{CWT}_{x(t - c)}(\lambda, t)|^2 dt $$

Shifting in time has no effect on power along frequency, so we remain invariant.

"Invariant" vs "equivariant"

Both are true but refer to different things. Formally, "equivariance" here means a coordinate transformation of the input identically transforms the same coordinate in the output:

$$ f(x, \tau(y), z) \Leftrightarrow \Phi(x, \tau(y), z) $$

which for CWT is

$$ f(t - c) \Leftrightarrow W_f (\lambda, t - c) $$

More generally equivariance is commutativity under action of a symmetry group.

"Invariance" refers to identity under group action - that is,

$$ F(\Phi(x, y, z)) = F(\Phi(x, \tau(y), z)) $$

where $F$ is a mapping, and $\tau$ a group action. For example, a hollow sphere, shifted, keeps all its descriptors identical relative to own center of mass, such as distance to each point. A similar mapping can be made for CWT.