An input signal x is filtered by a collection of dilated band-pass wavelets obtained from ψ, followed by a modulus and finally averaged by a dilation of φ. The wavelets we chose decompose the signal in a basis in which transient structure of a signal is represented more compactly.
In the context of deriving rotationally invariant representations, the Fourier Transform is particularly appealing since it exhibits invariance to rotational deformations up to phase (a truly invariant representa-tion can be achieved through application of the modulus operator).
It is implemented with a deep convolution net-work, which computes successive wavelet transforms and modulus non-linearities. Invariants to scaling, shearing and small deformations are calculated with linear operators in the scattering domain.
ScatNet [5] cascades wavelet transfor- m with nonlinear modulus and average pooling, to extract a translation invariant feature robust to deformations and preserve high-frequency information for image classifica-tion. The authors introduce ScatNet when they explore from mathematical and algorithmic perspective how to design the optimal deep ...
As a steerability consistent way of normalizing circular harmonics, we propose to ade-quately normalize their complex modulus. The proper scale follows from k k2 = kRe [ ]k2
For instance, while an individual CNT has an elastic modulus of around 1 TPa, a CNT forest’s compressive elastic modulus are frequently on the order of 1-10 MPa [26], akin to nat-ural rubber.
, Ktx], (7) Here, we note that the text embedding before augmentation [vj mod M, cy] is a concatenation of the prompt embedding vj mod M and the class embedding cy. Recall that M is the number of prompt embeddings, and the modulus operation j mod M is to ensure that the prompt embedding is selected cyclically, if Ktx exceeds M.
Unlike typical CNNs: 1) regarding the architecture, convolutional layers are defined by fixed wavelet filters, with modulus-based nonlinearity, but without subsequent pooling; 2) regarding the representation properties, the ar- chitecture yields the translation equivariance and certain robustness w.r.t. non-linear deformations.
The simple geometry of a clamped rod makes it easy to solve for vibration modes analytically as a function of length, diameter, density, and an elastic modulus.
Applying a modulus removes this variability. As a classical result of signal theory, ob-serve also that an averaged rectified signal is approximately equal to the average of its complex envelope [18].
