Gabor Transform = Projection of MOC under "Fixed Scale + Gaussian Window + Single Origin"

1. Intuitive Correspondence

- Fourier Transform: Global kernel e^{-i\omega t}, no localization.

- Gabor Transform: Fourier kernel with a Gaussian window → time-frequency localization.

- MOC: The local origin a_\alpha inherently provides “local reference”. If the window function is Gaussian and the curvature is constant (flat space), it naturally reduces to the Gabor kernel.

2. General Projection Form of MOC (retained)

\hat{f}(\xi, \tau) = \int_{\mathcal{M}} f(\mathbf{x}) \, e^{-i k_\alpha(\mathbf{x})\cdot(\mathbf{x}-\mathbf{a}_\alpha)} \, d\mathbf{x}

Allow the kernel to be multiplied by a real window (as amplitude modulation of the MOC kernel, i.e., attenuation envelope introduced by curvature).

3. Limiting Conditions for Gabor Transform

- Manifold reduces to the real line: \mathcal{M} \to \mathbb{R}

- Curvature is identically zero → globally constant wave number: k_\alpha(\mathbf{x}) \to \omega (real frequency)

- Multi-origins collapse to a translatable single origin: \mathbf{a}_\alpha \to b (translation parameter)

- Gaussian window introduced: Multiply the MOC kernel by e^{-(x-b)^2/(2\sigma^2)}, interpreted heuristically as a “local curvature-modulated envelope”.

Substitution:

\psi_{\omega,b}(x) = e^{-(x-b)^2/(2\sigma^2)} \cdot e^{-i\omega (x-b)}

(A phase factor e^{i\omega b} often appears in the standard Gabor kernel but can be absorbed into the transform.)

4. Standard Gabor Transform

\boxed{

G_f(\omega, b) = \int_{-\infty}^{\infty} f(x) \, e^{-(x-b)^2/(2\sigma^2)} \, e^{-i\omega (x-b)} \, dx

}

Summary: Unified MOC View of Three Transforms

Transform MOC Conditions Kernel Form 

Fourier Flat manifold + single origin + real constant wave number   

Wavelet Flat manifold + single origin + scale-dependent wave number + compact window   

Laplace Half-line manifold + single origin + complex wave number   

Gabor Flat manifold + single origin + Gaussian window   

Final Judgment

- As a geometrically intuitive unified framework: Complete and elegant. MOC serves as the common mother framework for Fourier, Wavelet, Laplace, and Gabor transforms.

- As rigorous mathematics: Still requires formalization (defining the curvature–wave number relation, geometric nature of origin shifting, MOC origin of window functions, etc.).