Wavelet Transform = Projection of MOC under Scale Shrinkage + Local Origin

Keep the same style as before, while marking heuristic steps.

---

1. Intuitive Correspondence

· Fourier: single origin + no scale + flat space
· Wavelet: single origin but with scale + local window + approximately flat
· MOC: multiple origins + curvature + arbitrary scale + global manifold

Therefore:
Wavelet transform ≈ projection of MOC under the limit of "single origin + variable scale + weak curvature approximation"

---

2. General Projection Form of MOC (following the previous form)

\hat{f}(\alpha,\mathbf{b})
=
\int_{\mathcal{M}} f(\mathbf{x})
\,\psi^*_{\alpha,\mathbf{b}}(\mathbf{x})\,d\mathbf{x}

where

\psi_{\alpha,\mathbf{b}}(\mathbf{x})
=
e^{-i k_\alpha(\mathbf{x})\cdot(\mathbf{x}-\mathbf{a}_\alpha(\mathbf{b}))}

· \mathbf{a}_\alpha(\mathbf{b}): local origin moving with translation parameter \mathbf{b} (heuristic)
· k_\alpha(\mathbf{x}): local wavenumber varying with scale \alpha (heuristic)
· curvature is implicit in the spatial dependence of k_\alpha(\mathbf{x})

---

3. Wavelet Limit Conditions

1. Space remains approximately flat: weak curvature, can be regarded as \mathbb{R}^d
2. Multiple origins degenerate into a single translatable origin: \mathbf{a}_\alpha(\mathbf{b})=\mathbf{b}
3. Wavenumber tied to scale: k_\alpha(\mathbf{x})=\dfrac{k_0}{\alpha}
4. Introduce window function (localization): \psi compactly supported or rapidly decaying

Substituting:

\psi_{\alpha,b}(x)
=
\frac{1}{\sqrt{|\alpha|}}
\,\psi\left(\frac{x-b}{\alpha}\right)
\,e^{-i \frac{k_0}{\alpha}(x-b)}

---

4. Obtaining the Standard Wavelet Transform

\boxed{
W_f(\alpha,b)
=
\int_{-\infty}^{\infty} f(x)\;
\frac{1}{\sqrt{|\alpha|}}\psi^*\!\left(\frac{x-b}{\alpha}\right)
e^{-i \frac{k_0}{\alpha}(x-b)}
dx
}

---