diff --git a/project.md b/project.md index ccd904f4dc98d20e9c74e5fd2e0195fd90926631..fdb1a63db0e698cf7f4d69f496439038f5d8f7f8 100644 --- a/project.md +++ b/project.md @@ -21,18 +21,18 @@ $$ Meanwhile, the Fourier transform of $$f$$ is $$ -\hat{f}(\hat{x}, \hat{y}) -= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i (\hat{x} x + \hat{y} y)} dx dy +\hat{f}(u, v) += \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i (u x + v y)} dx dy $$ -Note that the slice along the $$\hat{x}$$ axis in frequency space is described by +Note that the slice along the $$u$$ axis in frequency space is described by $$ \begin{align*} -\hat{f}(\hat{x}, 0) -&= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i \hat{x} x} dx dy \\ -&= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i \hat{x} x} dx \\ -&= \int_\mathbb{R} p(x) e^{-2 \pi i \hat{x} x} dx \\ -&= \hat{p}(\hat{x}) +\hat{f}(u, 0) +&= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i u x} dx dy \\ +&= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i u x} dx \\ +&= \int_\mathbb{R} p(x) e^{-2 \pi i u x} dx \\ +&= \hat{p}(u) \end{align*} $$