Smooth-Approximation-with-Gaussian-Convolution
所属分类：数学计算
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上传日期：2023-09-30 19:43:23
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sh-1993
说明： 使用高斯核卷积的分段常数函数的光滑逼近，
(Smooth approximation of piecewise-constant functions using Gaussian kernels convolution,)
文件列表:
Smooth Approximation.ipynb (409184, 2023-09-30)
images/ (0, 2023-09-30)
images/analytical.png (30889, 2023-09-30)
images/analytical_scale.png (90295, 2023-09-30)
images/anvnum.png (73753, 2023-09-30)
images/anvnumerr.png (45392, 2023-09-30)
images/function.png (17090, 2023-09-30)
# Smooth Approximation with Gaussian Convolution
*Original project done as part of MATH-GA - 2048. All rights reserved.*
## Introduction
This repository focuses on computing smooth approximations of piecewise constant functions using Gaussian convolution. The technique finds applications in image blurring, function interpolation, and data smoothing.
## Mathematical Background
Given a function $f$, we aim to find its smooth approximation through convolution with a Gaussian function, $\varphi_s(t)$, defined as:
$$
\varphi_s(t) = \frac{1}{s \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{t}{s} \right)^2}
$$
where $s$ is the scale factor and $F(x)$ is the cumulative distribution function (CDF) of $\varphi_s(t)$.
For a piecewise constant function $f$ (that takes the value $f_-$ before $x_0$, the values $f_i$ on $[x_i,x_{i+1}]$ and $f_+$ after $x_N$), the Gaussian convolution $\tilde{f}(x)$ can be computed analytically as:
To express the convolution $\tilde{f}(x)$ of a function $f(x)$ with a Gaussian kernel $\varphi_s(x)$, we consider the following integral:
$$
\tilde{f}(x) = \int_{R} f(t) \varphi_s(x-t) \, dt
$$
This integral can be broken down into separate regions:
$$
\tilde{f}(x) = \int_{-\infty}^{x_0} f_- \varphi_s(x-t) \, dt + \sum_{i=0}^{N-1} \int_{x_i}^{x_{i+1}} f_i \varphi_s(x-t) \, dt + \int_{x_N}^{\infty} f_+ \varphi_s(x-t) \, dt
$$
Applying a change of variables $u = x - t$, we obtain:
$$
= f_- \int_{x-x_0}^{\infty} \varphi_s(u) \, du + \sum_{i=0}^{N-1} f_i \int_{x-x_{i+1}}^{x-x_i} \varphi_s(u) \, du + f_+ \int_{-\infty}^{x-x_N} \varphi_s(u) \, du
$$
Finally, expressing the integrals in terms of the cumulative distribution function $F(x)$ of the Gaussian distribution, we have:
$$
= f_-(1 - F(x-x_0)) + \sum_{i=0}^{N-1} f_i (F(x-x_i) - F(x-x_{i+1})) + f_+ F(x-x_N)
$$
Thus, we can write the convolution $\tilde{f}(x)$ using $F(x)$ and the constant values of $f$.
## Example
Consider the piecewise constant function $f(x)$ defined as:
$$
f(x) =
\begin{cases}
\vdots\\
4 & \text{if } -3 \leq x < -2 \\
5 & \text{if } -2 \leq x < -1 \\
6 & \text{if } -1 \leq x < 0 \\
\vdots
\end{cases}
$$
Using the formula for $\tilde{f}(x)$, we obtain its smoothed approximation as shown below (scale=1):
### Effect of Different Scales
## Performance Comparison
The analytical method significantly outperforms brute-force numerical convolution:
- 10x faster in computation
- Slighty better in terms of error
## Conclusion
The analytical method significantly outperforms brute-force numerical convolution in terms of both speed and precision.
Additionally, further analysis yielded (see code) that it demonstrates comparable computational efficiency and error rates when benchmarked against FFT-based convolution methods from SciPy.
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