1D Bilateral Filter

hardPROConvolution and FilteringComputer Vision 2

Given a 1D signal, apply bilateral filtering. Unlike Gaussian filtering, the bilateral filter considers both spatial distance and intensity difference.

For each position $i$ , compute the filtered value as a weighted average over a window of radius $r$ :

$\text{output}[i] = \frac{\sum_{j} w(i,j) \cdot I[j]}{\sum_{j} w(i,j)}$

where the weight combines spatial and range kernels:

$w(i,j) = \exp\left(-\frac{(i-j)^2}{2\sigma_s^2}\right) \cdot \exp\left(-\frac{(I[i]-I[j])^2}{2\sigma_r^2}\right)$

The window ranges from $\max(0, i-r)$ to $\min(n-1, i+r)$ inclusive.

Round output to 4 decimal places.

Example

Input

signal = [10, 10, 10, 10]
sigma_s = 1.0, sigma_r = 1.0, radius = 1

Output

[10.0, 10.0, 10.0, 10.0]

Reasoning

The given signal is a constant signal with all values being 10, so the intensity difference $(I[i]-I[j])$ will always be 0.
For each position $i$ , the weight $w(i,j)$ simplifies to $\exp\left(-\frac{(i-j)^2}{2\sigma_s^2}\right)$ since the intensity difference term becomes $\exp(0) = 1$ .
The window for each position $i$ has a radius of 1, so it includes the current and neighboring positions, but since the signal is constant, the weighted average at each position will be the same as the original value, resulting in no change.
The final output is therefore the same as the input signal, which is [10.0, 10.0, 10.0, 10.0] after rounding to 4 decimal places.

Constraints

signal is a list of numeric values
sigma_s (spatial sigma) > 0
sigma_r (range sigma) > 0
radius is a positive integer
Return list of filtered values rounded to 4 decimal places

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Python 3.13

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1D Bilateral Filter

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