the cedar ledge

# Median Filtering

Date: April 25 2020

Summary: An overview on how to use and implement a median filter

Keywords: ##zettel #signalprocessing #noise #artifact #filtering #window #julialang #median #movingaverage #downsampling #archive

# Bibliography

Not Available

It is a non-linear digital filtering technique, used to remove noise from an image or signal. The main idea of the median filter is to run through the signal value by value, replacing each value with the median of neighboring values. The pattern of neighbors is called the "window", which slides, entry by entry, over the entire signal.

### Algorithm

An explanation of implementing the algorithm is given in example form:

Given the original signal, $x = (2, 3, 80, 6)$, a 1D Median Filter is applied as follows:

$y1 = med(2, 3, 80) = 3$ $y2 = med(3, 80, 6) = med(3, 6, 80) = 6$ $y3 = med(80, 6, 2) = med(2, 6, 80) = 6$ $y4 = med(6, 2, 3) = med(2, 3, 6) = 3$

Which yields the final filtered signal, $y = (3, 6, 6, 3)$.

In the example implementation, there is no value preceding the first value, thus the window "wraps" around the original to fulfill its window size.

There are other ways to handle filling the window other than wrapping such as:

• Avoiding selecting values directly located at either boundary of the signal.

• Use other values from within the signal.

• Shrink the window near the edges of the signal so that the window is always full.

Furthermore, it is very good for salt-and-pepper noise/impulse noise (e.g. noise that is caused by sharp and sudden disturbances in the image signal)

### Example Implementation

using Plots # IMPORT FOR PLOTTING
using Statistics # IMPORT FOR median FUNCTION
using LaTeXStrings # IMPORT TO ENABLE LaTeX FORMATTING
gr()

let

# CHOOSE WINDOW AND INPUT VALUES OVER WHICH TO CALCULATE
input = 0:0.001:1
window = 30
sampling_rate = 15

# GENERATE GENERIC SIGNAL - IN THIS CASE sin(2Î )
signal = [sin(2 * pi * i) for i in input]

# ADDING RANDOM NOISE TO FUNCTION
noisy_signal = [sin(2 * pi * i) + rand([-1, 1]) * round(rand(), digits = 2)
for i in input]

# FILTER THE SIGNAL USING A MEDIAN FILTER & DOWNFILTERING USING A MEDIAN FILTER
downsampled_signal::Array{Float32} = [noisy_signal[1]]
downsampled_input::Array{Float16} = [0]
median_signal::Array{Float32} = []
for i in 1:length(signal)
if length(noisy_signal) - (window + i - 1) < 0
forward = noisy_signal[i:end]
wrap = noisy_signal[1:abs(length(noisy_signal) - (window + i - 1))]
append!(median_signal, median(vcat(forward, wrap)))

# SETTING SAMPLING RATE AND CREATING DOWNSAMPLED OUTPUT
if i % sampling_rate == 0
append!(downsampled_input, input[i])
append!(downsampled_signal, median(vcat(forward, wrap)))
end
else
forward = noisy_signal[i:(window + i - 1)]
append!(median_signal, median(forward))

# SETTING SAMPLING RATE AND CREATING DOWNSAMPLED OUTPUT
if i % sampling_rate == 0
append!(downsampled_input, input[i])
append!(downsampled_signal, median(forward))
end
end
end

# PLOT SIGNALS
append!(downsampled_input, input[end])
append!(downsampled_signal, noisy_signal[end])
plot(input, noisy_signal, label = "Noisy Signal", title = "Example of Median Filter")
plot!(input, median_signal, label = "Median Filtered Signal", linewidth = 3)
plot!(input, signal, label = L"2\pi", linewidth = 5)
plot!(downsampled_input, downsampled_signal, linewidth = 3, color = :black,
label = "Downsampled Median Signal")

end

## How To Cite

Zelko, Jacob. Median Filtering. https://jacobzelko.com/04252020024813-median-filtering. April 25 2020.

## Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: May 19, 2024. Website built with Franklin.jl and the Julia programming language.