Nave Image Smoothing: Gaussian Blur Sylvain Paris MIT CSAIL - - PowerPoint PPT Presentation

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Nave Image Smoothing: Gaussian Blur Sylvain Paris MIT CSAIL - - PowerPoint PPT Presentation

Nov 30, 2023 393 likes •599 views

A Gentle Introduction A Gentle Introduction to Bilateral Filtering to Bilateral Filtering and its Applications and its Applications Nave Image Smoothing: Gaussian Blur Sylvain Paris MIT CSAIL Notation and Definitions Notation and

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SLIDE 1

A Gentle Introduction to Bilateral Filtering and its Applications A Gentle Introduction to Bilateral Filtering and its Applications

Naïve Image Smoothing: Gaussian Blur

Sylvain Paris – MIT CSAIL

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SLIDE 2

Notation and Definitions Notation and Definitions

  • Image = 2D array of pixels
  • Pixel = intensity (scalar) or color (3D vector)
  • Ip = value of image I at position: p = ( px , py )
  • F [ I ] = output of filter F applied to image I

x y

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SLIDE 3

Strategy for Smoothing Images Strategy for Smoothing Images

  • Images are not smooth because

adjacent pixels are different.

  • Smoothing = making adjacent pixels

look more similar.

  • Smoothing strategy

pixel → average of its neighbors

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SLIDE 4

Box Average Box Average

average

input square neighborhood

  • utput
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SLIDE 5

sum over all pixels q normalized box function intensity at pixel q result at pixel p

Equation of Box Average Equation of Box Average

∑

∈

− =

S

I B I BA

q q p

q p ) ( ] [

σ

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SLIDE 6

Square Box Generates Defects Square Box Generates Defects

  • Axis-aligned streaks
  • Blocky results

input

  • utput
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SLIDE 7

unrelated pixels unrelated pixels related pixels

Box Profile Box Profile

pixel position pixel weight

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SLIDE 8

Strategy to Solve these Problems Strategy to Solve these Problems

  • Use an isotropic (i.e. circular) window.
  • Use a window with a smooth falloff.

box window Gaussian window

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SLIDE 9

Gaussian Blur Gaussian Blur

average

input per-pixel multiplication

  • utput

*

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SLIDE 10

input

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SLIDE 11

box average

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SLIDE 12

Gaussian blur

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SLIDE 13

normalized Gaussian function

Equation of Gaussian Blur Equation of Gaussian Blur

( )

∑

∈

− =

S

I G I GB

q q p

q p || || ] [

σ

Same idea: weighted average of pixels.

1

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SLIDE 14

unrelated pixels unrelated pixels uncertain pixels uncertain pixels related pixels

Gaussian Profile Gaussian Profile

pixel position pixel weight

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− =

2 2

2 exp 2 1 ) ( σ π σ

σ

x x G

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SLIDE 15

size of the window

Spatial Parameter Spatial Parameter

( )

∑

∈

− =

S

I G I GB

q q p

q p || || ] [

σ

small σ large σ input limited smoothing strong smoothing

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SLIDE 16

How to set σ How to set σ

  • Depends on the application.
  • Common strategy: proportional to image size

– e.g. 2% of the image diagonal – property: independent of image resolution

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SLIDE 17

Properties of Gaussian Blur Properties of Gaussian Blur

  • Weights independent of spatial location

– linear convolution – well-known operation – efficient computation (recursive algorithm, FFT…)

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SLIDE 18

Properties of Gaussian Blur Properties of Gaussian Blur

  • Does smooth images
  • But smoothes too much:

edges are blurred.

– Only spatial distance matters – No edge term

input

  • utput

( )

∑

∈

− =

S

I G I GB

q q p

q p || || ] [

σ

space