1. Lecture Motivation Digital images Syllabus Date Title Link - - PowerPoint PPT Presentation

• 1. Lecture

Motivation Digital images

Syllabus

2 Date Title Link 23.02. Introduction, Properties of digital images [pdf] 01.03. Fourier transformation [pdf] 08.03. Fourier transformation/Sampling [pdf] 15.03. Image enhancement: Filtering [pdf] 22.03. Image enhancement: Filtering [pdf] 29.03. Image enhancement: Geometric transformations [pdf] 05.04. Image restoration: Spatial domain [pdf] 19.04. Image restoration: Frequency domain [pdf] 26.04. Color/Demosaicing [pdf] 03.05. Image compression/Texture segmentation (Manos Baltsavias) [pdf] 10.05. Feature extraction (Manos Baltsavias) [pdf] 24.05. Image segmentation (Manos Baltsavias) [pdf] 31.05. Image matching (Manos Baltsavias) [pdf]

Motivation

• Image data might suffer from distortions
• Transmission errors, compression errors,

sensor defects, motion blur …

• It is possible to remove some of these

distortions

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Transmission interference

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Compression artefacts

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Spilling

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Scratches, Sensor noise

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Bad contrast

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Removing motion blur

Original image Cropped part After motion blur removal [Images courtesy of Amit Agrawal]

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Removing motion blur

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Super resolution

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Super resolution

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Seeing through obscure glass

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[Shan et al.,2010]

Seeing through obscure glass

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[Shan et al.,2010]

Haze removal

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• riginal

haze removed [He et al. 2009]

Clear Underwater Vision

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[Schechner et al. 2004]

A 2D image

x y (x,y) f(x,y) (0,0)

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Concepts

• Continuous function: continuous codomain –

continuous domain

• Discrete function: continuous codomain – discrete

domain

• Digital function: discrete codomain – discrete

domain

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Image as 2D function

• Image: continuous function

2D domain: xy - coordinates 3D domain: xy + time (video)

• Brightness is usually the value of the function
• But can be other physical values too:

temperature, pressure, depth …

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Example for images

ultrasound temperature camera image CT

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Digitizing an image

• Approximating the continuous function by a

digital function

• Sampling: continuous domain will be

discretized

• Quantization: continuous co-domain will be

discretized

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Sampling 1D

Sampling in 1D takes a function, and returns a vector whose elements are values of that function at the sample points. We allow the vector to be of infinite length, and have negative as well as positive indices.

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Sampling 2D

Sampling in 2D takes a function and returns an array; we allow the array to be of infinite size and to have negative as well as positive indices.

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Sampling grids

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Retina-like sensors

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Quantization

• Real valued function will get digital values – integer

values

• Quantization is lossy!! Information is lost in this step
• After quantization the original signal cannot be

reconstructed anymore

• This is in contrast to sampling, as a sampled but not

quantized signal can be reconstructed.

• Simple quantization uses equally spaced levels with k

intervals

b

k 2 

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Quantization

00 01 10 11

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Usual quantization intervals

• Grayvalue image

8 bit = 2^8 = 256 grayvalues

• Color image RGB (3 channels)

8 bit/channel = 2^24 = 16.7Mio colors

• 12bit or 16bit from some sensors

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Properties

• Image resolution
• Geometric resolution: How many pixel per

area

• Radiometric resolution: How many bits per

pixel

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Image resolution

1024x1024 512x512 512x1024

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Geometric resolution

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Radiometric resolution

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Basic relationships between pixels

• Neighbourhood
• Connectivity
• Metric
• Distances

binary image

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Neighbourhood

4-Neighbourhood: Pixel p at position (x,y) has 4 neighbours S: (x+1,y), (x-1,y), (x,y+1), (x,y-1) The set S=N4(p) is called the 4-neighbourhood Diagonal Neighbourhood: The 4 diagonal neighbours ND(p) S: (x+1,y+1), x(-1,y+1),(x+1,y-1), (x-1,y-1) 8-Neighbourhood: Union of N4 and ND N8(p) = N4(p)+ND(p)

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Connectivity

• Connectivity allows to define

regions in an image or boundaries.

• Two pixels p,q are connected if

they are neighbours in one of the neighbourhoods, especially N4(p) and N8(p)

• We speak of 4-connectivity or 8-connectivity

The pixels p,q are not connected under 4- connectivity but under 8-connectivity

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Paradoxon of the 4-Connectivity

The black pixels are not 4-connected. However, they perfectly divide the two sets of white pixels (which are also not 4-connected) Using the 8-connectivity solves this problem

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Paradoxon of the 8-Connectivity

The most logical solution is (e): Foreground 8-neighbourhood + Background 4-neighbourhood

(Jordan theorem)

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Distance measures

Different distance metrics can be defined in an image:

– Euclidean distance – D4 distance (city-block) – D8 distance (chess-board)

Properties of a distance function or metric D:

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Euclidean distance

• The Euclidean distance between pixels p and q

is defined as: D = 5

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D4 distance

• The D4 or city-block-distance between pixels p

and q is defined as: D = 7

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D8 distance

• The D8 or chess-board-distance between

pixels p and q is defined as: D = 4

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Circle with radius T

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