Digital Image Fundamentals - Sharifee.sharif.edu/~dip/Files/DigitalImageFundamentalsForView.pdf•Digital Image Fundamentals: ... •Mathematical Representation: bits to store the

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E. Fatemizadeh, Sharif University of Technology, 2011

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Digital Image Processing

Digital Image Fundamental

• Digital Image Fundamentals:

– Elements of Visual Perception

– Light and the Electromagnetic Spectrum

– Image Sensing and Acquisition

– Image Sampling and Quantization

– Some Basic Relationships between Pixels

– An Introduction to the Mathematical Tools Used in Digital Image Processing

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• Eye Physiology:

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• Optical Sensors in retina: – Cones: Highly Sensitive to Color (6-7)×106

– Rods: Highly Sensitive to Low Levels of Illumination (75-150) ×106

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• Image Formation in the Eye:

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• Brightness Adaptation:

– Eyes can adapts a large dynamic ranges of intensity (1010) But NOT Simultaneously.

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• Brightness Discrimination:

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• Weber Ratio:

CΔI =Increment of illumination discriminable 50% times

Rods

Cons

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• Match Band Effect:

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• Simultaneous Contrast:

Appear Darker

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• Eye illusions:

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• 2.2: Light and the Electromagnetic Spectrum

• 2.3: Image Sensing and Acquisition

• Ignored!

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• A Simple Image Formation:

• Gray Level :Intensity of monochrome images.

, , ,

0 , : Illumination

0 , 1: Reflection

f x y i x y r x y

i x y

r x y

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• Image Sampling and Quantization (1):

Scan Line

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• Image Sampling and Quantization (2):

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• Image Sampling and Quantization:

– Spatial and Gray Level Resolution

– How to determine the sampling rate?

– Nyquist sampling theorem • If input is a band-limited signal with maximum frequency ΩN

• The input can be uniquely determined if sampling rate ΩS > 2ΩN

– Nyquist frequency : ΩN

– Nyquist rate : ΩS

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• Digital Image Representation:

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• Digital Image, Mathematical Definition:

– I = f(x,y)

– I: intensity (or color)

– (x,y): Position or Coordination

– When (x,y) and I are finite and discrete quantities -→ digital image

– pixels, picture elements, image elements

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• Mathematical Representation:

bits to store the image = M x N x k gray level = L = 2k

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• L- level digital image of size M×N

– Means: A digital image having: • A spatial resolution M×N pixels

• A gray-level resolution of L levels (0 … L-1)

• Spatial resolution in real-world space

line width=W cm space width=W cm

Resolution = 1/2W (line/cm)

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• L = 2k gray levels, gray scales [0,…,L-1]

• The dynamic range of an image

– [min(image) max(image)]

– If the dynamic range of an image spans a significant portion of the gray scale → high contrast

– Otherwise, low dynamic range results in a washed out gray look.

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• Saturation and Noise:

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• Number of Storage bits (M=N):

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• Spatial and Intensity Resolution:

1250 dpi 300 dpi 150 dpi 75 dpi

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• Bits Reduction (More Quantization):

– 8 bits to 1 bits (Left to Right-Top to Down)

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• Three types of image (Low/Medium/High Details):

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• Sampling-Quantization Tradeoff:

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• Image Interpolation: – Nearest Neighbor (NN)

– Bilinear (BL) using 4 nearest neighbor:

– Bicubic (BC) using 16 nearest neighbor:

– …

,f x y ax by cxy d

3 3

0 0

, i j

ij

i j

f x y a x y

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• Image Interpolation (Example): – Reduced to 72 dpi

– NN, BL, BC

– Reduced to 150 dpi

– NN, BL, BC

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• Basic Relationships Between Pixels:

– 4-Neighbors

– Diagonal Neighbors

– 8-Neighbors:

4 : 1, , 1, , , 1 , , 1N p x y x y x y x y

: 1, 1 , 1, 1 , 1, 1 , 1, 1DN p x y x y x y x y

8 4: DN p N p N p

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• Basic Relationships Between Pixels:

4 8

1 1 1 1 1 1

1 1 1 1

1 1 1

1

0 0 0

0 0

0

1

0 0

1

1 1 1

DN N N

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• Adjacency:

– p and q are 4-adjacent:

– p and q are 8-adjacent:

– p and q are m-adjacent:

4q N p

8q N p

4 4 4Dq N p or q N p and N p N q

8 m

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• Distance Measure:

• Examples:

– Euclidean:

– D4 (City Block or Manhattan):

– D8 (Chessboard):

a. D p,q 0 D p,q 0 iff

b. D p,q D q,p

b. D p,q D p,r D r,q

p q

2 2

,eD p q x s y t

4 ,D p q x s y t

8 , ,D p q Max x s y t

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• Constant Distance Contour:

– D4 (Left)

– D8 (Right)

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Mathematical Tools:

– Array and Matrix Operations

– Linear and nonlinear Operation • Fourier Filtering, Ordered Statistics Filtering, …

– Arithmetic Operation (+, -, *, /) • Averaging, Subtraction, …

– Set and Logical Operations • Fuzzy or Crisp Sets

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• Image Averaging: – Consider an additive noise condition:

g(x,y)=f(x,y)+η(x,y)

– Conditions: • Noise, η(x,y):

– Uncorrelated

– i.i.d

– Zero Mean

• Subject, f(x,y): – Physical Stationary

– Repeatable Experiments

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• Image Averaging:

2 2

1, ,

, ,1

, , 1

N

i

ig x y x y

E g x y f x y

g x y g x yN

N

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• Example:

– 5, 10,20, 50, and 100 averaging.

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• Image Subtraction:

– Original (Left), LSB set to zero (Center), Difference (Right)

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• Digital Subtraction Radiography (DSA):

– Pre and Post Imaging

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• Image Multiplication/Division:

– Shading Correction

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• Image Multiplication/Division:

– ROI Masking

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• Set and Logical Operation:

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• Example – Original (Left), Negative (Center), Right (union with Constant)

𝐴𝑈𝐵 = 𝑚𝑎𝑥 𝐴, 𝐵

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• Logical Operation:

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Mathematical Tools:

• Spatial Operations

– Single pixel

– Neighborhood

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• Single Pixel: 𝑠 = 𝑇 𝑧

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• Neighborhood Operation:

,

1, ,

xyr c S

g x y f r cmn

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• Geometric Spatial Transform

– General Formulation:

– Example (Affine Transform)

, ,x y T u v

1 0 0 1 1

x a b c u

y d e f v

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• Geometric Transform (1):

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• Geometric Transform (2):

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• Image Rotation:

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Image Registration

• General Formulation:

• Example (Bilinear Transform)

, ,x y T u v

1 1 1 1

2 2 2 2

x a u b v c uv d

y a u b v c uv d

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• Image Registration (Example):

– Using tie points

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• Vector and Matrix Operation:

– Multispectral Image Processing

– Image Transform (Fourier and etc.)

– Probabilistic Methods

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• Multispectral Image Processing:

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• Image Transform (Fourier and etc.):

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• Fourier Filtration:

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2 2 2

14.3, 31.6, 49.2

204.3, 997.8, 22424.9

L M H

L M H

m m m

• Probabilistic Approaches:

– Low-Medium-High contrast (From Left to Right)

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• Paradigm of image processing: – Low-level processing

• Inputs and outputs are images • Primitive operations: de-noise, enhancement,

sharpening, …

– Mid-level processing • Inputs are images, outputs are attributes extracted from

images • Segmentation, classification,…

– High-level processing • “Make sense” of an ensemble of recognized objects by

machines

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• Matlab Image Processing Read/Write:

– imformats

– imfinfo, imread, imwrite

– dicominfo, dicomread, dicomwrite

– analyze75info, analyze75read (Mayo Clinic)

– interfileinfo, interfileread

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• Matlab Image Processing Display:

– image, imagesc, imshow, imtool, subimage

– colorbar, montage

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• Matlab Image Processing Type Conversion:

– double, ind2gray, im2double

– uint16, uint8, gray2ind

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Digital Image Fundamentals - Sharifee.sharif.edu/~dip/Files/DigitalImageFundamentalsForView.pdf•Digital Image Fundamentals: ... •Mathematical Representation: bits to store the