Digital Image Fundamentals, Lecture notes of Digital Image Processing

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Week 1:

DIGITAL IMAGE FUNDAMENTALS

 Image formation model

 Image sampling and quantization

 Spatial and intensity resolution

 Image interpolation

 Some basic relationships between pixels

 Geomteric spatial transformations and image registration

OUTLINE

 Images are denoted by 2 - D functions of the form 𝑓(𝑥, 𝑦)

 𝑓(𝑥, 𝑦) is determined by the amount of source illumination incident on the

scene and the amount of illumination reflected by the objects in the scene.

 These are called the illumination 𝒊 𝒙, 𝒚 , and reflectance 𝒓(𝒙, 𝒚)

 Typical values of illumination and reflectance components:

Illumination: 90000 lm/m

2

(on a clear day), 10000 lm/m

2

(on a cloudy day)

Reflectance: 0. 01 (black velvet), 0. 65 (stell), 0. 93 (snow)

Image Formation Model

 The goal in image sampling and quantization is to convert the

continuous-time light energy at the output of the imaging sensor into

a digital image.

 To digitize the continous-time function f(x,y), both its coordinate

values and its amplitudes must be digitzed.

 Digitizing the coordinate values is called SAMPLING.

 Digitizing the amplitude values is called QUANTIZATON.

Image Sampling and Quantization

Image Sampling and Quantization

 Assume that the continuous image 𝑓 𝑥, 𝑦 is converted into digital image 𝑓(𝑚, 𝑛)

consisting of M columns and N rows after the sampling and quantization.

 We use integer values for coordinates ( m = 0 , 1 ⋯ 𝑀 − 1 , n = 0 , 1 , ⋯ , 𝑁 − 1 ).

 The value of the image at any coordinate 𝑚, 𝑛 is denoted by 𝑓(𝑚, 𝑛).

 Coordinates of the image are called the spatial coordinates, and the section of the

real plane spanned by the coordinates of an image is called the spatial domain.

 A digital image can be represented by one of three basic ways:

Plotted as a 2 - D surface

Displayed as a visual intensity array

Shown as a 2 - D numerical array.

Representing Digital Images

 When digitizing a digital image, we should decide on the dimensions ( M , N ) and

the number of intensity levels ( L ). Other than being positive, there are no

constraints on M and N. Since the processors perform algebraic expressions in base

2 , L is chosen such that it is an integer multiple of 2

L = 2

k

Intensity values are assumed to be an integer in the interval [ 0 , L- 1 ].

 The ratio of the maximum measurable intensity to the minumum discrenbile

intensity is called the dynamic range.

 The difference between the maximum and minimum intensity levels is called the

contrast.

 The number of bits b required to represent a digital image is computed from

b= M x N x k

If M = N , then b = N

2

k

Representing Digital Images

Representing Digital Images

The spatial resolutions are: 1250, 300, 150, 7 (dpi).

Spatial and Intensity Resolutions

Intensity resolutions are L = 256, 128, 64, 32.

Spatial and Intensity Resolutions

 Interpolation is the process of using known data to estimate values at unknown

locations.

 Suppose that an image of size 500 x 500 pixels has to be enlarged 1. 5 times to 750

x 750 pixels.

 An imaginary 750 x 750 grid with the same pixel spacing as the original is created.

Then, it is shrinked so that it fits excatly over the original image.

 The pixel spacing in the shrunken 750 x 750 grid will be less than the pixel spacing

in the original grid. Hence, the intensity values at newly created pixel locations

must be determined. For this purpose, there are several methods such as

Nearest-neighbor interpolation

Bi-linear interpolation

Bi-cubic interpolation

Image Interpolation

Image Interpolation

Example: Factor of 2 upsampling

Blue samples are retained in the interpolated image.

Orange samples are estimated from the surrounding blue samples.

image to be interpolated interpolated image

Example: Factor of 2 interpolation by NNI

𝐼[𝑚, 𝑛] 𝑂[𝑚, 𝑛]

Image Interpolation (Nearest Neighbor Interpolation)

𝑂[𝑚

′

′

] takes a weighted average of 4 samples

nearest to (

′

′

𝑀) in 𝐼 𝑚, 𝑛

𝑎: distance between [𝑚

′

′

] and column n

𝑏: distance between [𝑚

′

′

] and row m

Image Interpolation (Bi-linear Interpolation)

𝑂 𝑚

′

, 𝑛

′

= 1 − 𝑎 1 − 𝑏 𝐼 𝑚, 𝑛 + 𝑎 1 − 𝑏 𝐼 𝑚, 𝑛 + 1 + 1 − 𝑎 𝑏𝐼 𝑚 + 1 , 𝑛 + 𝑎𝑏𝐼 𝑚 + 1 , 𝑛 + 1