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Digital Image Processing
Lecture 2

Tariq Mahmood Khan

Image Processing (Computer Vision) - Recap

“Inverse Photography”

Stages in Computer Vision

  • Physics: Image Formation (Light, Reflectance)
  • Physics: Cameras: Optics (Lens), Sensors (CCD, CMOS)
  • Image Processing: Coding (Transmission, Compression)
  • Image Processing: Enhancement (Noise Cleaning, Colors)
  • IP-CV: Feature Detection (Objects, Actions, Motion)
  • Computer Vision: Scene recovery (3D, Reflectance)
  • Computer Vision: Object Recognition
  • Human and Machine Vision: Visual Perception
  • Robotics: Control Action (autonomous driving)

DIP Systems

All digital image processing systems consist of some means to

(1) digitise / acquire the images,

(2) process the images (computing capability),

(3) save the images

(4) produce human readable hardcopy, and

(5) communicate the images to other systems.

Image Acquisition

  • Light is emitted by light source
  • Light is reflected from objects
  • Reflected light is sensed (captured) by eye or by camera

In general, any sensor which can produce spatially-distributed intensity values of electromagnetic radiation is suited to image capturing.

Types of Image capturing system

  • In everyday life a number of image capturing systems are used, depending on the application field. They differ in the
  • acquisition principle
  • acquisition speed
  • spatial resolution
  • sensor system

Classification of Sensors

Sensors can be categorized into the following classes according to their sensitivity ranges:

  • Electromagnetic sensors - sensitive to a certain range of electromagnetic radiation
  • gamma radiation
  • X-ray radiation
  • the visual spectrum
  • the infrared spectrum
  • the radio wave range
  • Non- Electromagnetic sensors
  • ultrasonic sensors

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Image Acquisition

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Image description

f (x,y): intensity/brightness of the image at spatial coordinates (x,y)

0< f (x,y)<∞ and determined by 2 factors:

Illumination component i(x,y): amount of source light incident

Reflectance component r(x,y): amount of light reflected by objects

f (x,y) = i(x,y) r(x,y)

where

0< i(x,y)<∞: determined by the light source

0< r(x,y)<1: determined by the characteristics of objects

In case of X-rays, we would deal with a transmissivity instead of a reflectivity

The Digital Image Formation

The digital image is a numerical computer representation of the physical image. The physical image is divided into small regions called picture elements, or pixels. The number stored in each pixel represents the brightness of the scene in the designated region.

The Digital Image

The conversion process from physical to digital image is called digitisation. At each pixel location, the brightness of the physical image is quantized and converted into an integer number, called the grey level.

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Sampling and Quantization

Sampling and quantization

Digital line scan

quantization

The Digital Image

The image displayed is stored as an array of numbers in the computer memory.

Colour images are sampled 3 times, giving 3 digital images, 1 each for a primary colour variable (RGB, CMY or HSI).

Digital Image

Each pixel has an address in the digital image, i.e. row or line number and column or sample number. Typically, the origin (x,y)=(0,0) is at the top-left corner of the image. A digital image of 640 horizontal pixels and 400 vertical pixels will have address values of x=0-639 and y=0-399.

The Digital Image

The digital image should adequately resolve all spatial and intensity details of the original continuous tone image. The Nyquist (sampling) theorem requires that the pixel size should less than half the size of the finest detail in the original image. Likewise, the gray level brightness increments should be less than half the smallest tonal variation in the original image.

The Digital Image

Undersampling occurs when the number of pixels in a digital image is too low to accurately represent the fine details present in the original image.

The Digital Image

Undersampling results in spatial aliasing. The example shows this effect as Moire patterns.

Digital Image: Spatial and Intensity Resolution

  • Spatial resolution refers to the number of pixels in the digital image. Typically, 256x256 is the minimum acceptable spatial resolution.
  • Intensity resolution refers to the number of grey levels available in the digital image.

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Number of pixels is typically 2^N for computing convenience.

Spatial Resolution

Spatial Resolution

The Digital Image - Zooming

Although a digital image may appear smooth to the human eye, when zoomed up enough the individual pixels always become visible.

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Intensity Resolution / Grey level Resolution

Intensity resolution refers to the number of grey levels available in the digital image.

  • 256 grey levels

(b) 128 grey levels

(c) 64 grey levels

(d) 32 grey levels

Intensity Resolution / Grey level Resolution

For convenient computer storage, the number of grey levels is almost always 2N, N = number of bits.

(e) 16 grey levels

(f) 8 grey levels

(g) 4 grey levels

(h) 2 grey levels

Image (h) is a binary image.

Intensity Resolution / Grey level Resolution

Typically, the minimum number of acceptable grey levels is 16.

Note the introduction of false contouring when the brightness resolution is too low.

The Digital Image

N : N2 = number of pixels, square digital image.

k: 2k = number of grey levels.

The memory requirements to store digital images is large. One typical high-resolution image requires 1 Megabyte of memory. Colour images require 3X the memory of monochrome images.

Image Interpolation

  • It is a basic tool used extensively in tasks such as zooming, shrinking, rotating, and geometric corrections.
  • Fundamentally, Interpolation is a process of using known data to estimate values at unknown locations.

Original resampling shrinking zooming

Image Interpolation

Image Interpolation

Many methods exist in literature for interpolation such as:

  • Pixel Replication / Nearest Neighbor
  • Bilinear Interpolation
  • Bicubic Interpolation

Image Interpolation: Nearest Neighbor

  • Unknown pixel is assigned a value of its nearest neighbor

Image Interpolation: Nearest Neighbor

  • Unknown pixel is assigned a value of its nearest neighbor

Image Interpolation: Bilinear Interpolation

f(x, y) = ax + by + cxy + d

coefficients that need to be estimated

  • Unknown pixel is estimated using values of four neighbors.

Known pixels

Unknown pixels

Image Interpolation: Bilinear Interpolation

f(x, y) = ax + by + cxy + d

coefficients that need to be estimated

  • Unknown pixel is estimated using values of four neighbors.

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Known pixels

Unknown pixels

Image Interpolation: Bilinear Interpolation

f(x, y) = ax + by + cxy + d

coefficients that need to be estimated

  • Unknown pixel is estimated using values of four neighbors.

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Known pixels

Unknown pixels

Nearest Neighbor pixels

Image Interpolation: Bilinear Interpolation

f(x, y) = ax + by + cxy + d

coefficients that need to be estimated

  • Unknown pixel is estimated using values of four neighbors.

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ax1 + by1 + cx1y1 + d = f(x1, y1)

ax2 + by2 + cx2y2 + d = f(x2, y2)

ax3 + by3 + cx3y3 + d = f(x3, y3)

ax4 + by4 + cx4y4 + d = f(x4, y4)

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a, b, c, d

Image Interpolation: Bilinear Interpolation

  • Unknown pixel is estimated using values of sixteen neighbors.

Image Interpolation

Basic relationships between pixels

Neighbours of a pixel – 4-neighbors

A pixel p at coordinates (x, y) has four horizontal and vertical neighbors whose coordinates are given by

(x+1,y), (x-1,y), (x,y+1), (x,y-1)

This set of pixels, called the 4-neighbors of p, is denoted by N4(p).

Each pixel is a unit distance from (x, y), and some of the neighbors of p lie outside the digital image if (x, y) is on the border of the image.

Neighbours of a pixel – 8-neighbors

The four diagonal neighbors of p have coordinates

(x+1,y+1),(x+1,y-1),(x-1,y+1),(x-1,y-1)

and are denoted by ND(p).

These points, together with the 4-neighbors, are called the 8-neighbors of p, denoted by N8(p).

As before, some of the points in ND(p) and N8(p) fall outside the image if (x, y) is on the border of the image.

Some Definitions

  • Two pixels are said to connected if they are neighbors and if their gray levels satisfy a specified criterion of similarity (say, if their gray levels are equal)
  • 4-adjacency. Two pixels p and q with values from V are 4-adjacent if q is in the set N4(p).
  • 8-adjacency. Two pixels p and q with values from V are 8-adjacent if q is in the set N8(p).
  • m-adjacency (mixed adjacency). Two pixels p and q with values from V are m-adjacent if
  • q is in N4(p), or
  • q is in ND(p) and the set N4(p)  N4(q) has no pixels whose values are from V.

Basic relationships between pixels

Arrangement of pixels: 0 1 1
0 1 0
0 0 1

4 neighbors N4(p): 1
0 1 0
0

Diagonal neighbors ND(p): 0 1 1 0 1

8 neighbors N8 (p) = ND(p) U N4(p) : 0 1 1

0 1 0

0 0 1

Basic relationships between pixels

Mixed Connectivity:

Note: Mixed connectivity can eliminate the multiple path connections that often occurs in 8-connectivity

Pixel arrangement

8-adjacent to the center pixel

m-adjacency

Basic relationships between pixels

Path

Let coordinates of pixel p: (x, y), and of pixel q: (s, t)

A path from p to q is a sequence of distinct pixels with coordinates: (x0, y0), (x1, y1), ......, (xn, yn) where

(x0, y0) = (x, y) & (xn, yn) = (s, t),

and (xi, yi) is adjacent to (xi-1, yi-1) 1 i  n

Regions

A set of pixels in an image where all component pixels are connected

Boundary of a region

A set of pixels of a region R that have one of more neighbors that are not in R

Distance Measures

Given coordinates of pixels p, q, and z: (x,y), (s,t), and (u,v)

Euclidean distance between p and q:

  • City-block distance between p and q:

  • Chessboard distance between p and q:

Image Operation on a Pixel Basis

  • when we refer to an operation like “dividing one image by another,” we mean specifically that the division is carried out between corresponding pixels in the two images

  • Other arithmetic and logic operations are similarly defined between corresponding pixels in the images involved.

Liner and Nonlinear Operations

  • Let H be an operator whose input and output are images. H is said to be a linear operator if, for any two images f and g and any two scalars a and b,

H(af + bg) = aH(f) + bH(g).

  • An operator that fails the test of above equation by definition is nonlinear.

Reading Assignment

  • Chapter 2 (2.3-2.6) of “Digital Image Processing” by Gonzalez.

Assignment

Interpolate the following image of size 4x4 to size 8x8 by using:

Nearest Neighbor Interpolation

Bilinear Interpolation

3 1 2 1

2 2 0 2

1 2 1 1

1 0 1 2

Prob # 2.11 and 2.15 of textbook

Due Date 17/09/2012

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