IMAGE SAMPLING IMAGE SAMPLING

& &

QUANTIZATIONQUANTIZATION

M.V. RAGHUNADHM.V. RAGHUNADHAssistant Professor, Dept. of ECEAssistant Professor, Dept. of ECE

NIT, Warangal – 506004.NIT, Warangal – 506004.

raghu@nitw.ernet.inraghu@nitw.ernet.in

Introduction to Digital Image Processing Purpose of image processingPurpose of image processing

– Processing of image data for storage, display & transmission

– Improvement of pictorial information for Human Perception

– Improvement of pictorial information for Machine Perception

Typical application areasTypical application areas

Television DIP: brightness, contrast, hue & noise adjustment

Satellite DIP: Remote sensing, Climate, Geology, Land/Sea resource

Medical DIP: Ultra Sound, MRI/PET/CT Scans, X-Rays

Robot Control: Automatic inspection, Unmanned operations-Autonomous Vehicle (Mars Rover)

Visual Communications: Video-coding/ transmission/ conferencing, Tele-conferencing/shopping

Pattern Recognition: Law Enforcement, Biometric Identification: Fingerprint/face/iris/speech/writing/signature

Acquisition and display

• Image acquisition is the first key step in digital image processing.

• Hardware imaging devices: video cameras with frame grabbers, digital cameras, slide or flat bed scanners

• Issues: lower noise, greater dynamic range, high resolution, Sensitivity, color fidelity, Speed

• Acquisition programs: save image in various

formats, Plug-in program packages like Photoshop, Fovea Pro

What is an image?

• We can think of an image as a function, f, from R2 to R:– f( x, y ) gives the intensity at position ( x, y ) – Realistically, we expect the image only to be

defined over a rectangle, with a finite range:– f : [a,b]x[c,d] [0,1]

• A color image is just three functions pasted together. We can write this as a “vector-valued” function: (fR, fG, fB)

Images as functions

ANALOG Vs DIGITAL

TWO Ways to Process Information:TWO Ways to Process Information:

Analog:

Gramophone LP Record

Magnetic Tape Record

Conventional Radio

Slide Rule

Digital:

Optical CD ROM Record

Magnetic Memory- RAM&ROM

Web Radio

Digital computer

Digital Images• A natural image is a continuous, 2-dimensional

distribution of brightness (or some other physical effect).

• Conversion of natural images into digital form involves two key processes, jointly referred to as digitisation:– Sampling– Quantisation

• Both operations involve loss of image fidelity i.e. since they are approximation processes.

Image RepresentationAnalog image Sampling Quantization Digital image

2D array of picture elements (pixels)

Digital Image Coordinate Convention

Digital Image Representation

f(x,y) = Reflectance (x,y) * Illumination (x,y)Reflectance in [0,1], illumination in [0,inf]

Real Digital

•Image Intensity•Image Spectral Distribution (‘color’)•Image Geometry (Position, Angle, Distortion, …)•Image Resolution (‘sharpness’ or ‘focus’)

Real Images: Smooth …Continuous … Variable … Complete …Unlimited …

Digital Images:Sampled …Discrete …Quantized …Fixed …Limited …

What is a digital image?

• We usually operate on digital (discrete) images:

– Sample the 2D space on a regular grid

– Quantize each sample (round to nearest integer)

• If our samples are apart, we can write this as:

• f[i ,j] = Quantize{ f(i , j ) }

• The image can now be represented as a matrix of integer values

ј

і

DIGITAL IMAGES• Electronic Snapshots of a scene

or scanned texts/ documents/photographs/manuscripts/artwork

• Digital image is sampled (mapped) grid of dots/ pixels

• Each pixel is assigned a tonal value (black, white, shades of gray or color) represented in a binary code

• Bits of each pixel are stored in a sequence in memory

• These bits are then read and interpreted by the computer to produce an analog version for display or printing.

Pixel Values: As shown in this bitonal image, each pixel is assigned a tonal value, in this example 0 for black and 1 for white.

Sampling• Sampling represents the image by intensity

measurements at regularly spaced sample intervals.

• Two important criteria:-– Sampling interval

• distance between sample points or pixels– Tessellation

• the pattern of sampling points

• The number of pixels in the image is called the resolution of the image. If the number of pixels is too small, individual pixels can be seen and other undesired effects (e.g. aliasing) may be evident.

Discretizing the image -- Tessellation and Quantization

• There are only three ways to tessellate the plane with a regular polygons:

• Squares: The most common method.

• Regular hexagons: a useful method due to less

rotational dependence.

• Equilateral triangles: not commonly used.

Sampling and QuantizationContinuous image Continuous scan line from A to B

Sampling & QuantizationDigital scan line

Real vs. Digital

• Real Image == 2D Light Intensity map: I(x,y)

• Digital Image == 2D grid of numbers: I(m,n)

I(x,y)I(x,y)

x,yx,y

I(m,n)I(m,n)

m,nm,n

(pixels)

Digital Images As Vectors

• ‘Stack up’ pixel values: VERY LONG vector – 1 digital image == 1 point in N-dim. Space– Nearby points == Similar images– All possible digital images:

a grid of N-D points

– Space of all practical digital images:

• ~8Meg dimensions (2Mpix * RGBA)

• discrete, quantized

I(m,n)I(m,n)

m,nm,n

II0000

II0101

II0202

……II1010

II1111

II1212

……

I I = =

II00 00 I I01 01 I I02 02 II03 03 II04 04 II05 05

Sampling as a Grid / ArrayGrid / Array Formatting

‘Digital’ Images: 2D Grid of Numbers• NO intrinsic meaning, but widely assumed to represent

Point Samples of a “smoothed” 2D intensity surface

Uniform sampling pattern (but not always)

xx

yy

( weasel - word ! )( weasel - word ! )

Digital Images As Vectors

• Sensible element-by-element operations:– Add, subtract, scale two images:

0.5 + (I1 - I2 ) = out

-- ==II11 II22 outout

SAMPLING

More the samples Better the Approximation/Resemblance

The Samples

Digital images

SamplingPick a sample color from

each box/cell of grid to record

Each sample (cell) is called a pixel - picture element The finest the

sampling -the clearer the image

much finer sampling yet

Quantisation

• Quantisation uses an ADC (Analogue to Digital Converter) to transform brightness values into a range of integer numbers, 0 to M, where M is limited by the ADC and the computer.

- m= log2M is the number of bits used to represent the value of each pixel.

• This determines the number of grey levels.

• Too few bits results in steps between grey levels being apparent.

Quantization (bit allocation)

• Example: "rounding to the nearest integer“

• Non-uniform (variable bit allocation)

– Based on the statistics of the source

(Laplacian Quantizer)

– Based on the human visual system

(perceptually-tuned quantization)

– Either Rounding or Truncation operation on pixel values

QUANTIZATION•A computer can not record the pixel values with infinite precision.•Original pixel values are randomly valued•Hence the pixel values must be quantized

Quantization• Each sample value is replaced by the nominal value of its

quantization level.• a sample value a nominal value• More quantization levels gives a more accurate

representation

Quantization• Does not necessarily have to be Uniform or Linear

WiderQuantization level

NarrowerQuantization level

Coding

• Quantized samples could then be represented digitally by a string of 0’s and 1’s -- 0010 0011 0011 0000 1010 1011

• Assign a codeword to each quantized sample

010101100111100010011010101111001101

Image Resolution Ability to recognize small features and locate boundaries requires

enough pixels

Too few grey levels produce visible “ false contouring" artefacts in smoothly varying regions.

A Girl image with 256 and 4 grey levels.

left hand image produced by combining Bands 1 (B), 2 (G) & 3 (R)

right hand image produced by intensity channel of this color image mixed with Band 8 panchromatic image

improved sharpness of detail is evident

• SPATIAL RESOLUTION is the ability to distinguish fine spatial detail. The spatial sampling frequency is often a good indicator of resolution.

• dots-per-inch (dpi) or pixels-per- inch (ppi) are common to express resolution for digital images.

Pixels: Individual pixels can be seen by zooming in an image

The resolution of a digital camera refers to the number of samples (pixels) that the camera takes in

No. of samples Resolution Print Size

0.3 M 640 x 480 3x4 inches

1 M 1152 x 864 5x7 inches

1.2 M 1280 x 960 5x7 inches

2.1 M 1600 x 1200 8x10 inches

3.3 M 2048 x 1536 11x14 inches

You needat least 150 ppi(pixels per inch)For printing

• PIXEL DIMENSIONS are horizontal & vertical measurements

• Determined by multiplying width & height by the dpi.

• A digital camera has pixel dimensions expressed as its resolution (e.g., 2,048 by 3,072)

• Calculate the dpi dividing a document's dimension into the corresponding pixel dimension against which it is aligned

An 8" x 10" document that is scanned at 300 dpi has the pixel dimensions of 2,400 pixels (8" x 300 dpi) by 3,000 pixels (10" x 300 dpi).

Example:

BIT DEPTH is determined by the number of bits used to define each pixel. Digital images- black and white, (bitonal), grayscale, or color.• Bitonal image is represented by 1 bit pixels, which can represent two

tones (typically black and white), using ‘0’ for black and ‘1’ for white• Grayscale image is composed of multibit pixels( 2 to 8 bits)

– Ex: A 2-bit depth image has four possible representations: 00 - Black, 11-White, 01 - Dark gray and 10 - Light gray.

An 8 bit depth image has 256 tones assigned to each pixel.• Color image: has a bit depth ranging from 8 to 24.

– 8 bits are often used for Red, 8 for Green, and 8 for Blue. Combinations of those bits are used to represent other colors. Ex: A 24-bit image offers 16.7 million (2 24 ) color values.

Bit Depth: Left to right –

1-bit bitonal,

8-bit grayscale, and

24-bit color images.

• DYNAMIC RANGE is the range of tonal difference between the lightest light and darkest dark of an image.

• FILE SIZE is calculated by multiplying the surface area of a document (height x width) to be scanned by the bit depth and the dpi 2 . Divide this figure by 8 to get in Bytes.

• File size naming convention: in increments of 210 (1,024) or more:

• 1 Kilobyte (KB) = 1,024 Bytes• 1 Megabyte (MB) = 1,024 KB• 1 Gigabyte (GB) = 1,024 MB• 1 Terabyte (TB) = 1,024 GBFILE FORMATS: Pixel bits that comprise the image plus a header information on how to read and interpret the file. File formats vary in terms of resolution, bit-depth, color capabilities, and support for compression and metadata.

Color ImageColor Image

Color Dots image Red Channel image

Green Channel image

COLOR DIGITAL IMAGES

2D Arrays of pixels of varying color and intensity

Color Model: how to specify the color of a pixel (coding!)

R,G,B: colors represented by a number triplet, specifying the Red(R), Green(G) and Blue(B) intensities

Y, CR, CB: for digital images and videos

C,M,Y,K : for printing and arts

Cyan - no Red

Magenta - no Green

Yellow - no Blue

K - Black ink

Additive mixing

Subtractivemixing

Color difference

Color Quantization

This process is called

• Some display hardware stores 8 bits per pixel

• => it can display at most 256 distinct colors at a time

• To display a full-color image, the computer must choose an appropriate set of representative colors and map the image into these colors

Quantization phases• Sample the original image for color

statistics

• Select color map based on those statistics

• Map the colors to their representative in the color map• Redraw the image, quantizing each pixel Algorithm

Mapping…

The Median Cut Algorithm

• The concept – to use each of the colors in the color map to represent an equal number of pixels in the original image

• The algorithm repeatedly subdivides color space into smaller and smaller rectangular boxes