Image Definition and point operation g p p WHAT IS AN IMAGE? - - PowerPoint PPT Presentation

VIDEO SIGNALS VIDEO SIGNALS

Image Definition and point operation g p p

WHAT IS AN IMAGE? WHAT IS AN IMAGE?

Id ll hi k f i i 2 di di i l li li h

 Ideally, we think of an image

mage as a 2-di dimens mensiona

• nal

l li light int intensity function ity function, f(x,y), where x and y are spatial coordinates, and f at (x,y) is related to the brightness or , ( ,y) g color of the image at that point.

 In practice, most images are defined over a rectangle.

C ti i lit d ( ti t “)

 Continuous in amplitude („continuous-tone“)  Continuous in space: no pixels!

DIGITAL IMAGES AND PIXELS DIGITAL IMAGES AND PIXELS

A digit digit l i i g i i th th t ti f f ti ti

 A digit

digital i l ima mage i is th the represen representati tion

• n of

f a con conti tinuous nuous image image f(x,y) by a 2-d array of discrete samples. The amplitude of each sample is quantized to be amplitude of each sample is quantized to be represented by a finite number of bits.

 Each element of the 2-d array of samples is called a  Each element of the 2 d array of samples is called a

pix pixel l or pel pel (fr from “picture element”)

• m “picture element”)

 Pixels are point samples, without extent.

p p ,

 A pixel is not:

 Round, square, or rectangular

• u d, squa e, o

ecta gu a

 An element of an image sensor  m An element of a display

A DIGITAL IMAGE IS REPRESENTED BY NUMBERS A DIGITAL IMAGE IS REPRESENTED BY NUMBERS

AN IMAGE CAN BE REPRESENTED AS A MATRIX AN IMAGE CAN BE REPRESENTED AS A MATRIX

Th i l l f( ) t d i t th t i i t l“

 The pixel values f(x,y) are sorted into the matrix in „natural“

• rder, with x corresponding to the column and y to the row

index index.

 Matlab, instead, uses matrix convention. This results in f(x,y) =

fyx, where fyx denotes an individual element in common matrix fyx, where fyx denotes an individual element in common matrix notation.

 For a color image, f

f might be one of the components. g g p

IMAGE SIZE AND RESOLUTION IMAGE SIZE AND RESOLUTION

 These images were produced by simply picking

every n-th sample horizontally and vertically y p y y and replicating that value nxn times.

 We can do better

 prefiltering before subsampling to avoid aliasing  Smooth interpolation

IMAGE OF DIFFERENT SIZE IMAGE OF DIFFERENT SIZE

FEWER PIXELS MEAN LOWER SPATIAL RESOLUTION

COLOR COMPONENTS COLOR COMPONENTS

DIFFERENT NUMBERS OF GRAY LEVELS DIFFERENT NUMBERS OF GRAY LEVELS

HOW MANY GRAY LEVELS ARE REQUIRED? HOW MANY GRAY LEVELS ARE REQUIRED?

?

 How many gray levels are required?  Digital images typically are quantized to 256 gray

levels.

STORAGE REQUIREMENTS FOR DIGITAL IMAGES STORAGE REQUIREMENTS FOR DIGITAL IMAGES

 Image LxN pixels 2B gray levels c color  Image LxN pixels, 2B gray levels, c color

components Size = LxNxBxc Size = LxNxBxc

 Example: L=N=512 B=8 c=1 (i e monochrome) Size  Example: L=N=512, B=8, c=1 (i.e., monochrome) Size

= 2,097,152 bits (or 256 kByte)

 Example: LxN=1024x1280, B=8, c=3 (24 bit RGB

p 0, , ( image) Size = 31,457,280 bits (or 3.75 MByte)

 Much less with (lossy) compression!  For a video multiply by the frame rate and by the

number of seconds of its lenght.

BRIGHTNESS DISCRIMINATION EXPERIMENT BRIGHTNESS DISCRIMINATION EXPERIMENT

C h i l ?

 Can you see the circle?  Visibility threshold

CONTRAST WITH 8 BIT CONTRAST WITH 8 BITS A ACCORDING T CCORDING TO WEBER WEBER‘S LA S LAW CONTRAST WITH 8 BIT CONTRAST WITH 8 BITS A ACCORDING T CCORDING TO WEBER WEBER S LA LAW

A th t th l i diff b t t i

 Assume that the luminance difference between two successive

representative levels is just at visibility threshold

 For  Typical display contrast

 Cathode ray tube 100:1  Cathode ray tube 100:1 

Print on paper 10:1

 Suggests uniform quantization in the log(I) domain

MOST COMMON POINT OPERATIONS ON IMAGES MOST COMMON POINT OPERATIONS ON IMAGES

 Intensity scaling  Brightness/Contrast adjustment

Brightness/Contrast adjustment

 Gamma adjustment  Histogram equalization  Image averaging  Image averaging  High-dynamic range from multiple exposures  Image subtraction

HISTOGRAMS HISTOGRAMS

 Distribution of gray levels can be judged by measuring a histogram:  Distribution of gray-levels can be judged by measuring a histogram:

 For B-bit image, initialize 2B counters with 0  Loop over all pixels x,y

 When encountering gray level f(x,y)=i, increment counter #i 

Histogram can be interpreted as an estimate of the probability density function (pdf) of an underlying random process.

 You can also use fewer, larger bins to trade off amplitude resolution

against sample size.

EXAMPLE HISTOGRAM EXAMPLE HISTOGRAM

EXAMPLE HISTOGRAM EXAMPLE HISTOGRAM

HISTOGRAM COMPARISON HISTOGRAM COMPARISON

 Both these images present the same

Histogram

HISTOGRAM COMPARISON HISTOGRAM COMPARISON

 Histogram as an invariant feature

HISTOGRAM COMPARISON HISTOGRAM COMPARISON

HISTOGRAM EQUALIZATION HISTOGRAM EQUALIZATION

 Idea: find a non linear transformation  Idea: find a non-linear transformation

g g = = T(f) to be applied to each pixel of the input image f(x,y) f(x,y), such that a if di t ib ti f l l i th ti lt f th uniform distribution of gray levels in the entire range results for the

• utput image g(x,y).

g(x,y).

 Analyze ideal, continuous case first, assuming

0 ≤ f f ≤1 1 0 ≤ g g ≤1

 T(f)

T(f) is strictly monotonically increasing, hence, there exists f = T f = T −1(g) (g) 0 ≤ g g ≤1

 Goal: pdf (probability density function) pg(g)

(g) = cons

• const. over the range

 Goal: pdf (probability density function) pg(g)

(g) cons

• const. over the range

HISTOGRAM EQUALIZATION FOR CONTINUOUS CASE CONTINUOUS CASE

 From basic probability theory  Consider the transformation function  Then

HISTOGRAM EQUALIZATION FOR CONTINUOUS CASE

HISTOGRAM EQUALIZATION FOR DISCRETE CASE

 Now, f only assumes discrete amplitude values f0, f1, …,fL-1,

with probabilities:

 Discrete approximation of

 The resulting values gk are in the range [0,1] and need to be scaled and

rounded appropriately rounded appropriately

HISTOGRAM EQUALIZATION EXAMPLE HISTOGRAM EQUALIZATION EXAMPLE

HISTOGRAM EQUALIZATION EXAMPLE HISTOGRAM EQUALIZATION EXAMPLE

HISTOGRAM EQUALIZATION EXAMPLE HISTOGRAM EQUALIZATION EXAMPLE

HISTOGRAM EQUALIZATION EXAMPLE HISTOGRAM EQUALIZATION EXAMPLE

ADAPTIVE HISTOGRAM EQUALIZATION ADAPTIVE HISTOGRAM EQUALIZATION

 Apply histogram equalization based on a

histogram obtained from a portion of the image

 Must limit contrast expansion in flat regions of

the image, e.g. by clipping individual histogram the image, e.g. by clipping individual histogram values to a maximum

ADAPTIVE HISTOGRAM EQUALIZATION ADAPTIVE HISTOGRAM EQUALIZATION

ADAPTIVE HISTOGRAM EQUALIZATION ADAPTIVE HISTOGRAM EQUALIZATION

POINT OPERATIONS BETWEEN IMAGES POINT OPERATIONS BETWEEN IMAGES

 Image averaging for noise reduction  Combination of different exposure for high-

Combination of different exposure for high dynamic range imaging I g bt ti f h g d t ti

 Image subtraction for change detection  Accurat

ccurate ali alignment is nment is alw always a s a re requirement uirement g y g y q

IMAGE AVERAGING FOR NOISE REDUCTION IMAGE AVERAGING FOR NOISE REDUCTION

IMAGE AVERAGING FOR NOISE REDUCTION IMAGE AVERAGING FOR NOISE REDUCTION

T k N li d i

 Take N aligned images

A I

 Average Image:  Mean squared error vs. noise-free image g

HIGH DYNAMIC RANGE IMAGING HIGH-DYNAMIC RANGE IMAGING

 16 exposures, one f-stop (2X) apart

IMAGE SUBTRACTION IMAGE SUBTRACTION

 Find differences/changes

Find differences/changes between 2 mostly identical images f C

 Example from IC

manufacturing: defect detection in photomasks detection in photomasks by die-to-die comparison

WHERE IS THE DEFECT? WHERE IS THE DEFECT?

 Image A without defect

Image B with defect

ABSOLUTE DIFFERENCE BETWEEN TWO IMAGES ABSOLUTE DIFFERENCE BETWEEN TWO IMAGES

 Without alignment

With Alignment

DIGITAL SUBTRACTION ANGIOGRAPHY DIGITAL SUBTRACTION ANGIOGRAPHY