digital image representation...17/05/2010 compsci 373 - tutorial 4 pixel values data types...

17/05/2010 CompSci 373 - Tutorial 1

Digital Image Representation


Defining an Image

● Image Grid– cols * rows

● Pixels– Contains a single

code value

(R,G,B) = (128, 128, 255)“light purple”

rows

cols


Image Model

● A function – R is the rectangular image grid– V is the code value of a pixel (scalar or vector)– Maps an image grid to pixel values

● Thus can define a pixel uniquely with:

f :RV

x , y , f x , y


Pixel Values

● Data Types (bit-depth)– Binary – Unsigned integers– Floating points

● Channels– 1 channel (scalar code value) = binary or grayscale image– 3 channel (vector code value) = colour image (e.g. RGB)– Operation on channels are independent

Data type # of Values RangeBinary 2 [0 - 1]8-bit integer 256 [0 - 255]16-bit integer 65536 [0 - 65535]32-bit floating point

“lots” [0.0 - 1.0]

64-bit floating point

“lots lots” [0.0 - 1.0]


● Colour– Represented by separate, independent channel values– Different representations (colour spaces) possible, e.g.

● RGB (or BGR ordering)● HSV (or HSL)● Lab● CMYK

● Resolution– Number of pixels

● 800 cols * 600 rows– Spatial - Number of pixels per measurement unit

● pixels (dots) per inch, pixels per mm

Defining an Image


RGB Colour

R G B


More Image Types

● Videos– A sequence of images

● 3D images– The image grid is extended to the 3rd coordinate– Time varying images (similar to videos)

● 4D videos– Time varying 3D images


Example

● Defining an image– 1600 cols * 1200 rows– 16 bits per channel– 3 channels

● RGB colour model● Data stored in BGR order● (48 bits required per pixel)

– Displayed at 72 dots per inch (dpi)● i.e. physical size = 22.22 in * 16.67 in


Thresholding and Linear Mapping


General Idea

For every pixel's intensity as input, apply a (linear) function to generate an output intensity for that pixel

for each pixel (x, y)output_value

(x,y) = func ( input_value

(x,y) )


Threshold Functions

● Thresholding is like an if-else statement● Different types possible:

– Threshold to binaryif (input > threshold) output = max // 255 for 8-bit images, or 1 for binaryelse output = 0

– Threshold belowif (input < threshold) output = 0else output = input


Linear Functions

● Change intensity of a pixel linearly● In general:

– Gain (gradient) a– Bias (y-intercept) b

● In image processing, the output is clamped to the allowed pixel values (e.g. [0 - 255]).

out x , y = a⋅in x , y b

input

output


Examples

● Threshold below 128out = (in < 128) ? 0 : in


Examples

● Reducing contrastout = 0.60 * in


Examples

● Negation (8-bit)out = -1 * in + 255out = 255 - in


Histograms


Definition

● A histogram of an image is the frequency of each possible intensity value of the image.

●

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H i = count { f x , y=i} for 0≤i≤I max


Cumulative Histograms

● The cumulative histogram is the frequency of all pixels with intensity lower than a value.

●

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CH i = count { f x , y≤i} for 0≤i≤I maxCH i = ∑

j=0

i

H j


Example

2 6 9 2 68 9 3 7 72 6 3 8 96 0 2 1 35 0 9 6 7

Value Count0 21 12 43 34 05 16 57 38 29 4

Total 25

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H CH


Normalised Histograms

● Express the histogram as observed probability.–

where N = cols * rows is the number of pixels

● For the cumulative histogram, the normalised version is called the cumulative distribution function (cdf).

–

H i =H iN

cdf i = ∑j=0

iH j


Histograms

● Note:– CH(Imax) = N

– cdf has range [0.0 - 1.0]– cdf(Imax) = 1.0


Histograms

● Thought exercise:– What is the histogram of the clown image

flipped upside down?

● An image is not uniquely defined by its histogram.

● Any permutation (rearrangement) of pixels of an image will give an identical histogram.

– And thus, an identical cumulative histogram.


Linear Mapping and Histograms

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Linear Mapping and Histograms

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1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 201 206 211 216 221 226 231 236 241 246 251


Colour Histogram

● Different ways to express colours in histogram:– Convert to grey intensities– Graph each colour separately– Graph in 3D


Histogram Equalisation


Idea

● Stretch histogram such that frequencies are equally distributed.

– The integral of the histogram (the cumulative histogram) increases linearly.

–

– CH i ≈ i⋅KH i ≈ K


How to do it

● Apply the CH or cdf as a Look Up Table (LUT).–

● However, we need to normalize the result to the maximum range available. Use contrast stretching.

–

● When CH(input) = CHmin

, result = 0

● When CH(input) = CHmax

, result = Imax

result x , y = cdf input x , y ⋅I max

result x , y =CH input x , y −CH min

CH max−CH min⋅I max


How to do it

● It does not matter whether we use the normalized version or not.

● CHmin

is the first non-zero value.

● CHmax

is always N = cols * rows.

● Round the result to the nearest integer in the case of integer data types.


Example

2 6 6 25 7 3 72 6 3 56 7 2 4

Value H CH0 0 01 0 02 4 43 2 64 1 75 2 96 4 137 3 16


Example

0 5.25 5.25 02.92 7 1.17 7

0 5.25 1.17 2.925.25 7 0 1.75

Value CH out0 0 N/A1 0 N/A2 4 03 6 1.174 7 1.755 9 2.926 13 5.257 16 7

out x , y =CH in x , y−CH minCH max−CH min

⋅I max

out x , y = CH in x , y−416−4

⋅7


Example

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Demos


Demos

● Linear mapping with GIMP


Demos

● Building histograms with ImageJ macrowidth = getWidth();height = getHeight();histo = newArray(256);

for (y = 0; y < height; y++) { for (x = 0; x < width; x++) { v = getPixel(x, y); histo[v]++; }}

for (i = 0; i < 256; i++) { print(i, histo[i], "\n"); // equivalent to print(i + “ “ + histo[i] + “ “ + “\n”);}


Extras


LUT Linear Mapping

● Linear Mapping can be inefficient.– Recompute linear equation for every pixel.

● Use a Look Up Table (LUT)– One input value can only have one output, no

matter which pixel it is.– Build a LUT of all possible input values (e.g. [0 –

255]) and their output.● LUT is faster

if possible input values


LUT Linear Mapping

width = getWidth();height = getHeight();lut = newArray(256);

// build LUTfor (i = 0; i < 256; i++) { lut[i] = 0.95 * i + 25;}

for (y = 0; y < height; y++) { for (x = 0; x < width; x++) { v = getPixel(x, y); setPixel(x, y, lut[v]); }}

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digital image representation...17/05/2010 compsci 373 - tutorial 4 pixel values data types...

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