Einfhrung in Visual Computing U it 11 P i t O Unit 11: Point - PowerPoint PPT Presentation

Einführung in Visual Computing U it 11 P i t O Unit 11: Point Operations ti http:// www.caa.tuwien.ac.at/cvl/teaching/sommersemester/evc  Content:  Introduction to Point Operations Operations  Histogram  Histogram Normalization Histogram Normali ation  Histogram Equalization 1 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations p mainly applied in the pre ‐ processing step:  Intensity of a new pixel is dependent on the original pixel only:  Intensity of a new pixel is dependent on the original pixel only:  Example: Inversion, Threshold, Brightness Enhancement, Contrast Enhancement Contrast Enhancement… 2 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations p  Point Operations perform a mapping of the pixel values without changing the size, geometry, or local structure of the image g g , g y, g  Each new pixel value I’(u,v) depends on the previous value I(u,v) at the same position and on a mapping function f() at the same position and on a mapping function f()  The function f() is independent of the coordinates  Such operation is called “homogeneous” S ch operation is called “homogeneo s” 3 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations p 4 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations p Example of homogeneous point operations:  Modifying image brightness or contrast  Modifying image brightness or contrast  Applying arbitrary intensity transformation (curves)  Quantizing (posterizing) images Q ti i ( t i i ) i  Global thresholding  Gamma correction  Color transformations 5 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Definition of Point Operations p  New intensity value (gray ‐ level) of a pixel only depends on the previous value . l  Transformation is performed on the pixel intensity value using a pixel intensity value using a mapping function :  linear or  non ‐ linear.  Mapping is usually implemented pp g y p with look ‐ up tables (LUT).  Following functions are possible:  Linear  stepwise linear I‘(u,v) ← a · I (u,v) + b  non ‐ linear 6 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Identity Function p y Before:  The identity function does not alter any pixel values. y p I‘(u,v) ← a · I (u,v) + b After: a = 1, b = 0 7 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Inversion Function p Before:  Inversion means that dark become bright and vice versa. g I‘(u,v) ← a · I (u,v) + b After: a = ‐ 1, b = q = 255 8 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Example: Inversion Transformation p Fi Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 3) i f lid G l / W d DIP b k b i (Ch 3)

Inversion Transformation  Inverting Images I‘(u,v) ← − I(u,v) + q = q − I(u, v).

Threshold Operation p  Thresholding an image is a special type of quantization that separates the pixel values in two classes, depending on a given p p , p g g threshold value p th  The threshold function maps all the pixels to one of two fixed The threshold function maps all the pixels to one of two fixed intensity values p o , p 1 0 < p th ≤ p max  Example: binarization: p o =0 , p 1 =1 11 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Threshold Operation p p 0 p 1 p th Histograms 12 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Threshold Function p Before: Before:  Thresholding means a reduction to only two different color levels y upon a certain threshold value. After: 13 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Gray Level Reduction F nction Function Before: Before:  Gray ‐ level reduction reduces the number of intensity levels (e.g. 4 y ( g different levels). After: 14 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Brightness Increasing Function F nction Before: Before: Before: Before:  Generally the intensity values are increased. Note: Clipping may occur. pp g y I‘(u,v) ← a · I (u,v) + b After: After: a = 1, b = 90 15 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Brightness Reducing F nction Function Before: Before:  Generally the intensity values are reduced I‘(u,v) ← a · I (u,v) + b After: a = 1, b = ‐ 90 16 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Contrast Enhancing F nction Function Before: Before:  Intensity value that formerly were close together are now further apart (spreading). Note: Clipping may occur. I‘(u,v) ← a · I (u,v) + b After: a = 0,01, b = 1 17 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Contrast Reducing Function p g Before: Before:  Intensity value that formerly were wider apart are now closer (narrowing). The resulting ( g) g image does not contain the full intensity range. I‘(u,v) ← a · I (u,v) + b After: a = 90, b = 1 18 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Point Operations: Gamma Correcting F nction Function Before: Before:  Example of a non ‐ linear mapping function. After: 19 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Photoshop: Image ‐ Adjustments ‐ Curves p g j 20 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Histogram

Histogram g  The histogram function is defined over all possible intensity levels.  For each intensity level its value is equal to the number of the  For each intensity level, its value is equal to the number of the pixels with that intensity.  Consider a 5x5 image with integer intensities in the range  Consider a 5x5 image with integer intensities in the range between one and eight: 1 8 4 3 4 1 1 1 7 8 1 1 1 7 8 8 8 3 3 1 2 2 1 5 2 2 2 1 5 2 1 1 8 5 2 22 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Example p 1 8 4 3 4 1 1 1 7 8 8 8 3 3 1 2 2 1 5 2 1 1 8 5 2 1 2 3 4 5 6 7 8 23 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Histogram Function g H(x) = card{(u,v) | I(u,v)=k}, k  {0,...,q} n n n n n n n n 3 5 6 7 8 1 2 4  ( ( ) ) f f r n k k 1 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8

Histogram Function g   f f ( ( r r ) ) 8 8 1 n n n n n n n n  3 5 6 7 8 1 2 4 f ( r ) 4 2  f ( r ) 3 3  f f ( ( r ) ) 3 4 4  f ( r ) 2 5  f f ( ( r ) ) 0 0 6  f ( r ) 1 7 1 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 ( 8  f r ) 5

Histogram g  Assume that the digital image has q discrete gray levels and that n k , k = 0, ..., q ‐ 1 , is the number of pixel having intensity k . The histogram is given by: h ( r ) n   k k p ( r ) k n n where p is the normalized histogram function, n the total number of image pixels. n k are the number of pixels in the bin assigned to pixels with intensity level k .  It gives a measure of how likely is for a pixel to have a certain intensity. That is, it gives the probability of occurrence the intensity intensity.  The sum of the normalized histogram function over the range of all intensities is 1 all intensities is 1. 26 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Histogram g  The histogram function can be plotted graphically. The image histogram carries g p y g g important information about the image content. 27 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Histogram g  Distribution of gray ‐ levels can be judged by measuring a histogram: g g  For B ‐ bit image, initialize q=2 B counters with 0  Loop over all pixels x y  Loop over all pixels x,y  When encountering gray level f(r k )=i, increment co nter r increment counter r k  Histogram can be interpreted as an estimate of the probability density function (pdf) of the h b bili d i f i ( df) f h underlying random process.  You can also use fewer, larger bins to trade off amplitude resolution against sample size. 28 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Histogram g  Histogram on the right corresponds to the image on the left. It is a statistical measure of the occurrence of different color levels. 29 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations

Histogram g  Represents the relative frequency of occurrence of the various gray R h l i f f f h i levels in the image  For each gray level count the # of pixels having that level  For each gray level, count the # of pixels having that level  Can group nearby levels to form a big bin & count #pixels in it ( From Matlab Image Toolbox Guide Fig.10 ‐ 4 ) 30 Robert Sablatnig, Computer Vision Lab, EVC ‐ 11: Point Operations