Filtering (I) Dr. Chang Shu COMP 4900C Winter 2008 Agenda Getting to know images. Image noise. Image filters. Digital Images An image - rectangular array of integers Each integer - the brightness or darkness of the
image at that point N: # of rows, M: # of columns, Q: # of gray levels Q = 2q (q is the # of bits/pixel) Storage requirements: NxMxq (e.g., for N=M=1024 and q=8 storage = 1MB) f (0,0) f (1,0) ... f ( N 1,0) f (0,1)
f (1,1) ... f ( N 1,1) ... f (0, M 1) ... f (1, M 1) ... ... ... f ( N 1, M 1) Image File Formats
Header usually contains width, height, bit depth. Magic number identify file format. Image data stores pixel values line by line without breaks. Common Image File Formats GIF (Graphic Interchange Format, .gif) PNG (Portable Network Graphics, .png) JPEG (Joint Photographic Experts Group, .jpeg) TIFF (Tagged Image File Format, tiff) PGM (Portable Gray Map, .pgm) FITS (Flexible Image Transport System, .fits)
cvNamedWindow("Result", 1); // Show the images. cvShowImage("Source", image); cvShowImage("Result", gray); // Wait for a key stroke cvWaitKey(0); cvReleaseImage(&image); cvReleaseImage(&gray); cvDestroyWindow("Source"); cvDestroyWindow("Result"); return 0; }
Noise Noise in Computer Vision: Any entity (in images, data, or intermediate results) that is not interesting for the main computation Can think of noise in several contexts: Spurious pixel fluctuations, introduced by image acquisition Numerical inaccuracies, introduced by the computers limited precision, round-offs or truncations, etc Entities that do not belong to any meaningful region (edges that do not correspond to actual object contours) Image Noise
Image Noise Additive and random noise: I x, y I x, y n x, y I(x,y) : the true pixel values n(x,y) : the (random) noise at pixel (x,y) Random Variables A random variable is a variable whose values are determined by the outcome of a random experiment.
A random variable can be discrete or continuous. Examples 1. A coin is tossed ten times. The random variable X is the number of tails that are noted. X can only take the values 0, 1, ..., 10, so X is a discrete random variable. 2. A light bulb is burned until it burns out. The random variable Y is its lifetime in hours. Y can take any positive real value, so Y is a continuous random variable. Mean, Variance of Random Variable Mean (Expected Value)
E ( X ) xi p( xi ) (discrete) i E ( X ) xp( x)dx Variance
2 E (( X ) 2 ) 2 is called standard deviation (continuous) Gaussian Distribution Single variable 1 ( x ) 2 / 2 2 p( x) e
2 Gaussian Distribution 2 Bivariate with zero-means and variance x2 y2 1 G x, y exp
2 2 2 2 Gaussian Noise Is used to model additive random noise
n2 2 The probability of n(x,y) is e 2 Each has zero mean The noise at each pixel is independent Impulsive Noise Alters random pixels Makes their values very different from the true ones Salt-and-Pepper Noise: Is used to model impulsive noise
I h, k I sp h, k imin y imax imin x, y are uniformly distributed random variables l , imin,imax are constants x l x l
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3D CFD modelling based on RANS equations to understand complex 3D flows. However strong dependence on: Parameters (mesh, boundary conditions) Numerical options (turbulence models, wall functions, discretization schemes, etc.) Friday the 30th of August - 10.40 am - 11.00 am...
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