lecture02

lecture02 - ELEC317 Lecture 2 Digital Image Processing...

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ELEC317 Digital Image Processing Lecture 2 Image Model, sampling and quantization IMAGE SAMPLING & QUANTIZATION --- Continuous image needs be sampled on discrete grid, and --- The value of the sample quantized using finite number of bits Î The result is the representation of an image by a matrix of numbers. Example: An image of size 256x256 is actually a 256x256 matrix. Each picture element of the image is called a PIXEL . Suppose that each pixel is 8 bits, so there are 256 levels. The pixel value can therefore be any integer between 0 and 255. Total amount of data: 256x256 = 65.5 k bytes! Image Scanning A continuous image is 2-D. The sampler can only sample/quantize one pixel at a time Æ Need to scan the image. A common method is to scan row by row. Normally, the pixel locations are on a rectangular grid. However, it is worth noting that some scanning scheme have pixels on other type of grid. This will be
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discussed later. Image Display Normally, display of sampled image is achieved by D/A conversion. On CRT, after D/A conversion, an array of light spot is displayed on screen. The intensity of light spot is proportional to pixel value. If the spots are too small, image will look discontinuous: To achieve the effect of continuous image, interpolation is needed. Another display method is halftone display . This is used by Newspaper, Printers, and Fax machine. Here, each pixel is marked by a number of much smaller black or white dots. The number of dots is proportional to pixel value. The dots are placed pseudo-randomly:
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0,1,2,3 Two Dimensional Sampling Theory 1. Review of 1-D sampling theory Consider a 1-D continuous time signal x(t) with Fourier transform X(f) X(f) = dt e t x f j Π 2 ) ( The signal is called band-limited if X(f)=1 for |f| > B To sample x(t), the following scheme can be used: Where p(t) = −∞ = k kT t ) ( δ A pixel magnified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X(f) B -B f p(t) x s (t) x(t)
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Recall from ELEC 211, −∞ = Π = k T kt j e T t p 2 1 ) ( ) ( 1 ) ( 2 t x e T t x k T kt j s −∞ = Π = ) ( 1 ) ( −∞ = = k s T k f X T f X Thus, Xs(f) consists of replicas of X(f). If 1/T >= 2B, i.e. sampling frequency is greater than 2 x Max frequency, then replicas will not overlap. And X(f) [And hence x(t)] can be recovered from Xs(t) by LPF (low pass filter). The minimum sampling rate is called the Nyquist rate. 2. Reconstruction in 1-D case To reconstruct x(t) from xs(t), under the assumption 1/T >= 2B, we see t T x(t) x s (t) X s (f) X(f) k=1 k=-1 k=0 -1/T 1/T -B B
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) ( ) ( ) ( f X Tf rect T f X s × = )} ( { ) ( ) ( 1 1 Tf rect F t x T t x s × = df e Tf rect F T T tf j Π = 2 1 2 1 2 1 )} ( { ) ( sin 1 2 2 1 2 1 2 T t c T t j e T T tf j = Π = Π −∞ = = k s kT k kT x t x ) ( ) ( ) ( δ ) ( sin ) ( ) ( ) ( −∞ = = k T t c kT t kT x t x [ ) ( ) ( ) ( T t f T t t f = ] Î ) ( sin ) ( ) ( −∞ = = k T kT t c kT x t x [interpolation formula] 3. Sampling of 2-D signals Similar to 1-D case, a 2-D image is band-limited if ) , ( ) , ( 2 1 ξ F y x f f Rect(Tf) 1 -1/(2T) 1/(2T) T x(kT)
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and 0 ) , ( 2 1 = ξ F for 0 1 x > or 0 2 y > Sampling is achieved by where ∑∑ −∞ = −∞ = = mn y
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lecture02 - ELEC317 Lecture 2 Digital Image Processing...

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