Being edited by Prof. Sumana Gupta 1

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1 Being edited by Prof. Sumana Gupta 1 Introduction Digital Image Processing: refers to processing of 2-D picture by a digital Computer- in a broader context refers to digital proc of any 2-d data. A digital image is an array of real or complex numbers represented by a finite no. of bits. A typical digital image processing sequence consists of: Observer Image syst Digitization Digital storage Digital Computer On-line buffer Store Process Refresh /Store Display Record Output The image given in the form of a transparency slide or photograph is first digitized and stored in a computer memory.the digitized image is then processed and/or displayed on a high resolution TV monitor. For display, the image is stored in a rapid access buffer memory which refreshes the monitor at 30 frames/sec to produce a continuous display. Digital image processing has a broad spectrum of application such as : remote sensing via satellites or other space-crafts,image transmission, medical imaging, robotics, radar, sonar,automated inspection of industrial parts.we will consider the following classes of problems: 1. Image representation and modeling: This deals with characterization of the quality that each picture element (pixel) represents.it could represent luminance or absorption characteristics of a body or gravitational field in an area depending on the context. 2. Image enhancement: accentuate certain features of an image. 3. Image restoration: refers to removal of known degradation in an image,linear or non-linear methods used. 4. Image analysis:making qualitative measurements from an image to provide description. 5. Image reconstruction: relates to computer tomographies. 6. Image data compression: refers to transforms,quantization and coding.

2 In general any 2-D function that bears in forming can be considered as an image. Image models give a quantitative or a logical description of the properties of the function. An important consideration in image reptu is the fidelity or intelligibility criteria for measuring the quality of an image or the performance of a processing technique.specification of such measures requires models of perception of contrast, spatial frequencies, colors and so on; Knowledge of fidelity criteria helps in designing the imaging sensors because it tell us the variables that can be measured most accurately. An image model We define the term image as a 2D light intensity function denoted by f(x, y). The value or amplitude of f at spatial coordinates (x,y) gives the intensity (brightness) of the image at that point. Since light is a form of energy f(x, y) must be non-zero and finite, ie, 0 <f(x, y) < Now the images we perceive normally consists of light reflected from objects. The basic nature of f(x, y) can be considered as being characterized by two components. 1. The illumination component i(x, y) ie the component is the amount of source light incident in the scene been viewed. 2. The surface reflectance component r(x, y),amount of light reflected by objects in the scene. These forms combine to form f(x, y) givenby f(x, y) =i(x, y) r(x, y) where 0 <i(x, y) < nature of i(x, y) is determined by the light source. and 0 <r(x, y) < 1 determined by the characteristics of the objects in the scene. 0 total absorption 1 total reflectance 2

3 The range of values are theoretical bounds but some typical ranges are, for i(x, y): 9000 ft candles on a clear day on the surface of earth. :1000 ft candles cloudy day :0.01 ft candles clear evening, full moon day. and r(x, y):0.01 for black velvet :0.65 for steel :0.9 for silver :0.93 for snow The intensity of monochrome image f at (x,y) will be referred to as gray level (l) of image at the point. l lies in the range L min <l<l max is +ve finite The interval (L min,l max ) is called gray scale.it is common practice to use (0,1). in practice, L min = i min r min L max = i max r max For indoor image processing applications, L min L max 100 The interval (L min,l max ) is called the gray scale. It is common practice to shift this interval numerically to (0,L)wherel = 0 is considered black and l = L is considered white. All intermediate values are shades of gray varying continuously from black to white. Uniform Sampling and quantization In order to put the data in a form suitable for computer processing an image function f(x, y) must be digitized both spatially and in amplitude. Digitization of spatial coordinated (x, y) will be referred to as image sampling while amplitude digitization will be called gray level quantization. Suppose that a continuous image f(x, y) is approximated by equally spaced 3

4 samples arranged in the form of an N N array as: f(0, 0) f(0, 1) f(0,n 1) f(x, y) = f(n 1, 0)... f(n 1,N 1) N N where each element of the array refers to as pixel is a discrete quantity. The array represents a digital image. The above digitization requires a decision be made on a value for N as well as on the number of discrete gray levels allowed for each pixel. It is common practice in digital image processing to let N =2 n and G = nos of gray levels =2 m. It is assumed discrete levels are equally spaced between 0toLinthegrayscale. Therefore number of bits required to store a digitized image is b = N N m ie a image with 256 gray levels(ie 8 bits/pixel) requires a storage of 17,000 bytes. Equ f(x, y) f(0, 0).. f(0,n 1) is an approximation to a continuous image. Reasonable question to ask at this point is how many samples, gray levels are required for a good approximation? This bring up the question of resolution. The resolution (ie the degree of discernable detail) of an image is strangely dependent on both N and m. The more these parameters are increased, the closer the digitized array will approximate the original image. However eqn b = N N m clearly points out the unfortunate fact that storage consequently processing requirements increase rapidly as a... of N r m. In view of the above comments, it is of interest to consider the effect that variations in N r m have an image quality. A good image is difficult to define because quality requirements vary accordingly to application. The number of samples and gray levels required to produce a faithful reproduction of an original image depends on the image itself. As a basis for comparison, the requirements to obtain a quality comparable to that of monochrome TV pictures over a wide range of image types are of the order of pixels with 128 gray levels. As a rule, a minimum system for general image processing work should be able to display pixels with 64 gray levels. 4

5 Huang in (1965) considered this problem of effects produced on image quality by varying N and m independently. He tried to quantify experimentally the above effects. The experiment consisted of a set of subjective tests. Three images were considered 1. Image with relatively little detail 2. Image with an intermediate amount of detail. 3. Image with a large amount of detail(picture of crowd) Sets of these 3 images were generated by varying r m and observers were then asked to rank them according to their subjective quality. The results were summarized in the form of curves in the N-m plane called isopreference curves. Each point in this plane represents an image with values of N r m equal to the coordinates of that point. An isopreference curve is one in which the points represent images of equal subjective quality. Huang suggested several empirical conclusions: 1. As expected the quality of images increase as N r m increase. 2. There were a few cases where for fixed N the quality improved by decreasing m. By decreasing m we increase the apparent contrast of an image. 3. As the image detail increases, the curves tend to become vertical. This suggests that for images with a large amount of detail only few gray levels are needed. This is not true for images with less detail. Non-uniform sampling For a fixed value of N, it is possible in many cases to improve the appearance of an image by using an adaptive scheme where the sampling process depends on the characteristics of the image. In general fine sampling is required in the neighborhood of sharp gray level transitions while coarse sampling is required in relatively smooth regions. Example: A simple image consisting of a face superimposed on a uniform background. The background contains little detail... and can be quite adequately 5

6 represented by coarse sampling. The face contains more detail. Using in this region of the image the additional sample not used in the background will tend to improve the overall result, particularly if N is small. In general, in distributing the samples, greater sample concentration should be used in gray-level transition boundaries such as the boundary between face and background in the example. Drawbacks of non-uniform sampling: The necessity of having to identify the boundaries even if only on a rough basis is a definite drawback of this method. Also this method is not practical for images containing small uniform regions, example an image depicting a crowd. Non uniform quantization When the number of gray levels must be kept small, it is usually desirable to use unequally spaced levels in quantization process. This method is similar to the previous case. However, since the eye is relatively poor at estimating shades of gray near abrupt level changes, the approach is to use less number of gray levels near the boundaries. The remaining levels may be used in those regions where gray level variations are smooth, thus avoiding or reducing the false contours which often appear in these regions if they are coarsely quantized. Drawbacks are same as the previous method. Sampling and replication using a component frequency. The digitization process of images can be understood by modeling then as bandlimited signals. Consider a function f(x, y) that is bandlimited ie F (ξ 1,ξ 2 )=0 ξ 1 >ξ xo ξ 2 >ξ y0 6

7 ξ 2 y w of image ξ yo ξ xo ξ yo ξ xo x ξ 1 BW of image Consider an ideal sampling frequency which is a 2-D infinite array of Dirac delta functions situated on a rectangular grid with spacing x, y, ie. Comb(x, y, x, y) δ(x m x, y n y) The sampled image is m,n fs(x, y) =f(x, y) comb(x, y, x, y) y y x x FT[comb( )] is another component frequency with spacing 1/ x, 1/ y namely, Defining Comb(ξ 1,ξ 2 )=I[comb(x, y; x, y)] [ = 1 x 1 ] δ(ξ 1 k/ x, ξ 2 l/ y) y k,l= ξ xs 1 x and ξ ys 1 y 7

8 The FT of the sampled image fs(x, y) isgivenby Fs(ξ 1,ξ 2 )=F(ξ 1,ξ 2 ) comp(ξ 1,ξ 2 ) = ξ xs,ξ ys F (ξ 1,ξ 2 ) δ(ξ 1 kξ xs,ξ 2 lξ ys ) k,l= = ξ xs,ξ ys k,l= F (ξ 1 kξ xs,ξ 2 lξ ys ) If the x, y sampling frequencies are greater than twice the bandwidths ie ξ xs > 2ξ xo, ξ ys > 2ξ yo or or equivalently, if x < 1 2ξ xo and y < 1 2ξ yo Then F (ξ 1,ξ 2 ) can be recovered by a low pass filter with frequency response H(ξ 1,ξ 2 )= 1, (ξ (ξ xsξ ys) 1,ξ 2 ) ɛr 0 otherwise where R is the region of support of the ideal low pass filter R 2 R R 1 R 0 2ξ xo ξ xs ie. R is any region whose boundary δr lies in the annular region of two rectangles R 1 and R 2. F (ξ 1,ξ 2 ) H(ξ 1,ξ 2 )F s (ξ 1,ξ 2 ) =F (ξ 1,ξ 2 ) 8

9 Example. An image described by the frequency f(x, y) = 2 cos 2Π(3x, 4y) is sampled s t x = y =0.2 and the reconstruction filter has a rectangular region of support with cut off frequency at half the sampling frequency.find the reconstructed image. f(x, y)2 cos 2Π(3x +4y) is bandlimited since F (ξ 1,ξ 2 )=δ(ξ 1 3,ξ 2 4) + δ(ξ 1 +3,ξ 2 + 4) is zero for ξ 1 > 3, ξ 2 > 4 Hence ξ xo =3,ξ yo =4 Also ξ xs = ξ ys =1/0.2 = 5 which is less than the Nyquist frequency 2ξ xo, 2ξ yo. The sampled image spectrum is Fs(ξ 1,ξ 2 )=25 [δ(ξ 1 3 5k, ξ 2 4 5l)+δ(ξ k, ξ l)] k,l= } Now H(ξ 1,ξ 2 )= ξ ξ otherwise F (ξ1,ξ 2 )=δ(ξ 1 2,ξ 2 1) + δ(ξ 1 +2,ξ 2 +1) which gives the reconstructed image as ˆf(x, y)2 cos 2Π(2x + y) This shows that any frequency component in the input image that is above (1/2ξ xs, 1/2ξ ys )by( ξ x, ξ y ) is reproduced (or aliased) as a frequency component at (1/2ξ xs ξ x, 1/2ξ ys ξ y ) It can be shown that the PSD function S s (ξ 1,ξ 2 ) of the sampled image fs(x, y) is a periodic extension of S(ξ 1,ξ 2 )andisgivenby S s (ξ 1,ξ 2 )=ξ xs ξ ys S(ξ 1 kξ xs,ξ 2 lξ ys ) k,l= 1 when the image is reconstructed by an ideal low-pass filter with gain ξ xsξ ys, the reconstructed image PSD is S(ξ 1,ξ 2 )= S(ξ 1 kξ xs,ξ 2 lξ ys )W (ξ 1,ξ 2 ) k,l= 9

10 Where W (ξ 1,ξ 2 )= = S(ξ 1,ξ 2 ) { 1 (ξ1,ξ 2 ) ɛr 0 otherwise The aliasing power σa 2 is the power in the tails of power spectrum outside R. σa 2 = S(ξ 1,ξ 2 )dξ 1 dξ 2 = [(1 W (ξ 1,ξ 2 ))]S(ξ 1,ξ 2 )dξ 1 dξ 2 ) ξ 1,ξ 2 ɛr which is zero if f(x, y) is bandlimited with ξ xo 1 2 ξ xs; ξ yo 1 2 ξ ys This analysis is useful when bandlimited image containing wide band noise is sampled. The S/N ratio of the sampled image can deteriorate significantly unless it is low pass filtered before sampling. Sampling random fields In physical sampling environments, random noise is always present in the image so that it is important to consider sampling theory for random fields(a family of 2-D functions which itself is a r-v). A continuous stationary random field f(x,y) is called bandlimited if its PSD function S(ξ 1,ξ 2 ) is bandlimited, ie. if S(ξ 1,ξ 2 )=0 for ξ 1 >ξ xo ξ 2 >ξ yo Sampling theory for Random fields If f(x,y) is a stationary bandlimited random field then f(x, y) f(m x, n y) sinc(xξ xs m)sinc(yξ ys n) m,n= Converges to f(x,y) in the mean square sense ie E( f f 2 )=0 where ξ xs = 1 x ξ ys = 1 y ξ xs > 2ξ xo 10

11 Nyquist rate The lower bonds on the sampling rates that is given by ξ xs > 2ξ xo and ξ ys > 2ξ yo are referred to as Nyquist rates or Nyquist frequency. Their reciprocals are called Nyquist intervals. The sampling theory states that a bandlimited image sampled above its x and y Nyquist rates can be recovered without error by low pass filtering the sampled image. However if sampling frequencies are below Nyquist frequencies is ξ xs > 2ξ xo or ξ ys > 2ξ yo then periodic replication of F (ξ 1,ξ 2 ) will overlap resulting in a distorted spectrum of Fs(ξ 1,ξ 2 )fromwhichf (ξ 1,ξ 2 ) cannot be recovered. FIG 5 The frequencies above half the sampling frequency ie ξ xs /2andξ ys /2 are called foldover frequencies. ξ 2 2ξ yo ξ ys 2ξ xo ξ xs Aliasing ξ 1 The overlapping of the successive periods of the spectrum cases the foldover frequencies in the original image to appear below ξ xs /2andξ ys /2inthe sampled image. This phenomena is called aliasing. Aliasing errors cannot be removed by subsequent filtering. Aliasing can be avoided by low pass filtering the image first so that its bandwidth is less than one half of the sampling criteria(of equ (A)) is satisfied. In other words, if the region of support of the ideal low pass filter is the rectangular region [ ξxs R = 2, 1 ] 2 ξ xs [ 12 ] ξ ys,ξ ys centered at origin, then its impulse response is h(x, y) = sin c(xξ xs )sincyξ ys f(x, y) can be reconstructed as: IFT[ F (ξ 1 ξ 2 )] = IFT[H(ξ 1,ξ 2 )Fs(ξ 1,ξ 2 )] 11

12 f(x, y) = m,n= f(m x, n y)sinc(xξ xs m)sinc(yξ ys n) Interpolation formula = f(x, y) if x and y satisfy Nyquist criteria. 12