GNR607 Principles of Satellite Image Processing Instructor: Prof. B. Krishna Mohan CSRE, IIT Bombay bkmohan@csre.iitb.ac.in Slot 2 Lecture 10 Histogram and Image Enhancement August 12, 2014 10.35 AM – 11.30 AM
IIT Bombay Slide 1 August 12, 2014 Lecture 10 Histogram and Image Enhancement Contents of the Lecture • Histogram of an Image • Useful information from Histogram • Contrast in Satellite Image – Linear Contrast Enhancement GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 2 Shift Variant Systems ∫ α α α g(x) = f ( ) ( , ) h x d • The response of some image processing operators is shift variant, i.e., the form of the operation varies with position in the image • e.g., A projection system that is not working well will illuminate one part of the screen brightly while another part will be dull GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 3 Convolution • For a linear shift invariant system, we can write ∞ ∫ α − α α f ( ) ( h x ) d • g(x) = −∞ • This is known as the convolution integral • For a linear shift invariant system the output of the system is obtained by the convolution of the input and the impulse response of the system GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 4 Convolution • Convolution operation is compactly represented as g(x) = f(x) * h(x) ∞ ∫ α − α α f ( ) ( h x ) d • f(x) * h(x) ≜ −∞ In discrete case, ∑ = = − g n ( ) f n ( )* ( ) h n f n ( k h k ) ( ) k GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 5 Convolution f(x) f(x) g(x) g(x) ∞ ∫ ∗ = − f x ( ) g x ( ) f t g x ( ) ( t dt ) −∞ GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 6 Relevance of Linearity • Many image processing operations are linear • Linear operations are easier to analyze • They are often less time consuming to compute • Linear shift invariant operations can be efficiently performed using Fourier transforms (to be discussed later) GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 7 Non-linearity • Certain processes in image acquisition are inherently non-linear • For example, the deposition of silver on a photographic film in response to the amount of light incident is logarithmically related • Our perception of brightness in relation to the amount of incident light on the retina is also non-linear GNR607 Lecture 10 B. Krishna Mohan
Histogram and Its Role in Image Processing
IIT Bombay Slide 8 Concept of Histogram • Given a digital image Fm,n of size MxN, we can define f(j) = #{F m,n = j, 0 ≤ m ≤ M-1; 0 ≤ n ≤ N-1} • We refer to the sequence f(j), 0 ≤ j ≤ K-1, where K is the number of gray levels in the image, as the histogram of the image. • f(n) is interpreted as the number of times gray level n has occurred in the image. • Obviously, Σ n f(n) = M . N GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 9 Sample Histogram GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 10 Histogram • With digital images, we have a range of values that can be found at a given pixel. Depending on the resolution of the sensor from which the image is acquired, the gray level values may be [0-255], [0-1023], [0-2047], [0-63], [0-127] etc. in each band GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 11 Histogram • The normalized version of f(n) may be defined as p(n) = f(n) / (M.N) – p(n) probability of the occurrence of gray level n in the image (in relative freq. sense) � Σ n p(n) = 1 MIN = min n {f(n) | f(n) ≠ 0} MAX = max n {f(n) | f(n) ≠ 0} GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 12 Application of Histogram • Dynamic range of display system – min to max range of intensities that can be displayed • Normal range is 0 – 255 for gray scale; for color it is 0 – 255 for red, green and blue • If Min-Max range of data is comparable to dynamic range of display device, good quality display is possible GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 13 Histogram of image with good contrast GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 14 Image GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 15 Histogram of Low Contrast Image GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 16 Low Contrast Image GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 17 Good Contrast Image GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 18 Good Contrast Image GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 19 Information from Histogram • The information conveyed by the occurrence of an event whose probability of occurrence is p(n) is given by I(n) = ln{1/p(n)} = -ln{p(n)} • This implies that if the probability of occurrence of an event is low, then its occurrence conveys significant amount of information • If the probability of an event is high, the information conveyed by its occurrence is low GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 20 Average Information – Entropy • Average information conveyed by a set of events with probabilities p(i), i=1,2,…, is given by H = - Σ n p(n) log {p(n)} • H is called entropy and is extensively used in image processing operations • H is highest when all probabilities are equally likely. H max = - Σ n k.ln(k), where p(n) = k for all n • H is zero when p(j)=1 for some j, and p(k) = 0 for all k ≠ j GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 21 Role of Entropy • Indicator whether very few gray levels are actually present , or wide range of levels in sufficient numbers. • Entropy is also used for threshold selection • e.g., separating image into object of interest and background GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 22 Applications of histogram • Can be related to the discrete probability density function • The gray level corresponding to the highest frequency of occurrence is called the modal level, as seen in the histogram • If the image has two classes, the histogram may be bimodal with means µ 1 and µ 2 and standard deviations σ 1 and σ 2 respectively. GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 23 Applications of Histogram • A threshold or cutoff T between µ 1 and µ 2 • All gray levels below T – one class • Gray levels T and above – second class • Finding the value of T - Threshold selection. • Indicate the classes by 0 and 255 when displayed on the screen. GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 24 A Bimodal Histogram f(n) Peak 1 Width equal to σ Peak 2 Mode 1 Mode 2 =µ 1 =µ 2 n GNR607 Lecture 10 B. Krishna Mohan
IIT Bombay Slide 25 Image Statistics from Histogram • MIN gray level MIN = n: min n f(n) ≠0 • Max gray level MAX = n: max n f(n) ≠0 • Mean gray level µ = Σ n n.f(n) / (M.N) • Variance σ 2 = Σ n f(n)[n- µ ] 2 / (M.N) • Median Med = k: Σ k n=0 f(n) = (M.N)/2 GNR607 Lecture 10 B. Krishna Mohan
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