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![Distance Measures
• The Euclidean Distance between p and q is
defined as:
De (p,q) = [(x – s)2 + (y - t)2]1/2
Pixels having a distance less than or equal
to some value r from (x,y) are the points
contained in a disk of
radius r centered at (x,y)
p (x,y)
q (s,t)](https://image.slidesharecdn.com/mca-221031014909-23a3c317/75/mca-pptx-153-2048.jpg)
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mca.pptx
- 1. Digital Image Processing DrChethan H.K TEXT BOOK Digital Image Processing by Rafael C Gonzalez and Richard E Woods
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- 4. Description • Elephant isblack in colour • Gigantic • Has Four legs • A trunk and tusk
- 9. “One picture isworth more than ten thousand words” Visual Information is more powerful than textual information
- 10. Introduction • Digital imageplay an important role both in daily life applications such as – Satellite television – Magnetic resonance imagining – Geographical information system and astronomy – The term digital image generally refers to the manipulation /processing of an image by means of a processor/computer.
- 11. What is anImage? An image is a two dimensional function that represents a measure of some characteristics such as brightness or color of a viewed scene It Can be many things – photograph, CAD drawing, painting, a scene, a memory, and so on. An image is a projection of a 3D scene into a 2D projection plane.
- 12. Different types ofImages • Analog Image and Digital Image can be mathematically represented as a continuous range of values representing position and intensity example the television image. • A digital image is a two dimensional discrete signal (a matrix of small pixels and elements) For manipulating the images, there is a number of software and algorithms that are applied to perform changes
- 13. Why do weneed Image processing • It is motivated by three major applications –To improve pictorial information for human interpretation. –Image processing for autonomous machine application –Efficient storage and transmission
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- 24. • Now thesemethods mainly employ the different image processing techniques to enhance the pictorial information for human interpretation and analysis. • Typical applications of these kinds of techniques are noise filtering.
- 26. • To extractsome description or some features which can be used for further processing by a digital computer. And such a kind of processing can be applied in industrial machine vision, for product, assembly and inspection. • It can be used for automated target detection and tracking. • This can also be used for processing of aerial and satellite images, for weather prediction, crop assessment and many other applications. • This can be used for fingerprint recognition.
- 34. • We wantto process the image to reduce the space required to store that image or if want to transmit the image we should be able to transmit the image over a low bandwidth channel.
- 35. History of DigitalImage Processing •Early 1920s: One of the first applications of digital imaging was in the news- paper industry – The Bartlane cable picture transmission service – Images were transferred by submarine cable between London and New York – Pictures were coded for cable transfer and reconstructed at the receiving end on a telegraph printer Early digital image
- 36. • In early1920s image processing techniques were being used. And during those days, the image processing techniques or the digital images were used to transmit the newspapers pictures between London and New York. • And these digital pictures are carried by submarine cables; the system which was known as Bartlane Systems. • When you transmit these digital images via a submarine cable. • The transmitting side to have a facility for digitization of the image, similarly on the receiving side, to have a facility for reproduction of the image.. • The pictures are being reproduced by the telegraphic printers
- 37. History of DIP(cont…) •In 1921s: Improvements to the Bartlane system resulted in higher quality images – New reproduction processes based on photographic techniques – Increased number of tones in reproduced images Improved digital image
- 38. • And inthis case, on the receiver, instead of using the telegraphic printer, the digital images or the codes of the digital images were perforated on a tape and photographic printing was carried on using those tapes. • There are 2 images, the second image is obviously the image that we have shown in the earlier slide. • The first image is the image which has been produced using this photographic printing process. So here you find that the improvement both in terms of tonal quality as well as in terms of resolution is quite evident.
- 39. History of DIP(cont…) •1929s: Improvements to the Bartlane system resulted in higher quality images – New reproduction processes based on photographic techniques – Increased number of tones in reproduced images Improved digital image Early 15 tone digital image
- 40. History of DIP(cont…) •1960s: Improvements in computing technology and the onset of the space race led to a surge of work in digital image processing – 1964: Computers used to improve the quality of images of the moon taken by the Ranger 7 probe – Such techniques were used in other space missions including the Apollo landings A picture of the moon taken by the Ranger 7 probe minutes before landing
- 41. History of DIP(cont…) •1970s: Digital image processing begins to be used in medical applications – 1979: Sir Godfrey N. Hounsfield & Prof. Allan M. Cormack share the Nobel Prize in medicine for the invention of tomography, the technology behind Computerised Axial Tomography (CAT) scans Typical head slice CAT image
- 42. History of DIP(cont…) •1980s - Today: The use of digital image processing techniques has exploded and they are now used for all kinds of tasks in all kinds of areas – Image enhancement/restoration – Artistic effects – Medical visualisation – Industrial inspection – Law enforcement – Human computer interfaces
- 43. Examples: Image Enhancement •Oneof the most common uses of DIP techniques: improve quality, remove noise etc
- 44. Examples: The HubbleTelescope •Launched in 1990 the Hubble telescope can take images of very distant objects •However, an incorrect mirror made many of Hubble’s images useless •Image processing techniques were used to fix this
- 45. Examples: Artistic Effects •Artisticeffects are used to make images more visually appealing, to add special effects and to make composite images
- 46. Examples: Medicine •Take slicefrom MRI scan of canine heart, and find boundaries between types of tissue – Image with gray levels representing tissue density – Use a suitable filter to highlight edges Original MRI Image of a Dog Heart Edge Detection Image
- 47. Examples: GIS •Geographic InformationSystems – Digital image processing techniques are used extensively to manipulate satellite imagery – Terrain classification – Meteorology
- 48. Examples: GIS (cont…) •Night-TimeLights of the World data set – Global inventory of human settlement – Not hard to imagine the kind of analysis that might be done using this data
- 49. Examples: Industrial Inspection •Humanoperators are expensive, slow and unreliable •Make machines do the job instead •Industrial vision systems are used in all kinds of industries
- 50. Examples: Law Enforcement •Imageprocessing techniques are used extensively by law enforcers – Number plate recognition for speed cameras/automated toll systems – Fingerprint recognition – Enhancement of CCTV images
- 51. Examples: Law Enforcement •Imageprocessing techniques are used extensively by law enforcers – Number plate recognition for speed cameras/automated toll systems – Fingerprint recognition – Enhancement of CCTV images
- 52. Image Representation Digital imageis composed of M rows and N columns of pixels each storing a value Pixel values are most often grey levels in the range 0-255(black-white)
- 53. A digital imageis a representation of a two- dimensional image as a finite set of digital values, called picture elements or pixels
- 54. Definition • An imagemay be defined as a two dimensional function f(x,y) • Where x and y are spatial coordinates • Digital image processing refers to processing digital images by means of digital computer
- 55. Several fields dealwith images • Computer Graphics : the creation of images. • Image Processing : the enhancement or other manipulation of the image – the result of which is usually another images. • Computer Vision: the analysis of image content.
- 56. What is DIP? Thecontinuum from image processing to computer vision can be broken up into low-, mid- and high-level processes Low Level Process Input: Image Output: Image Examples: Noise removal, image sharpening Mid Level Process Input: Image Output: Attributes Examples: Object recognition, segmentation High Level Process Input: Attributes Output: Understanding Examples: Scene understanding, autonomous navigation
- 57. 3 types ofcomputerized process • Low-level : input, output are images Primitive operations such as image preprocessing to reduce noise, contrast enhancement, and image sharpening • Mid-level : inputs may be images, outputs are attributes extracted from those images Segmentation, Description of objects, Classification of individual objects • High-level : Image analysis
- 58. How are imagesrepresented in the computer?
- 59. Image Types RGB GrayscaleBlack & White
- 60. In digital imaging,a pixel, or pel, (picture element) is a single point in a raster image, or the smallest addressable screen element in a display device; it is the smallest unit of picture that can be represented or controlled. Each pixel has its own address. The address of a pixel corresponds to its coordinates. Basics about Pixels
- 61. A megapixel (MPor Mpx) is one million pixels, and is a term used not only for the number of pixels in an image, but also to express the number of image sensor elements of digital cameras or the number of display elements of digital displays. For example, a camera with an array of 2048 × 1536 sensor elements is commonly said to have "3.1 megapixels" (2048 × 1536 = 3,145,728). Basic about Pixels (contd…)
- 74. Common image fileformats • GIF (Graphic Interchange Format) - • PNG (Portable Network Graphics) • JPEG (Joint Photographic Experts Group) • TIFF (Tagged Image File Format) • PGM (Portable Gray Map) • FITS (Flexible Image Transport System)
- 75. Key Stages inDigital Image Processing Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 76. Image Acquisition • Itis the first process • The input image will be in digital form • It involves acquisition of source from real word images and storing them in computer for further pre-processing such as scaling.
- 77. Key Stages inDigital Image Processing: Image Aquisition Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 78. Image Enhancement • Processof manipulating an image so that the result is more suitable than the original for a specific application • Enhancement techniques are problem oriented • Means a method which is suitable for enhancing x ray Images may not be the best approach for enhancing satellite images
- 79. Key Stages inDigital Image Processing: Image Enhancement Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 80. Image Restoration • Whichdeals with improving the appearance of an image. • Restoration techniques are based on mathematical or probabilistic models of image degradation. • Where as enhancement is based on human subjective preferences
- 81. Key Stages inDigital Image Processing: Image Restoration Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 82. Morphological Processing • Dealswith tools for extracting image components that are useful in the representation and description of shape such as boundaries skeleton etc. • Input are images and output are attributes extracted from those images
- 83. Key Stages inDigital Image Processing: Morphological Processing Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 84. Segmentation • Partition ofan image into its constituent parts or objects • It is one of the most difficult stages in DIP • More accurate the segmentation the more likely the recognition to succeed
- 85. Key Stages inDigital Image Processing: Segmentation Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 86. Representation & Description •It is the output of the segmentation stage which usually is raw pixel data consisting either the boundary of a region • Description also called feature selection deals with extracting attributes that result in some quantitative information of interest or are basic differentiating once class of object from another
- 87. Key Stages inDigital Image Processing: Representation & Description Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 88. Object Recognition • Theprocess that assigns a label (eg Vehicle ) to an object based on its descriptors. • A pattern class is a family of patterns that share some common properties • Pattern classes are denoted by w1,w2….Ww Where W is the no of class patterns • Grouping into separate classes
- 89. Key Stages inDigital Image Processing: Object Recognition Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 90. Image Compression • Thetechnique deals with reducing the storage required to save an image or the bandwidth required to transmit it. Colour Image Processing • Use of color images has been gaining importance because of significant increase in the use of digital images over the internet
- 91. Key Stages inDigital Image Processing: Image Compression Image Acquisition Image Restoration Morphological Processing Segmentation Representation & Description Image Enhancement Object Recognition Problem Domain Colour Image Processing Image Compression
- 92. Components of anImage Processing System Network Image displays Computer Mass storage Hardcopy Specialized image processing hardware Image processing software Image sensors Problem Domain Typical general- purpose DIP system
- 93. Components of anImage Processing System 1. Image Sensors Two elements are required to acquire digital images. The first is the physical device that is sensitive to the energy radiated by the object we wish to image (Sensor). The second, called a digitizer, is a device for converting the output of the physical sensing device into digital form.
- 94. 2. Specialized ImageProcessing Hardware Usually consists of the digitizer, mentioned before, plus hardware that performs other primitive operations, such as an arithmetic logic unit (ALU), which performs arithmetic and logical operations in parallel on entire images. This type of hardware sometimes is called a front-end subsystem, and its most distinguishing characteristic is speed. In other words, this unit performs functions that require fast data throughputs that the typical main computer cannot handle.
- 95. 3. Computer The computerin an image processing system is a general- purpose computer and can range from a PC to a supercomputer. In dedicated applications, sometimes specially designed computers are used to achieve a required level of performance.
- 96. 4. Image ProcessingSoftware Software for image processing consists of specialized modules that perform specific tasks. A well-designed package also includes the capability for the user to write code that, as a minimum, utilizes the specialized modules.
- 97. 5. Mass StorageCapability Mass storage capability is a must in a image processing applications. An image of sized 1024 * 1024 pixels requires one megabyte of storage space if the image is not compressed. Digital storage for image processing applications falls into three principal categories: 1. Short-term storage for use during processing. 2. on line storage for relatively fast recall 3. Archival storage, characterized by infrequent access
- 98. 6. Image Displays Thedisplays in use today are mainly color (preferably flat screen) TV monitors. Monitors are driven by the outputs of the image and graphics display cards that are an integral part of a computer system.
- 99. 7. Hardcopy devices Usedfor recording images, include laser printers, film cameras, heat-sensitive devices, inkjet units and digital units, such as optical and CD-Rom disks.
- 100. 8. Networking Is almosta default function in any computer system, in use today. Because of the large amount of data inherent in image processing applications the key consideration in image transmission is bandwidth. In dedicated networks, this typically is not a problem, but communications with remote sites via the internet are not always as efficient.
- 101. Image Sampling andquantization • To create a digital image, we need to convert the continuous sensed data into digital form. This process includes 2 processes: – Sampling: Digitizing the co-ordinate value is called sampling. – Quantization: Digitizing the amplitude value is called quantization.
- 102. Image Sampling AndQuantisation A digital sensor can only measure a limited number of samples at a discrete set of energy levels Quantisation is the process of converting a continuous analogue signal into a digital representation of this signal
- 103. Analogue Vs digital •Thinkof the volume knob on your stereo •Records Vs CDs Vs Mp3s Remember that a digital image is always only an approximation of a real world scene Image Sampling And Quantisation
- 104. Image Sampling &Quantisation To create a digital image, we need to convert continuous sensed data into digital form. •This involves two processes: sampling and quantisation •The basic idea behind sampling and quantization is illustrated in Fig. 3.1.
- 105. Fig 3.1 Generatinga digital image (a) Continuous image. (b) A scan line from A to B in the continuous image. (c) Sampling & quantisation. (d) Digital scan line.
- 106. • Figure 3.1(a)shows a continuous image, f (x, y), that we want to convert to digital form. • To convert it to digital form, we have to sample the function in both coordinates and in amplitude. • An image may be continuous with respect to the x- and y-coordinates and also in amplitude.
- 107. • The one-dimensionalfunction shown in Fig. 3.1(b) is a plot of amplitute (gray level) values of the continuous image along the line segment AB in Fig. 3.1(a). • To sample this function, we take equally spaced samples along line AB, as shown in Fig. 3.1(c). • Location of each sample is given by a vertical tick mark in the bottom part of the figure.
- 108. • The samplesare shown as small white squares superimposed on the function. The set of these discrete locations gives the sampled function. • However, the values of the samples still span (vertically) a continuous range of gray-level values. • In order to form a digital function, the gray-level values also must be converted (quantized) into discrete quantities.
- 109. Fig. 3.2 (a)Continuous image projected onto a sensor array. (b) Result of image sampling and quantisation
- 110. Representing Digital Images •Let f(s,t) represent a continuous image function of two continuous variables s and t. • The result of sampling and quantisation is a matrix of real numbers. • The values of the coordinates at the origin are (x,y) = (0,0). • The next coordinate values along the first row are (x,y) = (0,1). • The notation (0,1) is used to signify the 2nd sample along the 1st row.
- 111. Representing digital images Fig.3.3. Coordinate convention used to represent digital images
- 113. A digital imageof size M x N
- 114. • It isadvantageous to use a more traditional matrix notation to denote a digital image and its elements. Fig. 3.5 A digital image
- 115. • The numberof bits required to store a digitised image is • b = M x N x k Where M & N are the number of rows and columns, respectively. • The number of gray levels is an integer power of 2: • L = 2k where k =1,2,…24 • It is common practice to refer to the image as a “k-bit image”
- 116. Spatial and Intensityresolution • The spatial resolution of an image is the physical size of a pixel in that image; i.e., the area in the scene that is represented by a single pixel in that image. • Dense sampling will produce a high resolution image in which there are many pixels, each of which represents of a small part of the scene. • Coarse sampling, will produce a low resolution image in which there are a few pixels, each of which represents of a relatively large part of the scene.
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- 118. • Spatial andgray-level resolution
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- 123. • N andk in different-details images
- 124. Interpolation • It isa basic tool used extensively in task such as Zooming, shrinking, rotating and geometric corrections. • Interpolation is the process of using known data to estimate values at unknown locations. • Bilinear interpolation in which we use the four nearest neighbour to estimate the intensity at a given location • Bicubic interpolation which involves the sixteen nearest neighbour of a point.
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- 126. Neighborhoods of aPixels • A Pixel ‘P’ at location (x,y) has two horizontal and two vertical neighbors • This set of four pixels is called 4-neighbor of P=N4(p) • Each of these neighbors is at a unit distance from P • If P is boundary pixel it will have less number of neighbors (x-1,y) (x,y-1) P (x,y) (x,y+1) (x+1,y)
- 127. Diagonal and 8-neighbors •A pixels ‘P’ has four diagonal neighbors = ND(P) • The points of N4(p) and ND(p) together are called 8-neighbors of P N8(p)=N4(p) U ND(p) • If p is a boundary pixel then both ND(p) and N8(p) will have less number of pixels (x-1,y-1) (x-1,y+1) P (x,y) (x+1,y-1) (x+1,y+1)
- 128. • Neighbors ofa pixel – 4-neighbors – diagonal-neighbors – 8-neighbors • Adjacency – 4-adjacency – 8-adjacency – m-adjacency (i-1,j-1) (i-1,j) (i- 1,j+1) (i,j-1) (i,j) (i,j+1) (i+1,j- 1) (i+1,j) (i+1,j+1 )
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- 130. Outline • In thislecture, we consider several important relationships between pixels in a digital image. – Neighborhood – • Adjacency – • Connectivity – • Paths – • Regions and boundaries
- 131. Metric and topologicalproperties of digital images • A digital image consists of picture elements with finite size-these pixels carry information about the brightness of a particular location in the image. • pixels are arranged into a rectangular sampling grid. • Such a digital image is represented by a two- dimensional matrix whose elements are natural numbers corresponding to the quantization levels in the brightness scale.
- 132. Neighbors of aPixel • A pixel p at coordinates (x,y) has two horizontal and two vertical neighbors whose coordinates are given by: (x+1,y), (x-1, y), (x, y+1), (x,y-1) This set of pixels, called the 4-neighbors or p, is denoted by N4(p). Each pixel is one unit distance from (x,y) and some of the neighbors of p lie outside the digital image if (x,y) is on the border of the image. (x, y-1) (x-1, y) P (x,y) (x+1, y) (x, y+1)
- 133. Neighbors of aPixel • The four diagonal neighbors of p have coordinates: (x+1, y+1), (x+1, y-1), (x-1, y+1), (x-1, y-1) and are denoted by ND (p). These points, together with the 4-neighbors, are called the 8-neighbors of p, denoted by N8 (p). As before, some of the points in ND (p) and N8 (p) fall outside the image if (x,y) is on the border of the image. (x-1, y+1) (x+1, y-1) P (x,y) (x-1, y-1) (x+1, y+1) (x-1, y+1) (x, y-1) (x+1, y-1) (x-1, y) P (x,y) (x+1, y) (x-1, y-1) (x, y+1) (x+1, y+1)
- 134. Adjacency and Connectivity •Let V: a set of intensity values used to define adjacency and connectivity. • In a binary image, V = {1}, if we are referring to adjacency of pixels with value 1. • In a gray-scale image, the idea is the same, but V typically contains more elements, for example, V = {180, 181, 182, …, 200} • If the possible intensity values 0 – 255, V set can be any subset of these 256 values.
- 135. Types of Adjacency 1.4-adjacency: Two pixels p and q with values from V are 4-adjacent if q is in the set N4(p). 2. 8-adjacency: Two pixels p and q with values from V are 8-adjacent if q is in the set N8(p). 3. m-adjacency =(mixed)
- 136. Types of Adjacency •m-adjacency: Two pixels p and q with values from V are m- adjacent if : • q is in N4(p) or • q is in ND(p) and the set N4(p) ∩ N4(q) has no pixel whose values are from V (no intersection) • Important Note: the type of adjacency used must be specified
- 137. Types of Adjacency •Mixed adjacency is a modification of 8- adjacency. It is introduced to eliminate the ambiguities that often arise when 8-adjacency is used. • For example:
- 138. Types of Adjacency •In this example, we can note that to connect between two pixels (finding a path between two pixels): – In 8-adjacency way, you can find multiple paths between two pixels – While, in m-adjacency, you can find only one path between two pixels • So, m-adjacency has eliminated the multiple path connection that has been generated by the 8-adjacency. • Two subsets S1 and S2 are adjacent, if some pixel in S1 is adjacent to some pixel in S2. Adjacent means, either 4-, 8- or m-adjacency.
- 139. Example 4-Connectivity • Setof color consists of color 1 ; C ={1} 1 1 1 0 1 0 0 0 0 1 1 1
- 140. Example 8-Connectivity • Setof color consists of color 1 ; C ={1} 1 1 1 0 1 0 0 0 0 1 1 1
- 141. Example M-Connectivity • Setof color consists of color 1 ; C ={1} 1 1 1 0 1 0 0 0 0 1 1 1
- 142. Labeling of connectedComponents • Scan an image pixel by pixel from left to right and top to bottom • There are 2 types of connectivity interested 4-connectivity 8-connectivity • Equivalent labeling
- 143. Step: 4-connected components •P is pixel scanned process • If pixel p is color value 0 move on the next scanning • If pixel p is color value 1 examine pixel top and left – If top and left were 0, assign a new label to p – If only one of them’re 1, assign its label to p – If both of them’re 1 and have • the same number, assign their label to p • Different number, assign label of top to p and make note that two label is equivalent • Sort all pairs of equivalent labels and assign each equivalent to be same type
- 144. Example 4-connected components 11 1 0 1 0 0 0 0 1 1 1 1 1 2 1 3 4 3 1 2 3 4 Equivalent Table
- 145. Step: 8-connected components •Steps are same as 4-connected components • But the pixel that are consider is 4 previous pixels ( top-left, top, top-right, and left )
- 146. Example 8-connected components 1 1 1 0 1 0 0 00 1 1 1 1 1 1 1 1 1 1
- 147. A Digital Path •A digital path (or curve) from pixel p with coordinate (x,y) to pixel q with coordinate (s,t) is a sequence of distinct pixels with coordinates (x0,y0), (x1,y1), …, (xn, yn) where (x0,y0) = (x,y) and (xn, yn) = (s,t) and pixels (xi, yi) and (xi-1, yi-1) are adjacent for 1 ≤ i ≤ n • n is the length of the path • If (x0,y0) = (xn, yn), the path is closed. • We can specify 4-, 8- or m-paths depending on the type of adjacency specified.
- 148. A Digital Path •Return to the previous example: In figure (b) the paths between the top right and bottom right pixels are 8-paths. And the path between the same 2 pixels in figure (c) is m-path
- 149. Connectivity • Let Srepresent a subset of pixels in an image, two pixels p and q are said to be connected in S if there exists a path between them consisting entirely of pixels in S. • For any pixel p in S, the set of pixels that are connected to it in S is called a connected component of S. If it only has one connected component, then set S is called a connected set.
- 150. Region and Boundary •Region Let R be a subset of pixels in an image, we call R a region of the image if R is a connected set. • Boundary The boundary (also called border or contour) of a region R is the set of pixels in the region that have one or more neighbors that are not in R.
- 151. Region and Boundary IfR happens to be an entire image, then its boundary is defined as the set of pixels in the first and last rows and columns in the image. This extra definition is required because an image has no neighbors beyond its borders Normally, when we refer to a region, we are referring to subset of an image, and any pixels in the boundary of the region that happen to coincide with the border of the image are included implicitly as part of the region boundary.
- 152. Distance Measures • Forpixels p, q and z, with coordinates (x,y), (s,t) and (v,w), respectively, D is a distance function if: (a) D (p,q) ≥ 0 (D (p,q) = 0 iff p = q), Identity (b) D (p,q) = D (q, p), and Symmetry (c) D (p,z) ≤ D (p,q) + D (q,z). Triangular inequality
- 153. Distance Measures • TheEuclidean Distance between p and q is defined as: De (p,q) = [(x – s)2 + (y - t)2]1/2 Pixels having a distance less than or equal to some value r from (x,y) are the points contained in a disk of radius r centered at (x,y) p (x,y) q (s,t)
- 154. Distance Measures • TheD4 distance (also called city-block distance) between p and q is defined as: D4 (p,q) = | x – s | + | y – t | Pixels having a D4 distance from (x,y), less than or equal to some value r form a Diamond centered at (x,y) p (x,y) q (s,t) D4
- 155. Distance Measures Example: The pixelswith distance D4 ≤ 2 from (x,y) form the following contours of constant distance. The pixels with D4 = 1 are the 4-neighbors of (x,y)
- 156. Distance Measures • TheD8 distance (also called chessboard distance) between p and q is defined as: D8 (p,q) = max(| x – s |,| y – t |) Pixels having a D8 distance from (x,y), less than or equal to some value r form a square Centered at (x,y) p (x,y) q (s,t) D8(b) D8(a) D8 = max(D8(a) , D8(b))
- 157. Distance Measures Example: D8 distance≤ 2 from (x,y) form the following contours of constant distance.
- 158. Distance Measures • Dmdistance: is defined as the shortest m-path between the points. In this case, the distance between two pixels will depend on the values of the pixels along the path, as well as the values of their neighbors.