IMAGE PROCESSING

Image Processing: Transforms, Filters and ApplicationsUnder Guidance Of:Dr. K RajAssociate ProfessorElectronics Engineering Dept.H.B.T.I-Kanpur

Submitted By:Ankur Singh (179/12)Deepak Dubey (191/12)Lecture OutlineHistorical DevelopmentIntroduction of 2D signalTransform of 2D signal 2D fourier transform 2D z transformSampling theorem in 2 Dsampling and aliasing of a image2D filterGeneral introduction of image processingFuture applicationReference

History and DevelopmentInitial ideas back to 1920 for cable transmission of pictures.First Computer Processing introduced about 1964 at JPL Used in images from Ranger-7 video images.Early work limited to space projects, due to cost of computer systems and especially display systemsAdvancements I Early/Mid 1980: Graphics workstations, (SUN, Apollo, VaxStation) Start of industrial inspection, scientific imaging, computer vision.

Advancements II Late 1980/early 1990s Supercomputer graphics workstations, (Sun 10/40, HP-9000, Dec-Alpha, Silicon-Graphics) Start of all-digital publishing, image processing going outside the research lab.

Advancements III 2000s The PC comes of age. Modern Pentium machines have power and memory of Supercomputer graphics workstations. Many big image processing packages written, or ported to PC. Vast growth in digital photography, all PC based. Digital imaging to PC systems now routine in many scientific applications

2003-onwards Digital imaging makes the mass market. Digital camera in mobiles, PDA. Image processing packages come free on PC. Self-service digital image printing and enhancement. All video and TV going digital.What is a transform?Transforms are decompositions of a function f(x) into some basis functions (x, u). u is typically the freq. index.

Illustration of Decomposition f = 11+22+33

Introduction of 2D signal1D signal has one independent variable - f(t)2D signal has two independent variables - f(x,y)Concepts of linearity, spectra, filtering, etc, carry over from 1-D. But concept of causality not relevant as image is a function of space, not time.2-D systems are more complex, e.g we can factor 1-D polynomials into a product of 1st and 2nd order polynomial, and thus study stability and system response. Stability for a 1-D system can be determined from system poles, but for 2-D system, poles are surfaces in 4-D space.2-D algorithms can offer more flexibility implementation, e.g one can process image data in a non-causal way, in parallel. We now extend 1-D results to 2-D where possible.A 2-D analog signal is a function of 2 continuous variables. A 2-D pulse is dened by :PA(x, y)={1 x, y A 0 x, y / A

It can be represented as product of two 1-D impulses:(x, y)=(x).(y)Response of a system to 2-D impulse is termed point-spread function.2D Fourier AnalysisIdea is to represent a signal as a sum of pure sinusoids of different amplitudes and frequencies.

In 1D the sinusoids are defined by frequency and amplitude.

In 2D these sinusoids have a direction as well e.g. f (x, y) = a cos(1 x + 2 y + )Spectral Analysis / Fourier TransformsNote the Euler formula : eit = cost + i sin t

The Fourier transform converts between the spatial and frequency domain.Real and imaginary components.Forward and reverse transforms very similar.

How do 1, 2 relate to direction?

In the direction : f() =a cos(0 + ), where is just some phase lag.

Given any point (x, y), = x cos +y sin Therefore:f(x, y) = a cos(0[x cos + y sin ] + )

Compare this withf(x, y) = a cos(1x + 2y + ) (1)

Therefore 1 = 0 cos and 2 =0 sin Here a = 1.0; = 20; =0 sin Here a = 1.0; = 20; =45; 0 = 0.05cycles per pel.The 2D Fourier Transform

The 2D Fourier Transform is separable!Can do 1D transform of rows rst then do 1-D transform of result along columns. Or vice-versa.Fourier transform of separable signals is also separable. Let f(x, y) = fx(x)fy (y) then F (1, 2) = Fx(1)Fy(2). [Note that this is dierent from convolution identity because the signal is SEPARABLE!].

The 2D z-transformRecall 1D Z Transform of signal xn

Just a polynomial in z (a complex number). Used to solve dierence equations. Also helps with stability of IIR lters. Z-transform of a sequence g(h, k) is denoted G(z1, z2).

The z1 and z2 components act on the sequence p(h, k) along vertical and horizontal directions in a sense. Z transform in 2D is a function of 2 complex numbers z1, z2. It exists in a 4D space so is very dicult to visualise. Z-transform of a sequence is related to the spectral content of the sequence as follows: G(1, 2) = G(ej1 , ej2 )

Sampling TheoryHow many samples are required to represent a given signal without loss of information?What signals can be reconstructed without loss for a given sampling rate?

A signal can be reconstructed from its samples, if the original signal has no frequencies above 1/2 the sampling frequency ShannonThe minimum sampling rate for bandlimited function is called Nyquist rate2D Sampling TheoremFor no overlap in the spectra to occur, 0 1 > 1 and 0 2 > 2. Thus we get Nyquists theorem for 2D (its similar to 1D).

for no aliasing to occur.

Sampling and Reconstruction

Aliasing (in general)In general: Artifacts due to under-sampling or poor reconstructionSpecifically, in graphics: Spatial aliasing Temporal aliasing

Sampling & AliasingReal world is continuousThe computer world is discreteMapping a continuous function to a discrete one is called samplingMapping a continuous variable to a discrete one is called quantizaionTo represent or render an image using a computer, we must both sample and quantizeAntialiasing(removing aliasing)Sample at higher rateNot always possibleDoesnt always solve problemPre-filter to form bandlimited signalForm bandlimited function (low-pass filter)Trades aliasing for blurringConvolve with sinc function in space domainOptimal filter - better than area sampling.Sinc function is infinite !!Computationally expensive.Cheaper solution : take multiple samples for each pixel and average them together supersampling.Can weight them towards the centre weighted average sampling

How is antialiasing done?We need some mathematical tools toanalyse the situation.find an optimum solution.Tools we will use :Fourier transform.Convolution theory.Sampling theory.

We need to understand the behavior of the signal in frequency domain2D Filter2D filter are used to process two dimensional signals such as a image. In 2d multidimensional filter can t be factored in polynomials in general hence by manipulation of transfer funtion coefficient required by particular network structure cannot be determined as it is case in 1D. In 2D FIR filter is implemented by using a non-recursive algorithm where as IIR filter is implemented by using recursive feedback algorithm structure.

Images and Digital ImagesA digital image differs from a photo in that the values are all discrete.Usually they take on only integer values.A digital image can be considered as a large array of discrete dots, each of which has a brightness associated with it. These dots are called picture elements, or more simply pixels.The pixels surrounding a given pixel constitute its neighborhood A neighborhood can be characterized by its shape in the same way as a matrix: we can speak of a 3x3 neighborhood, or of a 5x7 neighborhood.Recall: a pixel is a pointIt is NOT a box, disc or teeny wee lightIt has no dimensionIt occupies no areaIt can have a coordinateMore than a point, it is a SAMPLE

Aspects of Image ProcessingImage Enhancement: Processing an image so that the result is more suitable for a particular application. (sharpening or de-blurring an out of focus image, highlighting edges, improving image contrast, or brightening an image, removing noise)Image Restoration: This may be considered as reversing the damage done to an image by a known cause. (removing of blur caused by linear motion, removal of optical distortions)Image Segmentation: This involves subdividing an image into constituent parts, or isolating certain aspects of an image.(finding lines, circles, or particular shapes in an image, in an aerial photograph, identifying cars, trees, buildings, or roads.Types of Digital ImagesBinary: Each pixel is just black or white. Since there are only two possible values for each pixel (0,1), we only need one bit per pixel.Grayscale: Each pixel is a shade of gray, normally from 0 (black) to 255 (white). This range means that each pixel can be represented by eight bits, or exactly one byte. Other greyscale ranges are used, but generally they are a power of 2.True Color, or RGB: Each pixel has a particular color; that color is described by the amount of red, green and blue in it. If each of these components has a range 0255, this gives a total of (256)3 different possible colors. Such an image is a stack of three matrices; representing the red, green and blue values for each pixel. This means that for every pixel there correspond 3 values.Binary Image

BLACK0WHITE1Grayscale Image

Color Image

General Commandsimread: Read an imagefigure: creates a figure on the screen.imshow(g): which displays the matrix g as an image.pixval on: turns on the pixel values in our figure.impixel(i,j): the command returns the value of the pixel (i,j)iminfo: Information about the image.Spatial ResolutionSpatial resolution is the density of pixels over the image: the greater the spatial resolution, the more pixels are used to display the image.Halve the size of the image: It does this by taking out every other row and every other column, thus leaving only those matrix elements whose row and column indices are even.Double the size of the image: all the pixels are repeated to produce an image with the same size as the original, but with half the resolution in each direction.HistogramsGiven a grayscale image, its histogram consists of the histogram of its gray levels; that is, a graph indicating the number of times each gray level occurs in the image.We can infer a great deal about the appearance of an image from its histogram.In a dark image, the gray levels would be clustered at the lower endIn a uniformly bright image, the gray levels would be clustered a the upper end.In a well contrasted image, the gray levels would be well spread out over much of the range.Frequencies; Low and High Pass FiltersFrequencies are the amount by which grey values change with distance.High frequency components are characterized by large changes in grey values over small distances; (edges and noise)Low frequency components are parts characterized by littlechange in the gray values. (backgrounds, skin textures)High pass filter: if it passes over the high frequency components, and reduces or eliminates low frequency components.Low pass filter: if it passes over the low frequency components, and reduces or eliminates high frequency components.NoiseNoise is any degradation in the image signal, caused by external disturbance.Salt and pepper noise: It is caused by sharp, sudden disturbances in the image signal; it is randomly scattered white or black (or both) pixels. It can be modeled by random values added to an imageGaussian noise: is an idealized form of white noise, which is caused by random fluctuations in the signal.Speckle noise: It is a major problem in some radar applications. It can be modeled by random values multiplied by pixel values.Salt & Pepper Noise

Gaussian Noise

Speckle Noise

Color ImagesA color model is a method for specifying colors in some standard way. It generally consists of a 3D coordinate system and a subspace of that system in which each color is represented by a single point.RGB: In this model, each color is represented as 3 values R, G and B, indicating the amounts of red, green and blue which make up the color. HSV: Hue: The true color attribute (red, green, blue, orange, yellow, and so on). Saturation: The amount by which the color as been diluted with white. The more white in the color, the lower the saturation. Value: The degree of brightness: a well lit color has high intensity; a dark color has low intensity.Color Image

Color Conversion

Applications of Image ProcessingRemote Sensing: satellite of aircraft images for earth resource, weather, sea surface, etc.Inspection and Automation: robotic control, manufacture control, quality inspection, safety monitoring.Medical Imaging: X-ray, Computer tomography, MRI, PET, g-camera, thermal-IR, sample inspection.Astronomical Applications: main observation tool, photon camera, radio image formation, aperture synthesis, radio interferometry.Scientific: microscope sample analysis, confocal imaging, x-ray analysis, surface inspection,STM, AFM, etc.Data Compression: document storage, data reduction, JPEG/MPEG, digital image transmission.Communications: video telephone, multi-media computer links, document transmission, secure data links.Military Applications: target tracking, surveillance, smart weapons, automated guidance, secure data links.

Referenceshttp://www.cs.rit.edu/~jmg/cgIIhttp://www.nbb.cornell.edu/neurobio/land/OldStudentProjects/cs49096to97/ans/index.htmlhttp://www.jhu.edu/~Esignals/convolve/index.htmlhttp://ptolemy.eecs.berkeley.edu/eecs20/week13/aliasing.htmlJ.S. sim,Two Dimensional Signal and Image Processing,Prentice Hall International,1990