Review for Exam I, EE552 2/2009

Transcription

1 Gonale & Woods Review or Eam I, EE55 /009 Elements o Visual Perception Image Formation in the Ee and relation to a photographic camera). Brightness Adaption and Discrimination. Light and the Electromagnetic Spectrum Gamma ras, visible spectrum Violet Inrared), Radio waves. Visible spectrum spans the range rom ~0.4 µm violet) to about ~0.7 µm red). Light that does not have color is called monochromatic light i.e. each band o an RGB image). Attribute o monochromatic light is intensit. Chromatic color spans the range ~0.4 µm through ~0.7 µm. Image Sensing and Acquisition Depending on the source, illumination is relected rom, or transmitted through objects. Principal sensor arrangements used to transorm energ into digital images: single sensor, line sensor, arra sensor. Image acquisition using arra sensors. A simple image ormation model:, = i, r,. The relectance value is bounded b 0 total absorption) and1 total relectance) R. C. Gonale & R. E. Woods

2 Gonale & Woods Review or Eam I, EE55 /009 Image Sampling and Quantiation To create a digital image we need to convert the continuous sensed data into digital orm using two processes: sampling and quantiation. Digitiing the coordinates is sampling, digitiing the amplitude is called quantiation. Representing Digital Images. A real plane spanned b the coordinates o an image is called the spatial domain. The number o intensit levels in an image is L= k. The number o bits required to store a digitied image is b=m N k Image interpolation: nearest neighbors, bilinear v, = a b c d), bicubic interpolation 3 3 i j v, = aij ). i = 0 j = 0 Basic Relationships Between Piels A piel has 4-neighbors, diagonal neighbors, and 8-neighbors. Two piels can be 4-adjacent, 8-adjacent, and m-adjacent. Distance measures: Euclidean D p, q) = [ s) t) ] 1/ ), cit-block distance D p, q) = s t ), chessboard distance D p, q) = ma s t ) ) R. C. Gonale & R. E. Woods

3 Gonale & Woods Review or Eam I, EE55 /009 Mathematical Tools Used in DIP Arra versus matri operations. Linear versus nonlinear operations. Arithmetic operations: summation, subtraction, multiplication, division between corresponding piels. Applications o arithmetic operations: Reduction/removal o noise in a corrupted noise noise uncorrelated and ero mean). Shading correction multiplication b the inverse o the shading unction h,). Masking, also called region o interest ROI) operations. Scaling o images linear). Basic set and logical operations: A is a subset o B A B ), intersection A B ), union A B). Spatial operations: single-piel operations transormation unctions), neighborhood operations involves a neighborhood o m n piels), geometric spatial transormations scaling, rotation, translation, shear vertical) and shear horiontal)) R. C. Gonale & R. E. Woods

4 Gonale & Woods Review or Eam I, EE55 /009 Vector and Matri Operations Vector and matri operations are routinel used in multispectral image processing. Euclidean distance between a piel vector and an arbitrar point a in n-dimensional space is T D, a) = [ a) a)] 1/ Image transorms. Image processing tasks are best ormulated b transorming the input image, carring a speciied task, and then appling the inverse transorm. Forward and inverse transorms can be separable r,, u, = r1, u) r, ) and smmetric r,, u, = r1, u) r1, ) R. C. Gonale & R. E. Woods

5 Gonale & Woods Review or Eam I, EE55 /009 Intensit Transormation and Spatial Filtering Basic spatial domain process is g, = T[, ]. Intensit gra-level or mapping) transormation unction s = T r). Image negatives are obtained using the negative transormation s = L 1 r. Log transormations have the orm s = c log 1 r). γ Power-Law Gamma) transormations s = cr. Contrast stretching is used to epand the range o intensit levels in an image. Intensit-level slicing is used to highlight a speciic range o intensities in an image. Histogram Processing Transormation intensit mapping) o the orm s = T r) 0 r L 1. k A transormation unction o particular importance in DIP is sk = T rk ) = L 1) pr r j) which j = 0 perorms a histogram equaliation or histogram lineariation transormation. Histogram matching is used to generate a processed image that has a speciied histogram k q 1 sk = T rk ) = L 1) pr r j) and G q) = sk = L 1) p i) and q = G sk ). j = 0 i = 0 Global histogram processing can be adapted to local enhancement local histogram processing). Local mean and variance can be used to change an image based on local characteristics in a neighborhood S,, i.e. change the intensit o a piel i the local mean is larger/smaller than global mean R. C. Gonale & R. E. Woods

6 Gonale & Woods Review or Eam I, EE55 /009 Fundamentals o Spatial Filtering A spatial ilter consists o 1) a neighborhood, and ) a predeined operation. Spatial correlation and convolution. Correlation is the process o moving a ilter mask over the image and computing the sum o products. Convolution consists in a similar process but the ilter is irst rotated 180. T Vector representation o linear iltering; R = w, where w are the coeicients o the ilter and are the corresponding image intensities. Smoothing Spatial Filters Output o a smoothing, linear spatial ilter is simpl the average o the piels in a neighborhood. It is computationall more eicient to have coeicients valued 1, and then at the end o the process divide b a constant. Order-statistic nonlinear) ilters are based on ordering ranking) the piels in a neighborhood like the median ilter. Median ilters are eective in the presence o impulse noise R. C. Gonale & R. E. Woods

7 R. C. Gonale & R. E. Woods Gonale & Woods Sharpening Spatial Filters Sharpening can be accomplished b spatial dierentiation. The Laplacian is a sharpening ilter that uses second-order derivatives. Image sharpening using the Laplan. Sharpening using irst-order derivatives the gradient. Roberts cross gradient operators:. Sobel operators: Review or Eam I, EE55 /009 = g g = )], [ ), ), c g = ), M 7) ) ), M

8 Gonale & Woods Review or Eam I, EE55 /009 Filtering in the Frequenc Domain The Fourier transorm o a bo unction is a sinc unction. Convolution in the time domain represents a multiplication in the requenc domain: t) h t) H µ ) F µ ). Sampling and the Fourier Transorm o Sampled Functions The Sampling Theorem which states that that a band-limited unction can be recovered completel rom its samples i the are acquired at a rate eceeding twice the highest requenc in the unction 1 / T > µ ma. Aliasing or requenc aliasing is a process in which high requenc components o a unction masquerade as lower requencies. The DFT o One Variable The DFT o a sampled signal is continuous and ininitel periodic with period 1/ T. The discrete Fourier transorm pair allows transormation to and rom the requenc domain. Etension o the DFT to two variables. The Fourier transorm o a -D bo produces a -D sinc unction. Aliasing in images can be avoided i images are smoothed irst antialiasing) and then resampled R. C. Gonale & R. E. Woods

9 Gonale & Woods Review or Eam I, EE55 /009 The -D DFT and its inverse. Properties o the -D DFT: relationship between spatial and requenc intervals, translation and rotation, periodicit, smmetr properties. Table 4.1 allowed to bring a cop o it to the eam). jφ u, The -D DFT is comple in general and can be epressed as F u, = F u, e The ero-requenc term F0,0)) is proportional to the average value o,; F 0,0) = MN,. The Basics o Filtering in the Frequenc Domain Filtering in the requenc domain consists on modiing the Fourier transorm o an image. The 1 iltering equation is g, = I { H u, F u, }. Summar steps or iltering in the requenc domain Method in section 4.7.3). Correspondence between iltering in the spatial and requenc domains. A Gaussian lowpass in the requenc domain corresponds to a smoothing ilter in the spatial domain, a Gaussian highpass in the requenc domain corresponds to a sharpening ilter in the spatial domain R. C. Gonale & R. E. Woods

10 Gonale & Woods Review or Eam I, EE55 /009 Image Smoothing Using Frequenc Domain Filters Lowpass ilters: ideal ILPF), Butterworth BLPF), and Gaussian GLPF). Applications o lowpass iltering: machine perception, cosmetic processing, remote sensing. Image Sharpening Using Frequenc Domain Filters Highpass ilters: ideal IHPF), Butterworth BHPF), and Gaussian GHPF). Applications o lowpass iltering: machine perception, cosmetic processing, remote sensing. Laplacian in the requenc domain. Homomorphic iltering can be used to ilter an image which is product o two terms, i.e. illumination and relectance. Notch ilters The Fast Fourier Transorm FFT) Can reduce the number o computations o the DFT rom MN) multiplications and additions to MN log MN multiplications and additions. It uses the successive-doubling method to partition the 1-D transorm into hal and then into even and odd sequences R. C. Gonale & R. E. Woods