1. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lattice Sampling Spring ’09 Instructor: Min Wu Electrical and Computer Engineering Department, University of Maryland, College Park   bb.eng.umd.edu (select ENEE631 S’09)   minwu@eng.umd.edu ENEE631 Spring’09 Lecture 26 (5/6/2009)

2. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [2] Overview and Logistics Last Time: –Radon and inverse Radon transform for medical app. u Inverting Radon transform data based on Fourier properties vs. Filtered Back projections –Sampling theory: extending 1-D to 2-D Today: –Sampling on non-rectangular grid via the Lattice Theory Course evaluation (online) Project feedback UMCP ENEE631 Slides (created by M.Wu © 2004)

3. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [3] Recall: 2-D Sampling 2-D impulse function (a.k.a. 2-D comb) comb(x,y;  x,  y) =  m,n  ( x - m  x, y - n  y ) ~ separable function FT: COMB(  x,  y ) = comb(  x,  y ; 1/  x, 1/  y) /  x  y Sampling vs. Replication (tiling) –Nyquist rates (2  x0 and 2  y0 )  Aliasing Jain’s Fig.4.7 UMCP ENEE631 Slides (created by M.Wu © 2001)

4. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [4] 2-D Sampling: Beyond Rectangular Grid Sampling at nonrectangular grid –M ay give more efficient sampling density when spectrum region of support is not rectangular u Sampling density measured by #samples needed per unit area –E.g. interlaced grid for diamond shaped region of support u equiv. to rotate 45-degree of rectangular grid u spectrum rotate by the same degree From Wang’s book preprint Fig.4.2 UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

5. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [5] General Sampling Lattice Lattice  in K-dimension space R K –A set of all possible vectors represented as integer weighted combinations of K linearly independent basis vectors Generating matrix V (sampling matrix) V = [v 1, v 2, …, v k ] => lattice points x = V n e.g., identity matrix V ~ square lattice Voronoi cell of a lattice –A “unit cell” of a lattice, whose translations cover the whole space –Enclose all points that are closer to the origin than to other lattice points u cell boundaries are equidistant lines between surrounding lattice points From Wang’s book preprint Fig.3.1 UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

6. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [6] Sampling Density: d1 = 1 d2 = 2 /  3 From Wang’s book preprint Fig.3.1 Examples of Lattices UMCP ENEE631 Slides (created by M.Wu © 2004)

7. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [7] Periodicity in Lattice Representations Lattice represents a multi-dimensional periodic function –Repeats a basic pattern of a unit cell with all integer steps of translations –Voronoi cell describes the fundamental period of this periodic function Finding Voronoi cell (1) Draw straight line between origin and each of the closest nonzero lattice points; (2) Draw a perpendicular line that is half way between the two points => The polygon formed by these equidistant lines surrounding the origin is the Voronoi cell. u For 3-D or multi-dim, replace equidistant line with equidistant (hyper-)planes Sampling density d(  ) = 1 / |det(V)| |det(V)| measures volume of a cell; d(  ) is # lattice points in unit volume. UMCP ENEE631 Slides (created by M.Wu © 2007)

8. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [8] Frequency Domain View & Reciprocal Lattice Reciprocal lattice  # for a lattice  (with generating matrix V) –Generating matrix of  # is U = (V T ) -1 –Basis vectors for  and  # are orthonormal to each other: V T U = I –Denser lattice  has sparser reciprocal lattice  # : det(U) = 1 / det(V) Frequency domain view of sampling over lattice –Sampling in spatial domain  Repetition in frequency domain –Repetition grid in freq. domain can be described by reciprocal lattice –Intuition for “reciprocal” [e.g.] rectangular grid that sample faster horizontally than vertically =>the repetition in frequency domain is slower horizontally than vertically Aliasing and prefiltering to avoid aliasing –Aliasing happens when signal spectrum extends outside the Voronoi cell of reciprocal lattice UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

9. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [9] Generalized Sampling Theorem –Wang’s book Sec.3.2.2, Theorem 3.6 –(see also Sec.2.1-2.2 on multi-dimensional FT and DFT) UMCP ENEE631 Slides (created by M.Wu © 2004)

10. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [10] Sampling Efficiency Consider spherical signal spectrum support –Many real-world signals have symmetric freq. contents in many directions –The multi-dim spectrum can be approximated well by a sphere (with proper scaling of spectrum support) Voronoi cell of reciprocal lattice need to cover the sphere to avoid aliasing –Tighter fit of Voronoi cell to the sphere leads to lower sampling density What sampling lattice  gives best sphere covering? Sampling Efficiency  = volume(unit sphere) / d(  )  prefer close to 1 From Wang’s book preprint Fig.4.2 & 3.5 UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

11. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [11] Recall: 1-D Upsample and Downsample From Crochiere-Rabiner “Multirate DSP” book Fig.2.15-16 UMCP ENEE631 Slides (created by M.Wu © 2001)

12. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [12] General Procedures for Sampling Rate Conversion From Wang’s book preprint Fig.4.1 UMCP ENEE631 Slides (created by M.Wu © 2001)

13. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [13] Sampling Lattice Conversion From Wang’s book preprint Fig.4.4 Intermediate Original Targeted UMCP ENEE631 Slides (created by M.Wu © 2001)

14. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [14] Case Studies on Sampling and Resampling in Video Processing Reference Readings: Wang’s book Chapter 4 e.g. Interlaced 50 fields/sec  60 fields/sec UMCP ENEE631 Slides (created by M.Wu © 2004)

15. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [15] Example: Frame Rate Conversion Video sampling: formulate as a 3-D sampling problem Note: different signal characteristics and visual sensitivities along spatial and temporal dimensions (see Wang’s Sec.3.3 on video sampling) General approach to frame rate conversion –Upsample => LPF => Downsample Interlaced 50 fields/sec  60 fields/sec –Analyze in terms of 2-D sampling lattice (y, t) –Convert odd field rate and even field rate separately u do 25  30 rate conversion twice => not fully utilize info. in other fields –Deinterlace first then convert frame rate u do 50  60 frame rate conversion: 50  300  60 –Simplify 50  60 by converting 5 frames  6 frames u each of output 6 frames is from two nearest frames of the 5 originals u weights are inversely proportional to the distance between I/O –May do motion-interpolation for hybrid-coded video UMCP ENEE631 Slides (created by M.Wu © 2001/2004)

16. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [16] From Wang’s book preprint Fig.4.3 UMCP ENEE631 Slides (created by M.Wu © 2001)

17. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [17] Video Format Conversion for NTSC  PAL Require both temporal and spatial rate conversion –NTSC 525 lines per picture, 60 fields per second –PAL 625 lines per picture, 50 fields per second Ideal approach (direct conversion) –525 lines 60 field/sec  13125 line 300 field/sec  625 lines 50 field/sec 4-step sequential conversion –Deinterlace => line rate conversion => frame rate conversion => interlace UMCP ENEE631 Slides (created by M.Wu © 2001)

18. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [18] Video Format Conversion for NTSC  PAL Require both temporal and spatial rate conversion –NTSC 525 lines per picture, 60 fields per second –PAL 625 lines per picture, 50 fields per second Ideal approach (direct conversion) –525 lines 60 field/sec  13125 line 300 field/sec  625 lines 50 field/sec 4-step sequential conversion Deinterlace => line rate conversion => frame rate conversion => interlace –Food for thought: compare the computation of ideal and sequential approach u Hint: filtering done for X lines @ Y field/sec => examine X & Y UMCP ENEE631 Slides (created by M.Wu © 2001)

19. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [19] From Wang’s book preprint Fig.4.9 UMCP ENEE631 Slides (created by M.Wu © 2001)

20. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [20] Simplified Video Format Conversion 50 field/sec  60 field/sec –After deinterlacing, s implify to 5 frames  6 frames –Conversion involves two adjacent frames only 625 lines  525 lines –Simplify to 25 lines  21 lines –Conversion involves two adjacent lines only UMCP ENEE631 Slides (created by M.Wu © 2001)

21. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [21] UMCP ENEE631 Slides (created by M.Wu © 2001) From Wang’s book preprint

22. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [22] Interlaced Video and Deinterlacing Interlaced video Odd field at 0  Even field at  t  Odd field at 2  t  Even field at 3  t … Deinterlacing –Merge to get a complete frame with odd and even field Examples from http://www.geocities.com/lukesvideo/interlacing.html UMCP ENEE631 Slides (created by M.Wu © 2001)

23. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [23] De-interlacing: Practical Approaches Spatial interpolation –Vertical interpolation within same field (1-D upsample by 2) –Line averaging ~ average the line above and below D=(C+E)/2 Temporal interpolation –2-frame field merging => artifacts –3-frame field averaging D=(K+R)/2 u fill in the missing odd field by averaging odd fields before and after Spatial-temporal interpolation –Line-and-field averaging D=(C+E+K+R)/4 UMCP ENEE631 Slides (created by M.Wu © 2001) From Wang’s book preprint

24. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [24] De-interlacing: Practical Approaches Spatial interpolation –Vertical interpolation within same field (1-D upsample by 2) –Line averaging ~ average the line above and below D=(C+E)/2 u interpolation filter: [ ½, 1, ½ ] T u DTFT =  x(n) e -j2  n  y fy = 1 + ( e -j2  y fy + e j2  y fy ) / 2 = 1 + cos 2  y f y Temporal interpolation –2-frame field merging => artifacts –3-frame field averaging D=(K+R)/2 u fill in the missing odd field by averaging odd fields before and after Spatial-temporal interpolation –Line-and-field averaging D=(C+E+K+R)/4 UMCP ENEE631 Slides (created by M.Wu © 2001) From Wang’s book preprint

25. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [25] Motion-Compensated De-interlacing Stationary video scenes –Temporary deinterlacing approach yield good result Scenes with rapid temporal changes –Artifacts incurred from temporal interpolation –Spatial interpolation alone is better than involving temporal interpolation Switching between spatial & temporal interpolation modes –Based on motion detection result –Hard switching or weighted average –Motion-compensated interpolation UMCP ENEE631 Slides (created by M.Wu © 2001)

26. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [26] Example: De-interlacing From Woods’ book resource

27. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [27] Example: Temporal Upsampling (5 to 30 fps) From Woods’ book resource

28. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [28] Texture Analysis UMCP ENEE631 Slides (created by M.Wu © 2004)

29. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [29] Types of Image Processing Tasks Image in, Image out –Codec (compression-decompression) –Image enhancement and restoration –Digital watermarking => May require both analysis and synthesis operations u Intermediate output may be non-image like (coded stream), but end output should reconstruct into an image close/relate to the input Image in, Features out –Features may be used for classification, recognition, and other studies

30. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [30] Texture Observed in structural patterns of objects’ surfaces –[Natural] wood, grain, sand, grass, tree leaves, cloth –[Man-made] tiles, printing patterns “Texture” ~ repetition of “texels” (basic texture element) –Texels’ placement may be periodic, quasi-periodic, or random From http://texlib.povray.org/textures.html and Gonzalez 3/e book online resource

31. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [31] Properties and Major Approaches to Study Texture Texture properties: smoothness, coarseness, regularity Structural approach –Describe arrangement of basic image primitives Statistical approach –Examine histogram and other features derived from it –Characterize textures as smooth, coarse, grainy Spectral and random field approach –Exploit Fourier spectrum properties –Detect global periodicity

32. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [32] Statistical Measures on Textures x ~ r.v. of pixel value R = 1 – 1 / ( 1 +  x 2 ) ~ 0 for constant region; 1 for large variance E[ (X –  x ) K ] 3 rd moment: ~ histogram’s skewness 4 th moment: ~ relative flatness Uniformity or Energy ~ squared sum of hist. bins Figures from Gonzalez’s book resource

33. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [33] Characterizing Textures Structural measures –Periodic textures u Features of deterministic texel: gray levels, shape, orientation, etc. u Placement rules: period, repetition grid pattern, etc. –Textures with random nature u Features of texel: edge density, histogram features, etc. Stochastic/spectral measures –Mainly for textures with random nature; Model as a random field u 2-D sequence of random variables –Autocorrelation function: measuring the relations among those r.v. R(m,n; m’,n’) = E[ U(m,n) U(m’,n’) ] “wide-sense stationary”: R(m,n;m’n’) = R U (m-m’,n-n’) and constant mean –Fit into random field models ~ analysis and synthesis u Focus on second order statistics for simplicity u Two textures with same 2nd order statistics often appear similar

34. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [34] Examples: Spectral Approaches to Study Texture Figures from Gonzalez’s 2/e book resource

35. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [35] Texture Synthesis Recall: error concealment of small blocks –Exploit surrounding edge info. to interpolate Image in-painting –Filling in missing/occluded regions with synthesized version u Maintain structural consistency (in edge & overall color/brightness) u Maintain texture’s statistical continuity (such as oscillation pattern) for improved visual effect Ref: M. Bertalmio, L. Vese, G. Sapiro, and S. Osher: “Simultaneous Structure and Texture Image Inpainting,” IEEE Trans. on Image Proc., vol.12, no.8, August 2003. edge estimation edge-directed interpolation Image examples from Bertalmio et al. TIP’03 paper

36. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [36] Summary of Today’s Lecture Multi-dimension Lattice sampling –Sampling lattice and frequency-domain interpretation –Sampling rate conversion Texture analysis Next Lecture: –Continue on texture analysis and synthesis Readings –Lattice Sampling: Wang’s book Sec. 3.1-3.3, 3.5; Chapter 4 –Texture: Gonzalez’s book 11.3.3 (see also Jain’s 9.11) UMCP ENEE631 Slides (created by M.Wu © 2004)

37. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [37]

38. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [38] Image Inpainting: Basic Approach and Example Figures from Bertalmio et al. TIP’03 paper

39. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [39] Texture Synthesis Approach by Effros-Leung Model texture as Markov Random Field (MRF) –Probability distribution of pixel brightness given spatial neighborhood is independent of the rest of the image. –Model neighborhood of a pixel with a square window around Window size controls how stochastic the texture will be E.g. Choose window size on the scale of biggest regular feature 2-step estimate of cond’l p.d.f. –Match neighborhood with some allowable distortion –Build histogram of corresp. pixels from matched neighborhoods –Produce estimate based on the histogram Figures from Effros-Leung ICCV’99 paper

40. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [40] Recall: Characterize the Ensemble of 2-D Signals Specify by a joint probability distribution function –Difficult to measure and specify the joint distribution for images of practical size => too many r.v. : e.g. 512 x 512 = 262,144 Specify by the first few moments –Mean (1 st moment) and Covariance (2 nd moment) u may still be non-trivial to measure for the entire image size By various stochastic models –Use a few parameters to describe the relations among all pixels u E.g. 2-D extensions from 1-D Autoregressive (AR) model Important for a variety of image processing tasks –image compression, enhancement, restoration, understanding, … UMCP ENEE631 Slides (created by M.Wu © 2004)

41. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [41] Recall: Discrete Random Field We call a 2-D sequence discrete random field if each of its elements is a random variable –when the random field represents an ensemble of images, we often call it a random image Mean and Covariance of a complex random field E[u(m,n)] =  (m,n) Cov[u(m,n), u(m’,n’)] = E[ (u(m,n) –  (m,n)) (u(m’,n’) –  (m’,n’)) * ] = r u ( m, n; m’, n’) u For zero-mean random field, autocorrelation function = cov. function Wide-sense stationary (or wide-sense homogeneity)  (m,n) =  = constant r u ( m, n; m’, n’) = r u ( m – m’, n – n’; 0, 0) = r ( m – m’, n – n’ ) u also called shift invariant, spatial invariant in some literature UMCP ENEE631 Slides (created by M.Wu © 2004)

42. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [42] Recall: Special Random Fields of Interests White noise field –A stationary random field –Any two elements at different locations x(m,n) and x(m’,n’) are mutually uncorrelated r x ( m – m’, n – n’) =  x 2 ( m, n )  ( m – m’, n – n’ ) Gaussian random field –Every segment defined on an arbitrary finite grid is Gaussian i.e. every finite segment of u(m,n) when mapped into a vector have a joint Gaussian p.d.f. of UMCP ENEE631 Slides (created by M.Wu © 2004)

43. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [43] Recall: Spectral Density Function Spectral density function (SDF) is defined as the Fourier transform of the covariance function r x u Also known as the power spectral density (p.s.d.) –Example: SDF of stationary white noise field with r(m,n)=  2  (m,n)  S(  1,  2) =  2 SDF Properties: –Real and nonnegative: S(  1,  2 ) = S*(  1,  2 ); S(  1,  2 )  0 u By conjugate symmetry of covariance function: r (m, n) = r * (-m, -n) u By non-negative definiteness of covariance function –SDF of the output from a LSI system w/ freq response H(  1,  2 ) S y (  1,  2 ) = | H(  1,  2 ) | 2 S x (  1,  2 ) UMCP ENEE631 Slides (created by M.Wu © 2004)

44. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [44] More on Image Modeling Good image model can facilitate many image proc. tasks –Coding/compression; restoration; estimation/interpolation; … Image models we’ve seen/used so far –Consider pixel values as realizations of a r.v. u Color/grayscale histogram –Predictive models u Use linear combination of (causal) neighborhood to estimate –Random field u(m,n) u Characterized by 2-D correlation function or p.s.d. Generally can characterize u(m,n) = u’(m,n) + e(m,n) –u’(m,n) is some prediction of u(m,n); e(m,n) is another random field u Minimum Variance representation(MVR): e(m,n) is error of min. var. prediction u White noise Driven representation: e(m,n) is chosen as a white noise field u ARMA representation: e(m,n) is a 2-D moving average of a white noise field.

45. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [45] Recall: Linear Predictor Causality required for coding purpose –Can’t use the samples that decoder hasn’t got as reference Use last sample u q (n-1): equiv. to coding the difference (DPCM) p th –order auto-regressive (AR) model –Linear predictor from past samples Prediction neighborhood –Line-by-line DPCM u predict from the past samples in the same line – 2-D DPCM u predict from past samples in the same line and from previous lines –Non-causal neighborhood u Use samples around as prediction/estimation => for filtering, restoration, etc Predictor coeff. in MMSE sense: get from orthogonality condition (from Wiener filtering discussions) UMCP ENEE631 Slides (created by M.Wu © 2001)

46. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [46] Commonly Used Image Model Gaussian model and Gaussian mixture model Every segment defined on an arbitrary finite grid is Gaussian -or- follows a distribution of linear combining several Gaussian u Reduce the modeling to estimate mean(s), variance(s), and weightings (Ref: Prof. R. Gray’s IEEE talk S’07 http://www-ee.stanford.edu/~gray/umcpqcc.pdf) Markov random field –Markovianity: conditional independence u Define past, present, future pixel set for each pix location u Given the present, the future is independent of the past –2-D/spatial causal AR model (under Gaussian noise or MVR) Gauss-Markov random field model –For Gaussian: conditional indep. => conditional uncorrelateness Bring in multi-scale and wavelet/multi-resolution ideas Ref: Section 4.1-4.5 of Bovik’s Handbook

47. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [47] Comparison of Various Predictive Models –Ref. Jain’s pp495 From Jain’s Fig.11.12 UMCP ENEE631 Slides (created by M.Wu © 2001)

48. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [48] Recursive Estimation: Basic Ideas Kalman filter: recursive Linear MMSE estimator 1-D example: provide linear estimation for an AR signal –System model AR(M): x(n)=c 1 x(n-1)+… c M x(n-M) + w(n) u State equation: x(n) = C x(n-1) + w(n) for n = 0, 1, …, x(-1)=0 ~ AR sig u Observation equation: y(n) = h T x(n) + v(n) u M-dimension state vector x(n); model noise w(n) and observation noise v(n) are white & orthogonal –Signal model is Mth-order Markov under Gaussian noise w –Linear MMSE estimator is globally optimal if model and observation noise w and v are both Gaussian –C can be time-variant and physics motivated by applications General MMSE solution: equiv. to find conditional mean u Filtering estimate: E[ x(n) | y(n), y(n-1), … y(0) ]  x a (n) u One-step predictor estimate: E[ x(n) | y(n-1), … y(0) ]  x b (n)

49. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [49] Recursive Estimation (cont’d) 1-D Kalman filter equations u Prediction: x b (n) = C x a (n-1) u Update: x a (n) = x b (n) + g(n) [ y(n) – h T x b (n) ] –Error variance equations P b (n) = C P a (n-1) C T + Q w initialize x(0) = [w(0), 0, … 0 ] T, x b (0) = 0 P b (n) = (I – g(n) h T ) P b (n) P b (n) = Q w all zero except 1 st entry  w 2 –Kalman gain vector: g(n) u g(n) = P b (n) (h T P b (n) h +  v 2 ) -1 2-D Kalman filtering: define proper state vector u Raster scan observations & map to equiv. 1-D case u Restrict Kalman gain terms to be just surround current observation to reduce computational complexity past present state future

50. M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec26 – Lattice Sampling [50] Summary of Today’s Lecture Readings –Texture: Gonzalez’s book 11.3.3 (see also Jain’s 9.11) –Image modeling: Wood’s book Chapter 7 & 9.4 u See also Jain’s Chapter 6 and Bovik’s Handbook 4.1-4.5 u Recursive/Kalman estimation: Woods’ book Chapter 7; EE621 (Poor’s book) UMCP ENEE631 Slides (created by M.Wu © 2004)