CRYPTOGRAPHY MATH 0209A
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This 20 page Class Notes was uploaded by Kaylin Wehner on Friday September 4, 2015. The Class Notes belongs to MATH 0209A at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 130 views. For similar materials see /class/177832/math-0209a-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.
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Date Created: 09/04/15
mum mmsmmn scamv MlihScIN Mathemutka Review on m Web This quzry umk mus saunas temz um r Hem Remeve mauumnW arm Erma mums 11E7n 11F72 11099 14st ZZESS Lzumng Gerald FVFARISH Cnhmmlngy nfDrinfEl l Imrlularvariz zs Pan 1 Geumelry cuunung urpmms enema harmumcanalysxs CambndgesmmesmAdmcedlxIammucs 1 Cambwdgiz InWVth PVIZSSCW17YIngZ 199a x1v344pp 54 95 SBA9752174795979 9811am 11m 11F72 1109914035 HESS Lzumng Gerard FVFARISH Wth anappmdlxby Jamme Waldwurger Cambridge smmes mAdmcedMammucs 55 Cambwdgiz InWVth Pra0ambndga 1997 mHK pp 54 95 ISBNEIVSZIA7EI6177 FEATURED REVIEW supmsucaled mums hm represmtauun Lheury 39heury ur enduscupy me Langlands guup nh m a a r w m y Langlands conjectures for GL2 over a function eld The two volumes under review take up the project of extending Drinfeld s theory from GL2 to This is a formidable task which brings together heavy amounts of local and global harmonic analysis on GLn with a careful study of the cohomology of the moduli spaces for Drinfeld modules Fortunately for those wishing to learn this beautiful eld the author has produced an outstanding work which is equally a mathematical and an expository achievement Whether one is dealing with Shimura varieties attached to a reductive group G or Drinfeld mod ular varieties the basic goal is always the same to decompose the appropriate ladic cohomology eg cohomology with compact supports intersection cohomology etc relative to the commut ing actions of the Galois group and the ring of Hecke operators In fact after passing to the limit the Hecke operators give way to a representation of the adelic group G A00 Here A00 denotes the ring of adeles away from 00 where 00 is a set of places of the global eld F If F is a number eld 00 is the set of Archimedean places and in the function eld case it is the singleton consisting of the arbitrary place of F that must be xed to de ne Drinfeld modules The joint action of Galois and Hecke yields a correspondence 7r gt 07r between the nite parts 7Tf W oo 7rv of certain cuspidal representations of G A and ladic representations of the Galois group GalF It is then of interest to describe 07r in terms of 7r This goal is achieved in the case at hand namely for GLn over a function eld in Chapter 12 of Volume II The cuspidal representations 7r intervening in the description of the cohomology with compact supports of the Drinfeld modular varieties are precisely those whose local component at the place 00 are Steinberg representations Theorem 1241 asserts that as expected the corre spondence 7r gt 07r is the Langlands correspondence ie it is characterized by the condition that at almost all places 12 the characteristic polynomial of the conjugacy class of a Frobenius element coincides under a suitable normalization with the characteristic polynomial of the Langlands class attached to the local component m The Ramanujan Petersson conjecture for 7r is also de rived using work of Jacquet Shalika Grothendieck and Deligne We might add that it is precisely this sort of easily stated relation between 7r and 07r which does not hold when the group is not The correct relation can be formulated only after the introduction of endoscopic groups and this is what substantially complicates the conjectural statements of LanglandsKottwitz for general Shimura varieties The hypothesis that W00 is a Steinberg representation is intrinsic to the approach via Drinfeld modules taken by Laumon To treat arbitrary cuspidal representations it is necessary to use Drinfeld s theory of shtukas This was carried out by Drinfeld himself for GL2 and is currently being pursued for GLn in work of Laurent Lafforgue Although Eichler Shimura theory rests on the congruence relation for Hecke correspondences most higherdimensional versions including the present one use a different approach originating in work of lhara and Langlands namely a comparison of the Arthur Selberg trace formula and the GrothendieckLefschetz xed point formula This method of proof involves two main stages the rst being concemed with preparations for the trace formula and the second with the trace formula itself The material is correspondingly divided between the two volumes The rst three chapters of Volume I introduce the Drinfeld modular varieties and derive a formula familiar from the case of Shimura varieties for the Lefschetz numbers of an operator of the form Frobg X T where FrobO is a power of the Frobenius at an unrami ed place 0 and T is a Hecke operator away from o and 00 Laumon rst writes the formula as a sum over an appropriate set of elliptic elements of certain terms each of which is a product of volume factors a twisted orbital integral at o and an orbital away from o and 00 The nal formula only involves ordinary orbital integrals but the idea of introducing the twisted orbital integral at this point is due to Kottwitz in the Shimura variety case who realized that a suitable case of the Fundamental Lemma as formulated by Langlands would then show that the twisted orbital integral is equal to an ordinary orbital integral The case of the Fundamental Lemma needed here is proved in Chapter 4 The rest of Volume I is devoted to the construction of a particular function f00 on GLnFOO whose orbital integrals give the ooadic contribution to the volume factors Laumon introduces the notion of a very cuspidal function and proves that the f00 he has constructed is very cuspidal and is also a pseudocoef cient for the Steinberg representation The end result is that the Lefschetz numbers are expressed as global orbital integrals of a certain function f A on GLnA Volume I also contains an excellent discussion of the unrami ed principal series and four useful appendices devoted to simple algebras Dieudonne modules some combinatorics and representation theory Most of Volume II is devoted to computing what happens when you plug f A into Arthur s non invariant trace formula This trace formula asserts the equality of two complicated expressions called the geometric and spectral sides of the trace formula Although the noninvariant trace formula depends on the choice of an arbitrary truncation parameter Laumon develops the trick of Kazhdan which asserts that when the power 7quot of Frobenius is su iciently large this dependence disappears and the geometric side of the trace formula reduces precisely to the Lefschetz number computed in Volume I It then remains to compute the terms on the spectral side to achieve the desired goal of relating the Lefschetz numbers to the trace of Hecke operators This is done in the dif cult Chapter 11 of Volume II at least for the terms attached to a socalled regular cuspidal datum The result for general cuspidal data due to Lafforgue is stated without proof and is used to derive the main theorems in Chapter 12 The last chapter of Volume II discusses some conjectures related to the intersection cohomology of Drinfeld modular varieties This volume also contains four very useful appendices on aspects of global harmonic analysis For some reason they are labelled D E F G even though the fourth appendix of Volume I is also D As described above these two volumes contain many results that are new and important How ever they are also the best source available for learning about the approach to zeta functions via the theory of automorphic representations They contain a wealth of information theorems and calculations laid before the reader in Laumon s superb expository style A wide range of topics from arithmetic geometry and local or global harmonic analysis are handled with ease and com pleteness In short these two volumes are a welcome addition to the literature on automorphic representations and are highly recommended Reviewed by Jonathan David Rogawski Review for Final Pic 20 1 Write a syntactically correct method that will take a string replace all the o39s in it by a replace all the e39s by a and leave all other symbols in the string alone Thus if it would chance quotlovequot to IIlavaquot Assume all letters are lower case 2 Write a method that input an integer between 0 a d 51 will return a string that is the name of a card from a standard deck The rules are a integer between 0 and 12 is to be a club a integer between 13 and 25 is to be a diamond a integer between 26 and 38 is to be a spade a integer between 39 and 51 is to be a heart a integer congruent 12 mod 13 is to be an ace a integer congruent 11 mod 13 is to be a king a quot 10 mod 13 quot quot quot a queen a quot quot 9 mod 13 quot quot quot a jack 3 u u 3 mod 13 u u u a 10 etc Thus for example 38 will correspond to the ace of spades 0 will correspond to the 2 of clubs 3 Write a class to be called SumOfPowers that will have two constructors The first will be used to compute sums of the form 1quot 2quot 3quot nquot the sum of the k powers of the integers from 1 to n where k is an integer and n is a positive integer The second constructor is to compute sums of the form aquota1quota2quotbquot the sum of the k powers of all the integers between a and b where k is an integer and a and b are positive integers with a lt b All the variables of the class are private and it is to contain a getSum method 4 Write a driver application for the class in problem 3 that will print out via Systemoutprintln the two sums 132333 103 502 512 1002 5 what problems m1ght be oreseen if you used the methods of problems 3 and 4 to calculate the sum ff mm2 r nlscuss the the methods you would employ to solve the card shu llng as are to be sh in a 13x 1 grld wlth clubs commg nst dlamonds second spades thlrd ea 11 u e cards in each Sun are to be dlsplayed 1n lncreaslng order as you go down the grld You may assume that a permutatlon of the 52 lntegers from o to 51 represents a shuffle of the deck 7 The collowlng program just m1m1cs lnputz publlc class eee publlc strlng Ise1 srlng st return st publlc statlc vold malnstrlng args 1 eee app new eee strlng test quotMy country 15 of thee strlng result applItselfuzest Systemloutlprlnt1n nnquot test Systeml exlt 0 and Command Plump Mg country 15 of thee Z11Pic20ReUieuFinalgt 4 Modlty 1 so that if the lnput strlng has three or more lower case e39s 1 wlll throw an exceptionl That ls 1f the strlng entered ls changed to strlng test quotMy country 15 of theeequot 1 wlll prlnt out Commond Plump Too mang e39s in Mg country tis of theee Z11Pic20ReUieuFinalgt 4 3 You are oo wr1e an apple ha provldes a drlll 1n mullpllca1on n 9 When for any wo one 9 numbers each of whlch 1s heween 2 a 1 he apple nrs opens he screen W111 look llke this ml u App APmE staned When the user clicks on the quotNextquot button the next screen will presen a mul1pllca1on problem in he nrs ex meld u u n Apple APmE staned In addluon he second ex eld he one ha had quotclch Mex gt o sarquot wlll be cleared lnulng he user w com in he answer and press enterl I he rlgh ans er 1s 111 type c e ed he rlgh he answer 1s 45quot W111 appear in he hlrd ex 21 u u n Apple 5 5 o5 omenswwa m APp E staned e user con1nues by presslng on he Mex buttonl When hls 1s done ne roblem appears in he rs eld and he seclnd and hlrd elds are cleared Applelv we Mu pl at nclass EEI APPM m um App etstaned If he answer enered 1s wrong he apple wlll say so m n Apple APp E staned The user then continues by pressing the Next button Specifucations text fields The first three elements of the l X 4 grid are The first and third are set ineditable You may assume that the entry in the second field is syntactically correct that is it the IntegerparseInt method will return an integer when applied to the second field entry Your program must clear the second and third fields when the Next button is cleared When he is done closing icon he exits by pressing the window An outline of the program I wrote is listed below The initO block is correct Your problem is to complete the actionPerformedActionEvent e and setProblanO methods import javaxswing import javaawtevent import javaawt public class Multiplication extends JApplet implements ActionListener private JTextField textFieldltextField2 textField3 private JButton next String problem int product public void initO Container c getContentPane csetLayout new GridLayoutl4 textFieldl new JTextField textFieldlsetEditablefalse caddtextFieldl textField2 new JTextField textField2setTextquotClick Next gt to startquot textField2addActionListenerthis caddtextField2 textField3 new JTextField textField3setEditablefalse caddtextField3 next new JButtonquotNextquot nextaddActionListenerthis caddnext end init public void actionPerformedActionEvent e if egetSource next problem setProblem ifegetSource textField2 int n IntegerparseIntegetActionCommand public String setProblem C int a 2 intMathrandom8 int b 2 intMathrandom8 product ab end class Multiplication 9 Write a method public double evaluateString st double u where st is a string that has 5 elements 3567xA7 where the first element is a double it need not be 3567 x is a variable and the last element is a positive integewr it need not be 7 The double u is any double number The value the method is to return is the value you get when you substitute u for x and then calculate the numerical value of the expression 10 A threads run method lncludes the followlng 3T sleep1oo 4T 5T catch Interruptedzxceptlon 1e Assumng the thread ls not lnterrupted whlch of the followlng statements ls correct a The code wll not complle b u c eca se exceptlons may not be aught in a threads run methodl b At llne 3 the thread wlll stop runningl nxecutlon wlll resume in at most 100 millisecondsl 1 At llne 3 the thread wlll stop runningl nxecutlon wlll resume in exactly 100 milliseconds 1 At llne 3 the thread wlll stop runningl nxecutlon wlll resume some time after 100 millisecondsl 1u True or False If a thread has a wa1t Instruction in the try block of lts run method then 1 must also have a notlcyo Instruction in Its 211110 methodOl If false why 12 once u create an lnstance of a thread in an y l True or False yo appllcatlon the run method beginsl If false wh 13 The Painterljava appllcatlon we dld for the mouse permltted the user to make free hand drawlngs on the screen Dvau me muuse m mew The code ls on the next pager import javaxswing import javaawtevent import javaawt public class Painter extends JFrame private int xValue 10 yValue 10 public PainterO getContentPaneadd new LabelquotDrag the mouse to drawquot BorderLayoutSOUTH addMouseMotionListener new MouseMotionAdapterO public void mouseDraggedMouseEvent e xValue egetX yValue egetY repaint setSize300300 setLocation300300 show end PainterO public void paintGraphics g gfillOvalxValue yValue 4 4 This is a set of directions on how to install scanned materials on your web site It will also contain a brief review of setting up a we page The Directory SetUp 1 Everything will be done on your unix account I assume you know the elementary unix commands for listing the elements of a directory ls creating a directory mkdir copying a file to another directory cp changing directories cd moving up one directory cd in the hierarchy and finding your present working directory pwd For simplicity we will use the name of my main unix directory rjm In addition all directory and file names and commands will be printed in red An Overview of the Directories The directories of interest in this conversation are sketched below rjm publicihtml demo 31b100f 2 We assume that there is a subdirectory of rjm called publicihtml This name is the default directory that contains all material you wish to display on your web site If you don39t have such a subdirectory then create one by means of the unix command gt mkdir publicihtml Permissions to Read 3 After creating it give the public the permission to read it by the command gt chmod 755 publicihtml In general you must set permissions for all subdirectories directories and files This is done by the instruction gt chmod 755 fileioridirectoryiname If you fail to do this you will get a IIyou don39t have accessquot when you try to view it in the browser Contents of public html 4 The publicihtml will contain one file indexhtml and subdirectories holding material that is to be displayed on your website For example the contents of the subdirectory rjmpublicihtml include demo 31bl00f indexhtml The slash denotes a directory The TopMost indexhtml File The contents of the file indexhtml just listed are ltHTMLgt ltBODYgt ltPgt ltA HREFquotdemodirectionspdfquot gt scanned material ltAgt ltA HREFquotdemoindexhtmlquot gt pdf stuff ltAgt ltPgt ltA HREF quot3lbl00findexhtml gt 313 Fall 2000 ltAgt ltBODYgt ltHTMLgt When you opened this browser you saw three lines displayed scanned material pdf stuff 313 Fall 2000 Clicking on quot scanned materialquot opened this file If you backtrack and click on IIpdf stuffquot above you will see a new screen with two lines homework 4 ratsch39s stuff If you click on either of these you will see a file that was scanned in It may take some time for the adobe reader to load when you click on the first scanned file indexhtml Files in the Subdirectories Note that the topmost indexhtml displayed above contains a reference to the file directionspdf in the directory de o two references to indexhtml files contained in the two subdirectories demo an 31b100f The contents of the subdirectory rjmpublicihtmldemo are four files directionspdf hw4pdf indexhtml ratschpdf The contents of this indexhtml file are ltHTMLgt ltBODYgt ltPgt ltA HREFquothw4pdfquot gt homework 4 ltAgt ltPgt ltA HREFquotratschpdfquot gt ratsch39s stuff ltAgt ltPgt ltBODYgt ltHTMLgt In sum the various indexhtml files as used here are the IInervous systemquot of the network of directories and files The indexhtml file at the bottom of the hierarchy contains references to material that is actually displayed on the web site A Note on HTML Syntax The HTML language is very sensitive to blanks If the instruction ltA HREFquothw4pdfquot gt homework 4 ltAgt contains a blank at the beginning say lt A HREFquothw4pdfquot gt homework 4 ltAgt the reference may not appear on the web page Displaying a Scanned File We assume you are in your main unix directory I would be in rjm You will be using the program quotpinequot to read yiur email for it can save pdf files that have been sent to you So 1 Open pine gt pine 2Select the message with the pdf file attached The message will have been sent by rjm and will be described as IIyour scanned materialquot Try opening it 3The following message will be displayed IICannot display this part Press 39V39 then ISI to save in a filequot Follow the instructions and the file will be saved in in your home directory under a name that pine will tell you 4Exit pine Supposing the file has been saved as quotmyscanpdfquot move it to publicihtmldemo by the instruction cp myscanpdf publicihtmldemo 5 Move to the demo directory open or create an indexhtml file in itadd the line ltA HREFquotmyscanpdfquot gt my scanned material ltAgt Image denoising using diffusion on curvelet scaled Gabor lter responses Jim Bremer Yoel Shkolnisky I Arthur Szlam I October 29 2007 1 Introduction In 5 a general framework for adaptive function regularization was introduced and this framework was demonstrated in several applications including image denoisingr The basic idea of the method applied to image denoising is to choose a set of features and consider the pixels of the image as lying in feature space We then try to use the heat equation on the points in feature space to smooth the image In this note we would like to give a more in depth account of image denoising in this framework using curvelet scaled Gabor filter responses as features 11 Weights from images To build the imagedependent weights we first associate a feature vector to each location I in the image I by convolving I with a filter bank Here we will be mostly concerned with the Gabor filters described in 12 If g gl gd are our filters Q 01 gtlt 01 and I Q gt gt IR is our noisy image map Q into by ng Q A Rd I H 191I7 719d1 Once we have features fI let May llfg1r fg1yll7 where is the Euclidean norm in Rail Now we pick a usually exponentially decreasing function h and variance parameter 6 and de ne Wzy h i 2 Depaltment of Mathematics UC Davis IDepaltment of Mathematics Yale University IDepaltment of Mathematics UCLA 1 A common choice is ha exp7ai The idea is that we expect that very close data points with respect to p will be similar but do not want to assume that far away data points are necessarily different The exponential weight h in 2 gives a large preference to very close points It is computationally prohibitive to nd Wzy for all pairs of pixels In addition unless there are patterns repeated in many locations in the image far away pixels are unlikely to be useful in determining the denoisingi So we modify p by choosing sets 5 51 C Q so that Many 5179 ify 651 3 otherw1sei A simple and effective choice for 51 is a square of xed side length centered at 1 Even within the coarse search box 5 there may be many pixels that are not too similar to I To further decrease the computational complexity and to insure that I only communicates with pixels very similar to it we x a small number k and set pz y 00 for any y which is not one of the k nearest p neighbors of 1 Finally we follow 4 and modify p so that the distance between I and its four nearest spatial neighbors is not set to 00 regardless of whether they are among the k nearest points to z in feature space this leads to a dramatic reduction in artifactsi With h as above building W with the modi ed p results in a matrix with at most k 4 entries per rowi 12 Construction of curveletscaled Gabor lters To build the lters we partition the frequency plane into radial bands and then the radial bands into polar rectangular tiles doubling the number of tiles every two bands as in In each frequency tile we place a Gaussian bump with mean in the centroid of the tile and variances scaled as the ratio of the radial length of the tile to the angular width of the tile To make real lters we also place a bump in the tile re ected through the origin More precisely the radial variable is split into N3 1 bands B l 0 WW Bj mmw for j 2MNB giving NB 1 annulii For each annulus 77r is split into N9 pieces where N9B0 1 N9 B1 4 and then N9B1 is doubled every two bands after N9 B1 giving the sequence 144 88 a An elliptical Gaussian is placed in each polar rectanglei To do this rst we de ne center points 99 CF a mp 2 Pick a number 6 and choose variances such that the value of the Gaussian centered at C9Cp is 6 at the points W62gt and p2 932 i In our ex periments below we take NB 6 and 6 3 13 Evolving the heat equation in feature space We interpret the weight Wij as a measure of similarity between the pixels 239 and j A natural averaging lter acting on functions on Q can be de ned by normalization of the weight matrix as follows let mmZwuma yEV and let the lter be KI7y D 1IWI7 9 4 Kz y 1 This lter acts on a function f on Q via mmZMmma zeQ so that zer and hence it is a local averaging operation with locality measured by the sim ilarities Wi One can also think of the matrix K 1 as a diffusion or random walk on Q which is run for one step by multiplying from the other side This lter can be iterated several times by considering the power K from the point of view of the diffusion process this corresponds to taking n steps of the random walk whose transition probabilities are the transpose of Ki We can think of applying the powers of K as running the heat equation on Q embedded in the feature coordinatesi This heat equation is nonlinear because of the use of the image in the de nition of K but is linear in the sense that we do not update K after applying it Thus as t gt gt 00 an tends to a constant We can balance smoothing by K with delity to the original noisy function by setting m1 Kfn f1 5 where B gt 0 is a parameter to be chosen and large 6 corresponds to less smoothing and more delity to the noisy image This is a standard technique in PDE based image processing see 3 and references therein If we consider iteration of K as evolving a heat equation the delity term sets the noisy function as a heat source with strength determined by Note that even though when we smooth in this way the steady state is no longer the constant function we still do not usually wish to smooth to equilibriumi 2 Experiments We now show the results of some denoising experiments We build the Gabor lters and K as above setting ha far We choose the parameters 6 and n by 1Note that Dm 0 if and only if m is not connected to any other vertex in which case we trivially de ne D 1z O or simply remove m from the graph Figure 1 Top left clean boat image Top right noisy boat with a 20 Middle righttdenoising using diffusion in NL means type patch emloedding7 SNR1675 Middle left denoising using diffusion on curvelet Gabor features averaged with the NL means type denoising7 SNR1713 Bottom right residual from the NL means type denoising Bottom left is the residual from the denoising using diffusion on curvelet Gabor features averaged With the NL means type denois ing Figure 2 Top left clean Lena image Top right noisy Lena With a 20 Middle rightdenoising using diffusion in NL means type patch emloedding7 SNR17 65 Middle left denoising using diffusion on curvelet Gabor features averaged with the NL means type denoising7 SNR1793 Bottom right resid ual from the NL means type denoising Bottom left is the residual from the de noising using diffusion on curvelet Gabor features averaged With the NL means type denoising Figure 3 Top left clean peppers image Top right noisy peppers with a 20 Middle rightdenoising using diffusion in NL means type patch em loedding7 SNR1788 Middle left denoising using diffusion on curvelet Gabor features averaged with the NL means type denoising7 SNR1828 Bottom right residual from the NL means type denoising Bottom left is the resid ual from the denoising using diffusion on curvelet Gabor features averaged with the NL means type denoising hand to maximize recovered SNRi As a baseline in our experiments we will use a 7 X 7 NLmeans type denoisingi This NLmeans denoising is obtained exactly as the Gabor lter denoising except with a different choice of lters eg for 3 X 3 patches use the 9 lters 1 0 0 0 l 0 0 0 0 fro 0 0 0 7 f12 0 0 0 f33 0 0 0 6 0 0 0 0 0 0 0 0 l for 7 X 7 patches we get 49 ltersi Once we have the lter responses from the image which are just the 7 X 7 patches surrounding a given pixel we proceed as above with the construction of pp using the lters f and KP using pp again choosing all parameters to maximize recovered SNRi We note that in terms of SNR and in our subjective assesment in terms of visual quality this form of the NLmeans algorithm using a coarse search with a small xed number of neighbors for each pixel chosen from inside the coarse search multiple iterations with a relatively small 6 in equation 2 and forcing the four nearest spatial neighbors of a pixel to be neighbors in the weights outperforms 1 and gives results equivalent to has been reported that experimentally 4 that choosing the parameter 6 to be 0 is optimal for the NL means type denoising our experience corrobrates this In terms of SNR the Gabor feature denoising tends to perform better with a nonzero delity but visual artifacts are more pronounced For simplicity of comparison for between both sets of weights we choose 6 in equation 5 to be 0 The images cleaned by diffusion on the Gabor lters and by the patch lters have essentially equivalent SNR si But more important than a simple ranking is the fact that the two choices of lters distill different and complementary information For example the Gabor lters diffusion does a much better job preserving the thin faint cables in the boats image and the soft edged pole in the Lena imagei On the other hand the patch lters smooth content free portions of the images much more completely and are superior for preserving repeating textures such as the rapidly alternating light and dark stripes in the bottom left of Lenals hati This suggests taking the additive mean of the denoisings obtained by the two methods and in fact doing so results in a signi cant and consistent increase in recovered SNR as well as in subjective image quality over either of the methods alonei Figures 1 2 and 3 display examples of denoising with a diffusion on the curveletGabor features averaged with an NLmeans denoising and the NL means denoising for comparison On the top left of each gure we have the clean imagei On the top right is the noisy image f0 with a 20 On the middle right of the each gure is a denoising using diffusion in NLmeans type patch embedding and then on the middle left is the denoising using diffusion on curveletGabor features averaged with the NLmeans denoisingi On the bottom right of the each gure is the residual from the NLmeans type patch embedding and on the bottom left is the residual from the denoising using diffusion on curveletGabor features averaged with the NLmeans denoisingi
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