3d Intenational Confeence on Multimedia echnology(icm 03) A Simple Compessive Sampling Mode and the Recovey of Natue Images Based on Pixel Value Substitution Wenping Shao, Lin Ni Abstact: Compessive Sampling (CS) is a new technique fo infomation acquisition and pocessing. In this pape we popose a new algoithm based on Pixel Value Substitution (PVS) to econstuct the natue images when the model of compessive sampling is vey simple. By indiectly utilizing the fact that usually the gadient of a natual image is spase, we divide the image into many small blocks, and fo each block, we use only one typical value to epesent all the pixels' values of that block. And thus we can constuct a new matix with full column ank fom the Gaussian measuement matix, and get a new system of equations that can be solved by the least squae method which is also used in the geedy algoithms. And though analyzing we find out the statistical featue of the econstucted signal, and the factos that influence the econstuction quality, which tell us that, in ode to get the most appopiate value fo each pixel, the econstuction needs to be epeated as is shown in the concete steps of the algoithm, and also the block s size should be appopiate. At last, expeimental esults ae given to demonstate the viewpoints in this pape and they show that, in addition to impoving the quality, PVS can also significantly educe the time consumption. Keywods: Compessive Sensing Pixel Value Substitution Gaussian Measuement Matix Natual Images. W. Shao, L. Ni ( ) Depatment of Electonic Engineeing and Infomation Science, Univesity of Science and echnology of China, Jinzhai Road, Hefei, Anhui Povince, P.R.China, e-mail: nilin@ustc.edu.cn 03. he authos - Published by Atlantis Pess 68
intoduction aditional signal pocessing is based on the Nyquist sampling theoem, which means the signal can be accuately ecoveed when the sampling fequency is at least two times of the signal fequency. In ecent yeas, a new signal pocessing technique has appeaed, which is called Compessed Sensing (CS) [, ]. And by using the CS theoy, signal sampling fequency can be geatly educed. By using the fact that many natual signals ae spase in some tansfomed domains, the basic idea of CS is that we can tansfom the signal into two incoheent domains [3]. If the oiginal signal is x, the tansfomation opeato is f, and obsevation opeato is g, then the tansfomed signals will be f(x) (which is spase) and g(x). CS theoy tell us we can choose just a small pat of the elements set of g(x) to ecove f(x) and x accuately, and the choice is abitay [4], which means we can use any pat of the set, pemising that the numbe of elements is sufficient. hee ae many specific methods based on CS theoy, and main types ae geedy algoithms [5, 6, 7] and l convex optimization []. oday, signal and image pocessing technology has been able to pocess an image in vaious ways, but in some occasions, when collecting images infomation o when obseving images, the equipment can not match the needs of complex pocessing, and so that the amount of captued data should be small, and the sampling mode should be simple. his is closely elated with the CS theoy, and it s necessay to eseach how to ecove an image when it was sampled in some vey simple ways. Suppose that x is the oiginal image signal, and we obseve x diectly by using a Gaussian measuement matix A, and then we can get the obseved signal: y = Ax. () his is a vey simple sampling mode, and this aticle popose an algoithm about how to ecove x fom y in this condition. Pixel Value Substitution In this section we popose the PVS algoithm, discuss the econstuction quality, and show the concete steps of the algoithm.. Pinciple We can see fom () that, it s an undedetemined system of equations, and if we know nothing about x, then we cannot get the solution of these equations. But fo- 683
tunately, natual images do have some ules, and one ule is that the gadient of a natual image is spase [8], which means most pixels of the gadient image have the values that ae close to zeo. And then we know that, usually, values of adjacent pixels ae almost the same, and we can use just one value to epesent them. We know that, if we multiply a pixel value of x by a column of A, then we can get a pat of y.so we can ewite the obsevation equations on the following fom: m n y = Ax = x I = x I i= j=. () Hee, A R M N, y R M, and m n = N. And x denote the pixel in the ith ow and jth column of an m n image x, and I denote a column of the measuement matix A, which coespond to x. Now, we divide the image x into small blocks, and the th block is denoted by D. Assume that D has k pixels, and then: k... N + k + + k =. We know that,usually, all pixel values of D ae almost the same,thus: ( y = x I = x I ) (3) = i, = [ α ( I i, )] = = α Ψ = Ψα (4) = I α = k ( ) βi = Bβ. (5) k = i, () Hee, x denote the pixel in the ith ow and jth column of x, and it belongs to D; B R M, α, β R ; andα epesent a typical value that can substitute fo each x,and obviously, α is close to them; and β is the nomalization of α. () And (3), (4), and (5) tell us that we can educe the numbe of unknown paametes by substituting one value fo seveal values. Accoding to the knowledge of andom distibution, I is a andom vecto, and elements of I and I have the same distibutional paametes. So we know that B is a Gaussian matix too. And accoding to the Machenko-Pastu law [9], it's easy to know that: M Rank( B) =. (6) And (6) tell us if is less than M, then B would be a full column ank matix. heefoe, we can use the least-squae method to solve (5), and can get the solutions: = 684
ˆ β = ag min y Bβ. (7) hen we can get αˆ and the econstucted image signal xˆ : β xˆ ( ) ˆ α = = ˆ β k. (8). Analysis of the Reconstuction Quality Assume that we have got xˆ, and now we analyze the diffeence between xˆ and x, which is also called esidual x.we know that: x ( =< x xˆ, x xˆ >= ( x ˆ α ) ) = i, [ ( kx ) ( = i, j And equalities hold when and only when: i, j { x ) ]}. (9) k ˆ α = x, =,,...,. (0) k i, In othe wods, when αˆ is the aveage value of D, < x, x > has a minimum enegy, and unde the condition (0), we use ˆβ, ˆα, ˆx to ewite βˆ,αˆ, xˆ. On the othe hand, if xˆ is the oiginal signal, then the obsevation signal will be yˆ = Axˆ, and the econstucted signal will exactly be xˆ. But if the obsevation signal is y, then it will be almost impossible that the econstucted signal is x. So we need to know the diffeence between y and ŷ. Let y denotes the diffeence, and we know that: ( y = y yˆ = ( ( x ˆ α ) I ) = x I ) = x I. () = i, = i, So y can be consideed as a andom vecto, and accoding to knowledge of chi-squae distibution, when A is a lage dimensional matix, thee is an appoximate popotional elationship between < y, y > and < x, x >. So in the statistical sense, when x ˆ = xˆ, < y, y > eaches the minimum value, o the diffeence between y and ŷ is the smallest. We know that if the diffeence of obsevation signals is smalle, then the diffeence of econstucted signals will be smalle too. So ˆx is the neaest signal to xˆ which is ecoveed fom y, o the econstucted signal tends to be ˆx. 685
Let a denote the aveage value of a small block, b denote the value of a pixel in the cente of this block, and c denote the value of a pixel on the edge of this block, and then usually the diffeence between a and b is smalle than the diffeence between a and c. So in the cente of a block, ˆx and x have a elatively high degee of similaity, while on the edge, have a elatively low degee of similaity. So in ode to get the most appopiate value fo each pixel, we can epeat the dividing and the econstucting fo seveal times, and each time use a diffeent way to divide the image, as is shown in.4. he conclusion that the econstucted signal tends to be ˆx is coect only in the statistical sense, which means we can find fom the esults of lage numbe of expeiments that, geneally speaking, ˆx is most close to x. But it is not tue in a specific expeiment, because y also leads to a econstucted signal. We know that: y = A xˆ + A( x xˆ ) = Ax ˆ + e. () So y can be divided into two pats: one pat is A ˆx and (7) tell us we can get the econstucted signal ˆx fom this pat; the othe pat is e, and also, we can get the econstucted signal δ, which is something we don t expect to see because it can cause some uncetain intefeence. Accoding to (7), we define: ˆ β = ag min e Bβ. (3) β And so we can know that the value of < e, e > o < ˆ β, ˆ β > should be as small as possible. Divide e into two pats: e P = B βˆ, and e V = e B βˆ, then, accoding to the elated knowledge of matix theoy, we can know that ep ev, and : And if we define: e, e >=< e P, e P > + < ev, e >. < V < e P, e P > λ =, (4) < ev, ev > then, λ should also be as small as possible. Fo the M matix B, accoding to the knowledge of subspace pojection, λ inceases when M deceases and maintains constant, o when M maintains constant and inceases. Now we can get some factos that affect the quality of the econstucted signal. he fist one is the image s own featues, such as the smoothness of the image, and they will affect e. he second one is the compession atio M / N which is also called q. If othe conditions emain unchanged, inceasing q is equivalent to inceasing M while letting emain unchanged, and so it is equivalent to deceasing λ which can impove the quality. And q also affects the uppe limit of, because the deivation (6) should be tenable. he thid facto then, is. Deceasing is equivalent to deceasing λ and this could help to impove the quality. Howeve, a 686
smalle means a lage aveage size of the blocks, and usually < e, e > will be lage too, and this will educe the quality. So should be an appopiate value. Of the thee factos, the fist two ae usually unalteable, while the last one is what we can take advantage of..3 Compaison with the Geedy Algoithms Actually, this algoithm is simila to the geedy algoithms, such as OMP, ROMP, etc, because both of the two kinds use some paticula column vectos called key column vectos to solve poblem. We can see fom (3)(5)() that: y = A xˆ + A( x xˆ ) = B ˆ β + A xˆ = [ B, A] w = Cw. (5) And it s easy to know that usually: wi >> wj, i =,,..., ; j = +, +,..., + N. (6) So, the column vectos of B ae moe impotant than the column vectos of A, and they ae the key column vectos. he diffeence, then, is that in the PVS algoithm we can constuct all of these vectos diectly and quickly, while in the geedy algoithms we obtain them by using iteative methods which ae usually timeconsuming jobs. And the expeimental esults will show that the PVS algoithm can save much time..4 Concete Steps In ode to get a bette esult, we need to epeat the dividing and econstucting. Usually, we divide the image into many small blocks and each non-edge block is of the same size called the basic size. Assume that the size of the oiginal image is m n ( m n = N ), the basic size is s t, and the size of the block at the top-left end of the image is u u, and meanwhile define: v = + mod( m u, s), v = + mod( n u, t). (7) K = + ( m u v) / s, L = + ( n u v) / t. (8) hen, it s easy to know that the m n image can be divided into = K L blocks, and the block at lowe ight end of the image has v v pixels. Fo the th block D, we have know that each x has a diffeent coelation degee with size α, so we can constuct a elative weight coefficient matix ρ with the m n, and () ρ epesents the elative weight coefficient between () () x and 687
α. And also we can constuct a total weight coefficient matix p to accumulate ρ, because the econstuction will be epeat fo s t times, and each epetition needs a diffeent ρ. he following is the concete steps: Algoithm:Pixel Value Substitution (PVS) Input: y, A, m, n, s, t. Initialization: u = s, u = t, p = 0, x ˆ = 0. Steps: (i) Divide the image accoding to (7)(8); (ii) Constuct B accoding to (3)(4)(5)(i); and set ρ accoding to (i); (iii) Calculate βˆ and αˆ accoding to (7)(8); (iv) xˆ ( ) = xˆ ( ) + ρ ( ) α ; p = p + ρ ; (v) u = u ; if u > 0, go to (i); else u = s ; (vi) u = u ; if u > 0, go to (i); else end. Output: x ˆ = xˆ / p and xˆ is the output. 3 Expeiments, Results and Discussion In ode to show the effect of the block s size on the quality of the econstucted image, in the expeiments we choose thee diffeent basic sizes including 5 5, 3 3, and fo the dividing of the image, and so we can get 3 specific methods of the PVS algoithm called 5 5 PVS, 3 3 PVS, and PVS. Of couse we should make sue that M. Now we discuss how to set ρ. Fo the 5 5 o 3 3 PVS, we know that usually, D has a cente pixel, and if we use Ω to denote all the cente pixels mentioned above, and all the edge pixels of the while image, then we could have:, x Ω ρ =. 0, x Ω And fo the PVS, then, always we have: ρ =. ake the 5 5 gay LENA test image as the oiginal image, and in ode to incease the speed, it is divided into 56 sub-images [0], and the size of each sub-image is 3 3. hese sub-images ae pocessed independently. 688
34 3 30 5 5PVS 3 3PVS PVS ROMP PSNR(dB) 8 6 4 0 0. 0.5 0. 0.5 0.3 0.35 0.4 0.45 0.5 M/N Fig. he PSNR of each method. Red, geen, blue and black cuves denote espectively the PSNR of the 4 methods as ae shown As a contast, the ROMP algoithm is used too, and the following is the specific pocess. Fist, 5 level wavelet tansfom is applied to the oiginal image, and so we can get the spase image. hen, the spase image is divided into 56 subimages, and the size of each sub-image is 5, which means each sub-image also contains 04 pixels. And they ae pocessed independently too. Fig shows the peak signal to noise atio (PSNR) of each of the fou methods when q vaies fom 0. to 0.5. We can see fom Fig that, fo the PVS algoithm, a pope basic size o can help to impove the PSNR and the quality of the econstucted image, if the othes conditions keep constant. he expeimental esults also show that a pope should satisfy / M < 0. 8. Again, we can see fom Fig that, whateve the value of q is, the best method is not ROMP but one of the 3 PVS methods, which means that compaed with ROMP, PVS can get a bette esult when the basic size is the most pope one. And one eason is that, what the ROMP algoithm econstucts diectly is not the oiginal image itself but the spase image tansfomed fom it, and just one pixel of this spase image may have a geat influence on the whole oiginal image. Fig shows the oiginal image and the econstucted images, and we can see fom Fig (c) that subjective visual effect is good. 689
(a) he oiginal image (b) M/N=0., PSNR=4.9 (c) M/N=0.5, PSNR=33.03 Fig. he oiginal image and the econstucted images based on PVS. (a) denotes the 5 5 LENA test image. (b) denotes the econstucted image when the basic size is 5 5 and M/N= 0., and hee PSNR=4.9. (c) denotes the econstucted image when the basic size is and M/N=0.5, and hee PSNR=33.03 80 ime consumption (second) 70 60 50 40 30 0 5 5PVS 3 3PVS PVS ROMP 0 0 0. 0.5 0. 0.5 0.3 0.35 0.4 0.45 0.5 M/N Fig. 3 he time consumption of each method. Red, geen, blue and black cuves denote espectively the time consumption of the 4 methods as ae shown Fig 3 shows the time consumption of the fou methods. We can see that compaed with ROMP, PVS can significantly educe the time consumption. In fact, ROMP needs many times of iteation to seach those key columns, and each itea- 690
tion needs much moe time than the pevious one. But PVS is diffeent, because each epetition needs the same time which is in invese popotion to the numbe of the epetitions. So, the time consumption of PVS is usually much lowe than that of ROMP, which means PVS can save much time. 3 Conclusion In this pape, we have discussed the econstuction of the natue images when the sampling model is vey simple. We used a new way to solve the poblem, and poposed the PVS algoithm including its concete steps. We also analyzed the econstuction quality and the similaities and diffeences of the two algoithms. he expeimental esults told us it s a feasible method and in some degee is bette than the geedy algoithms. Howeve, we do have some futue wok to do. Fo example, we can use blocks of diffeent sizes to divide the image, o we can popose an adaptive selection method so that the algoithm can automatically choose the most suitable basic size. Refeences. Candès, E. J. (006). Compessive sampling. In Poceedings oh the Intenational Congess of Mathematicians: Madid, August -30, 006: invited lectues (pp. 433-45). Donoho, D. L. (006). Compessed sensing. Infomation heoy, IEEE ansactions on, 5(4), 89-306. 3. Candes, E., & Rombeg, J. (007). Spasity and incoheence in compessive sampling. Invese poblems, 3(3), 969 4. Candès, E. J., & Wakin, M. B. (008). An intoduction to compessive sampling. Signal Pocessing Magazine, IEEE, 5(), -30 5. Mallat, S. G., & Zhang, Z. (993). Matching pusuits with time-fequency dictionaies. Signal Pocessing, IEEE ansactions on, 4(), 3397-345 6. opp, J. A., & Gilbet, A. C. (007). Signal ecovey fom andom measuements via othogonal matching pusuit. Infomation heoy, IEEE ansactions on, 53(), 4655-4666 7. Needell, D., & Veshynin, R. (009). Unifom uncetainty pinciple and signal ecovey via egulaized othogonal matching pusuit. Foundations of computational mathematics, 9(3), 37-334 8. Rudin, L. I., Oshe, S., & Fatemi, E. (99). Nonlinea total vaiation based noise emoval algoithms. Physica D: Nonlinea Phenomena, 60(), 59-68 9. Bai, Z. D. (999). Methodologies in spectal analysis of lage-dimensional andom matices, a eview. Statist. Sinica, 9(3), 6-677 0. Gan, L. (007, July). Block compessed sensing of natual images. In Digital Signal Pocessing, 007 5th Intenational Confeence on (pp. 403-406). IEEE 69