Optimum LMSE Discrete Transform

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1 Image Trasformatio Two-dimesioal image trasforms are extremely importat areas of study i image processig. The image output i the trasformed space may be aalyzed, iterpreted, ad further processed for implemetig diverse image processig tass. These trasformatios are widely used, sice by usig these trasformatios, it is possible to express a image as a combiatio of a set of basic sigals, ow as the basis fuctios. I case of Fourier trasform of a image these basis sigals are siusoidal sigals with differet periods which describe the spatial frequecies i a image. Thus such trasforms, such as the Fourier trasform, reveal spectral structures embedded i the image that may be used to characterize the image. The term Image trasform are usually refers to a class of uitary matrices used for represetig images. large class of image processig trasformatios is liear i ature; a output image is formed from liear combiatios of pixels of a iput image. Such trasforms iclude covolutios, correlatios, ad uitary trasforms. Liear trasforms have bee utilized to ehace images ad to extract various features from images. For example, the Fourier trasform is used i highpass ad lowpass filterig as well as i texture aalysis. Aother applicatio is image codig i which badwidth reductio is achieved by deletig low-magitude trasform coefficiets. The followig table illustrate the complexity of trasformatios Trasform Processig ad Ecodig TYPE OPERATIOS COMMETS Karhue Loeve 4 Optimum LMSE Discrete Trasform Discrete Fourier HADAMARD log log Asymptotic to K.L.Trasform Biary Basis Matrices Discrete Cosie More Complex Useful i sca directio i real time, Better tha Fourier ad Hadamard Slat Trasform Complex Suited for vertical ecodig, Better tha Fourier ad Hadamard HAAR Trasform Geeralized Differetial Ecodig Simply Implemeted Fourier Trasform Fourier trasform is oe of the most importat tools which have bee extesively used ot oly for uderstadig the ature of a image ad its formatio but also for processig the image. Usig Fourier trasform, it has bee possible to aalyze a image as a set of spatial siusoids i various directios, each siusoid havig a precise frequecy.

2 Oe-Dimesioal Fourier Trasform The oe-dimesioal cotiuous Fourier trasform,(cft) of a cotiuous fuctio f(x) is Two-Dimesioal Fourier Trasform Extedig the cocept of oe-dimesioal Fourier trasform, the two-dimesioal Fourier trasform of a cotiuous fuctio f(x, y) is deoted by The variable i Eq. 4.6 idicates the frequecy,i.e., the umber of waves per uit legth i X directio, ad idicates the umber of waves alog the Y directio. The correspodig iverse two-dimesioal Fourier trasform is Discrete Fourier Trasform (DFT) Sice the sigal is discretized, the operatio of itegratio i cotiuous Fourier trusfom (CFT) is replaced by summatio operatios i DFT. We preset the oedimesioal DFT ad also the two-dimesioal DFT i the followig subsectios. Oe-Dimesioal DFT The oe-dimesioal discrete Fourier trasform of a fuctio f(x) of size with iteger idex x ruig from 0 to -, is represeted by

3 [error : replace y i summatio by x] The correspodig oe-dimesioal iverse DFT is [replace M by ] Two-Dimesioal DFT The two-dimesioal discrete Fourier trasform of a two-dimesioal sigal f ( x, y ) of dimesio M x with iteger idices x ad y ruig from 0 to M - ad 0 to -, is represeted by The equivalet two-dimesioal iverse DFT is Trasformatio Kerels As we have already oted the geeral forward ad iverse Fourier trasformatio ca be expressed as I the above equatios g(x, y, u, v) is ow as forward trasformatio erel ad h(x, y, u, v) is the iverse trasformatio erel. Here u ad v assume values i the rage (0,,..., - ). As ca be further observed, these erels are depedet oly 3

4 o the spatial positios ad frequecies alog X ad Y directios, ad are idepedet of either f ( x,y ) or F(u,w ). If the erel g ( x, y, u, v ) = g(x,u)g(y,v), we say that the erel is separable. If i additio the fuctioal forms of g ad g are idetical, the the erel is said to be symmetric. It easy to show that the erel g(x, y, u, v) is separable ad symmetric sice Properties - Traslatio: The traslatio of a Fourier trasform pair is The above implicatio idicates the correspodece betwee a two dimesioal image fuctio ad its Fourier trasform. - Rotatio: Assumig that the fuctio f ( x,y ) udergoes a rotatio of, the correspodig fuctio f (x,y ) i polar coordiates will the be represeted as f ( r, ), where x= r cosc ad y = r si. The correspodig DFT F(u,w) i polar coordiates will be represeted as The above implies that if f ( x,y ) is rotated by, the F ( u,w ) will be rotated by the same agle ad hece we ca imply that correspods to i the DFT domai ad vice versa. 3- Separability : The separability property of a two-dimesioal trasform ad its iverse esures that such computatios ca be performed by decomposig the twodimesioal trasforms ito two oe-dimesioal trasforms. From Eqs. 4.3 ad 4.4 describig the DFT ad iverse DFT of a two-dimesioal fuctio f ( x,y ), we ca express them i separable form as follows. 4- Distributive property: The DFT of sum of two fuctios f( x,y ) ad f ( x, y ) is idetical to the sum of the DFT of these two fuctios, i.e., 4

5 5- Scalig property: The DFT of a fuctio f ( x, y ) multiplied by a scalar() is idetical to the multiplicatio of the scalar with the DFT of the fuctio f(x,y), i.e., 6- Covolutio: The DFT of covolutio of two fuctios is equal to the product of the DFT of these two fuctios, i.e., 7- Correlatio: The correlatio betwee two fuctios f(x,y) ad f ( x,y) i cotiuous domai is deoted as whereas the correlatio i the discrete domai is deoted as 8- Periodicity: The DFT of a two-dimesioal fuctio f ( x, y ) ad its iverse are both periodic with period, i.e., Fast Fourier Trasform The umber of complex multiplicatios ad additios to compute Eq. 4. for DFT is O(). However, we ca adopt a divide-ad- coquer approach to reduce the computatioal complexity of the algorithm to O( log ). This algorithm is popularly ow as the Fast Fourier Trasform (FFT). Here we preset a geeral idea of the divide-ad-coquer approach toward implemetatio of the FFT. The geeral priciples is based o successive divisio method usig divide-ad-coquer approach as explaied below. As show i Eq. 4., the oe-dimesioal DFT of a oedimesioal 5

6 sigal f(x) is computed The Discrete Cosie Trasform ( DCT ) This importat trasform (DCT for short) has origiated by. The DCT i oe dimesio is give by The iput is a set of data values pt (pixels, audio samples, or other data), ad the output is a set of DCT trasform coefficiets (or weights) Gf. The first coefficiet G0 is called the DC Coefficiet, ad the rest are referred to as the AC coefficiets. otice that the coefficiets are real umbers eve if the iput data cosists of itegers. Similarly, the coefficiets may be positive or egative eve if the iput data cosists of oegative umbers oly. This computatio is The IDCT i oe dimesio is give by the most of the trasform coefficiets produced by the DCT are zeros or small umbers, ad oly a few are large (ormally the first oes). We will see that the early coefficiets cotai the importat (low-frequecy) image iformatio ad the later coefficiets cotai the less-importat (high-frequecy) image iformatio. Compressig data with the DCT is therefore doe by quatizig the coefficiets. Experiece idicates that = 8 is a good value, ad most data compressio methods that employ the DCT use this value of. The followig example illustrates the differece i performace betwee the DCT ad the DFT. We start with the simple, highly correlated sequece of eight umbers (8, 6, 4, 3, 40, 48, 56, 64). It is displayed graphically i followig Figure Applyig the DCT to it yields (00, 5, 0, 5, 0,, 0, 0.4). Whe this is quatized to 6

7 (00, 5, 0, 5, 0, 0, 0, 0) ad trasformed bac, it produces (8,5,4, 3, 40, 48,57, 63), a sequece almost idetical to the origial iput. Applyig the DFT to the same iput, o the other had, yields (36, 0, 0, 6, 6, 4, 4, 4). Whe this is quatized to (36, 0, 0, 6, 0, 0, 0, 0) ad is trasformed bac, it produces (4,, 0, 3,40,5, 59, 48). This output is show i the followig Figure, ad it illustrates the tedecy of the Fourier Trasform to produce a periodic result. The DCT i two dimesio is give by for 0 i ad 0 j m ad for Ci ad Cj defied by Equatio (4.3). The first coefficiet G00 is agai termed the DC coefficiet, ad the remaiig coefficiets are called the AC coefficiets., The IDCT i two dimesio is give by We illustrate the performace of the DCT i two dimesios by applyig it to two blocs of 8 8 values. The first bloc (Table 4.3a) has highly correlated iteger values i the rage [8, ], ad the secod bloc has radom values i the same rage(table 4.4) The first bloc results i a large DC coefficiet, followed by small AC coefficiets (icludig 0 zeros, Table 4.3b, where egative umbers are uderlied). Whe the coefficiets are quatized (Table 4.3c), 7

8 The result, show i Table 4.3d, is very similar to the origial values. I cotrast, the coefficiets for the secod bloc (Table 4.4b) iclude just oe zero. Whe quatized (Table 4.4c) ad trasformed bac, may of the 64 results are very differet from the origial values (Table 4.4d). The ext example illustrates the differece i the performace of the DCT whe applied to a cotiuous-toe image ad to a discrete-toe image. We start with the highly correlated patter of Table

9 9 This is a idealized example of a cotiuous-toe image, sice adjacet pixels differ by a costat amout except the pixel (uderlied) at row 7, colum 7. The 64 DCT coefficiets of this patter are listed i Table 4.6. It is clear that there are oly a few domiat coefficiets. Table 4.7 lists the coefficiets after they have bee coarsely quatized, so that oly four ozero coefficiets remai! The results of performig the IDCT o these quatized coefficiets are show i Table 4.8. It is obvious that the four ozero coefficiets have recostructed the origial patter to a high degree. The oly visible differece is i row 7, colum 7, which has chaged from to 7.55 (mared i both figures). [ page 5-54 of ail] properties of dct بولطم Sie trasform The third equatio represet the two dimesio (,) 0.5 provided that KLT, DST iscloseto the fast algorithm, 0, si, 0, si 0, si 0 0 S S S S v u u v T

10 Hartley Trasform The discrete two-dimesioal Hartley trasform is defied by the trasform pair Hadamard Trasform [Ray] The basis vector of the trasform tae oly the biary value +,- The symmetric WHT for =,, 4 ad 8 are show i Eqs respectively: H = [I] (4.6) 0

11 Haar Trasform [scott page 5-30] KARHAUE-LOEVE TRASFORM OR PRICIPAL COMPOET AALYSIS [ray] Karhaue.-Loeve Trasform, or Pricipal Compoet Aalysis (PCA) has bee a popular techique for may image processig ad patter recogitio applicatios. This trasform which is also ow as Hotellig Trasform is based o the cocepts of statistical properties of image pixels or patter features Pricipal compoet aalysis (PCA) forms the basis of the Karhue-Loeve (KL) trasform for compact represetatio of data. The KL trasform ad the theory behid the pricipal compoet aalysis are of

12 fudametal importace i sigal ad image processig. The priciple has also foud its place i data miig for reductio of large-dimesioal datasets. It has bee successfully applied to text aalysis ad retrieval for text miig as well, Before describig the PCA, we would briefly preset the cocepts of covariace matrix. Covariace Matrix I practical patter recogitio problems there are usually more tha oe feature. Durig the process of statistical aalysis of these data, we have to fid out whether these features are idepedet of oe aother. Otherwise there exists a relatioship betwee each pair of features. For example, while extractig the features of huma face, oe may choose two features such as () X to deote the distace betwee the ceters of the two irises, ad () Y to deote the distace betwee the ceters of the left ad right eyebrows. From a large set of huma faces, we ca determie the mea ad the stadard deviatio of the above two features. The stadard deviatio for each of the above two dimesios of the face data set may be computed idepedetly of each other. To uderstad whether there exists ay relatioship betwee these two features, we have t o compute how much the first feature X of each of the patters i our data set varies from the mea of the secod feature Y. This measure, which is computed similar to variace, is always measured betwee two features. The covariace is computed as follows: the covariace is measured betwee each pair of features. Eigevectors ad Eigevalues Before we discuss pricipal compoet aalysis, we will briefly explai the cocept of eigevectors ad eigevalues of a matrix. Let us assume that we have a square matrix A of dimesio x, which whe multiplied by a vector X of dimesio x yields aother vector Y of dimesio x, which is essetially the same as the origial vector X that was chose iitially. Such a vector X is called a eigevector which trasforms a square matrix A ito a vector, which is either the same vector X or a multiple of X. Every eigevector is associated with a correspodig eigevalue. The cocept of a eigevalue is that of a scale which whe multiplied by the eigevector yields the same scaled vector i the same directio. Pricipal Compoet Aalysis While computig the pricipal compoet aalysis, we represet a x image as a oe-dimesioal vector of * elemets, by placig the rows of the image oe after aother. The we compute the covariace matrix of the etire data set. ext we compute the eigevalues of this covariace matrix. The eigevectors correspodig

13 to the most sigificat eigevalues will yield the pricipal compoets. To get the origial data bac we have to cosider all the eigevectors i our trasformatio. If we discard some of the less sigificat eigevectors i the fial trasformatio, the the retrieved data will lose some iformatio. However, if we choose all the eigevectors, we ca retrieve the origial data. The Slat Trasform [Ail page 6 6] The x slat trasform matrices are defied by the recursio. Table 5.3 Summary of image trasform [Ail page 79] 3