pixel

1. pixel c

2. 16x16 YCbCr 4:2:0 8x8 8x8 Y Cb Cr v 16x16 16x16 16x16 R G B

3. Wavelet Transform • Alternative to Fourier transform • Localized in time and frequency • No blocking/windowing artifacts • Compact support • Sums of dilations and translations of (mother) wavelet function

4. 2 2 2 2 2 2 2 H1 H1 H2 H1 H2 H2 Hi 1-Level Wavelet Decomposition (2D DWT) LL Component (Low pass) HL Component (Low pass) Input Image (High pass) LH Component (Low pass) HH Component (High pass) (High pass) Row-wise operations Column-wise operations Filter Decimator x[n] y[n] Keep one out of two pixels

5. LL HL2 HL1 HL1 LL LH2 HH2 LH1 HH1 LH1 HH1 2D-DWT 2D-DWT c

6. Wavelet-Transformation (1D) • Die diskrete Wavelettransformation (DWT) basiert auf einer schrittweisen Filterung von Signalen(oder Datenreihen). Filterung ist die Faltung einer Funktion f mit einem Filter h. Im diskretenFall: • Filter sind in der Lage, gewisse Frequenzbereiche einer Funktion zu unterdr. ucken. In der Wavelettransformation nden immer zwei Filter (Filterpaar) Anwendung. Ein Hochpasslter g und einTiefpasslter h. Die Frequenzg.ange dieser Filter sollen sich ausschlie¡en, sodass die Daten nachder Filterung in einen tief- und einen hoch-frequenten Teil getrennt ist.

7. c Beim Ergebnis l.asst man jeden zweiten Wert aus: Downsampling. Dadurch bleibt die Datenmenge gleich. Die gelterten Datenreihen nennt man Subbands. Danach f.ahrt man mit demtiefpassgelterten Subband (approximation subband, i.Gs. zu detail subband) rekursiv fort.

8. C Wavelettransformation (2D) Bei der zweidimensionalen DWT f. uhrt man eine horizontale und eine vertikale Filterung (aufZeilen und Spalten) hintereinander aus. Dadurch ergeben sich 4 neue Subbands. Davon wird nureines (LL, das in beide Richtungen tiefpassgeltert worden ist) weiterzerlegt.

9. The original signal, S, passes through two complementary filters and emerges as two signals Discrete Wavelet Transform Analysis Filter Pair The Approximation – (f-domain) Low frequency components (t-domain) Average of the data at previous stage The Detail – (f-domain) High frequency components, (t-domain) The difference of the data at the previous stage and the averaged data at the current stage A and D are orthogonal. Thus the decomposition of the data is a change of basis to an orthogonal basis – the wavelet basis. The condition of perfect inverse tranformation

10. Discrete Wavelet Transform The signal’s wavelet decomposition tree can yield valuable information

11. Discrete Wavelet Transform The Whole Picture of the process of Discrete Wavelet Transform

12. DWT in Image N*M N/2*M N/2*M/2

13. Inverse DWT in Image

14. The image in DWT

15. DWT separates function into averages and details global and local info Two filters: highpass and lowpass lowpass: low frequency (averages) highpass: high frequency (details) Highpass filter: decimates constant signal (no detail info) Lowpass filter: decimates oscillating signal (no global info) Result: two signals, half length of original most info in lowpass signal

16. JPEG2000 & Wavelet Compression • New JPEG standard wavelet-based • Wavelet compression studied extensively for years • JPEG2000 first attempt at standardizing • WSQ: used to compress fingerprints for FBI • used in place of JPEG, which quickly blurred important information • Similar compression ratios to JPEG, but with higher quality

17. Standard Decomposition 2dim - V2 L H LL HL H LLL LHL LH Row HLL HHL HH Col LL LL LL HL HLL HL LL HL HL HHH HLL HHL HH

18. Bilde

19. Komprimering 1:50JPEG Wavelet

20. JPEG2000 Verification Model • Wavelet transform • Mallatt (dyadic), SPACL, Packet decompositions • Filters: several options available • floating point -(9,7) tap biorthogonal (Daubechies) • integer - (5,3) Daubechies, (13,7) CRF or TUB, (2,10), S+P

21. Two Kinds of Bases for DWT • Reversible basis: 5/3 filter • Analysis Filters: hL: 1/8 [-1 2 6 2 –1], hH : 1/2 [-1 2 –1] • Synthesis Filters: gL: 1/2 [1 2 1], gH : 1/8 [-1 -2 6 –2 –1] • Irreversible basis: Daubechies 9/7 filter • Analysis Filters: hL: [0.027 –0.017 –0.08 0.267 0.603 0.267 –0.08 –0.017 0.027], hH : [0.09 –0.06 –0.59 1.12 –0.59 –0.06 0.09]. • Synthesis Filters: gL: [-0.09 –0.06 0.59 1.12 0.59 –0.06 -0.09], gH :[0.027 0.017 –0.08 -0.267 0.603 -0.267 –0.08 0.017 0.027].

22. 5/3 Filer • Include Reversible and Irreversible Filters • Odd value are high frequency value • Even value are low frequency value • Here below is the equations for the reversible one

26. wavelet ~ „hullámocska” (wave = hullám ~ szinusz) • A wavelet egy véges intervalumon definiált függvény, melynek átlagértéke zérus. • A wavelet transformáció egy tetszőleges ƒ(t) függvény és a wavelet függvényekből álló készlet szuperpoziciója. • A wavelet függvények készletét „baby wavelet”-eknek is nevezik, mivel egyetlen prototipusból az un „anya wavelet”-ből vannak származtatva egyrészt zsugorítással (scaling) másrész eltolássalm (shift). • DWT in Terms of Filters • The basic idea of the wavelet transform is to represent any arbitrary function ƒ(t) as a superposition of a set of such wavelets or basis functions. These basis functions or baby wavelets are obtained from a single prototype wavelet called the mother wavelet, by dilations or contractions (scaling) and translations (shifts). • The Discrete Wavelet Transform of a finite length signal x(n) having N components, for example, is expressed by an N x N matrix. For a simple and excellent introduction to wavelets, see [6]. For a thorough analysis and applications of wavelets and filterbanks, see

27. WAvelet • Despite all the advantages of JPEG compression schemes based on DCT namely simplicity, satisfactory performance, and availability of special purpose hardware for implementation, these are not without their shortcomings. Since the input image needs to be ``blocked,'' correlation across the block boundaries is not eliminated. This results in noticeable and annoying ``blocking artifacts'' particularly at low bit rates as shown. Lapped Orthogonal Transforms (LOT) attempt to solve this problem by using smoothly overlapping blocks. Although blocking effects are reduced in LOT compressed images, increased computational complexity of such algorithms do not justify wide replacement of DCT by LOT.

28. c

29. Wavelet • The Haar Wavelet The Haar Wavelet is probably the simplest wavelet to understand. Consider two numbers a and b, neighboring samples of a sequence. a and b have some correlation (hopefully) of which we'd like to take advantage. We use a simple linear transform to replace a and b by their average s and difference d. • s = (a + b)/ 2 d = b - a • We can think of the averages as a coarser resolution version of the original signal, and the differences as the higher resolution details. If the original signal is highly correlated, the coarse representation of the signal is very close to the original, and the details can be represented efficiently. • We can then apply the same average/difference transform to the coarser signal, splitting it into a yet coarser signal and more details. This transform can be applied until their is but a single signal value and a whole lot of detail values. The remaining signal is the average of the entire signal, and can be thought of as the DC or zero frequency of the original signal.

30. Wavelet theory is also a form of mathematical transformation, similar to the FT in that it takes a signal in time domain, and represents it in frequency domain. Wavelet functions are distinguished from other transformations in that they not only dissect signals into their component frequencies, they also vary the scale at which the component frequencies are analyzed. Therefore wavelets, as component pieces used to analyze a signal, are limited in space. In other words, they have definite stopping points along the axis of a graph--they do not repeat to infinity like a sine or cosine wave does. As a result, working with wavelets produces functions and operators that are "sparse" (small), which makes wavelets excellently suited for applications such as data compression and noise reduction in signals. The ability to vary the scale of the function as it addresses different frequencies also makes wavelets better suited to signals with spikes or discontinuities than traditional transformations such as the FT.