几 igure 2 The box basis for I Figure 3 The Haar wavelets for W1 A more graphical representation is Orthogonality (x)=9x The haar basis es an important onaliry, which is not always shared by ot thogonal basis is one in which all of the 40, 0, v0, 1i,... are orthogonal to one another. Note that orthoge nality is stronger than the minimum requirement for wavelets thaty/ be orthogonal to all scaling functions at the same resolution level Note that the coefficie c, are just the four original pixel values 9735] that is sometimes desirable is normalization. A We can rewrite the expression for I(x)in terms of basis functions )is normalized if (uu)=1. We can normaliz in v and w, using pairwise averaging and differencing: replacing our earlier definitions with I(x)=c0 0(x)+c oi(x)+do w0(x)+divx) u(x):=22u(x-0 where the constant factor of 2/2 is chosen to sati the standard inner product. with these modified definitions, the new 4 normalized coefficients are obtained by multiplying each old coef- ficient with superscript by 271/2. Thus, in the example from the previous section, the unnormalized coefficients 62 1-1] becom the nor ed coefficients These four coefficients should look familiar as well Finally, we'll rewrite I(x)as a sum of basis functions in r, w As an alternative to first computing the unnormalized coefficients and then normalizing them, we can include normalization in the de- and n composition algorithm. The following two pseudocode procedures (x)=c0 90()+d0/0(x)+d6 w0(x)+ divi(x) accomplish this normalized decomposition procedure Decomposition Step(C: array [l..h of reals) fori←-ltoh/2do C←(C[2i-1]+C2) C[h/2+←(C12i-1-C[2)/ end procedure procedure Decomposition(C: array [1.. h of reals) CC/vh (normalize input coefficients) Once again, these four coefficients are the Haar wavelet transform DecompositionStep(CIl.h the Haar basis for wge The four functions shown above constitute f the original Instead of using the usual four box functions end wl we can use 0, w 0, v 0, and w i to represent the overall average,the end procedure broad detail, and the two types of finer detail possible in a function y. The Haar basis for D with j>2 includes these functions Now we can work with an orthonormal basis well as narrower translates of the wavelet l(x) both orthogonal and normalized. Using an orthonormal basis turns1 0 0 1 1 2 1 0 0 1 1 2 1 0 0 1 1 2 2 1 2 2 2 3 1 0 0 1 1 2 2 0 Figure 2 The box basis for V2. 1 0 1 2 1 0 1 -1 1 2 1 0 -1 1 1 1 Figure 3 The Haar wavelets for W1. A more graphical representation is I(x) = 9 + 7 + 3 + 5 Note that the coefficients c2 0, : : : , c2 3 are just the four original pixel values [9 7 3 5]. We can rewrite the expression for I(x) in terms of basis functions in V1 and W1 , using pairwise averaging and differencing: I(x) = c1 0 1 0(x) + c1 1 1 1(x) + d1 0 1 0 (x) + d1 1 1 1 (x) = 8 + 4 + 1 + ￾1 These four coefficients should look familiar as well. Finally, we’ll rewrite I(x) as a sum of basis functions in V0 , W0 , and W1 : I(x) = c0 0 0 0(x) + d0 0 0 0 (x) + d1 0 1 0 (x) + d1 1 1 1 (x) = 6 + 2 + 1 + ￾1 Once again, these four coefficients are the Haar wavelet transform of the original image. The four functions shown above constitute the Haar basis for V2 . Instead of using the usual four box functions, we can use 0 0, 0 0 , 1 0 , and 1 1 to represent the overall average, the broad detail, and the two types of finer detail possible in a function in V2 . The Haar basis for Vj with j > 2 includes these functions as well as narrower translates of the wavelet (x). Orthogonality The Haar basis possesses an important property known as orthog￾onality, which is not always shared by other wavelet bases. An or￾thogonal basis is one in which all of the basis functions, in this case 0 0, 0 0 , 1 0 , 1 1, : : :, are orthogonal to one another. Note that orthogo￾nality is stronger than the minimum requirement for wavelets that j i be orthogonal to all scaling functions at the same resolution levelj. Normalization Another property that is sometimes desirable is normalization. A basis function u(x) is normalized if hu j ui = 1. We can normalize the Haar basis by replacing our earlier definitions with  j i (x) := 2j=2 (2j x ￾ i) j i (x) := 2j=2 (2j x ￾ i), where the constant factor of 2j=2 is chosen to satisfy hu j ui = 1 for the standard inner product. With these modified definitions, the new normalized coefficients are obtained by multiplying each old coef- ficient with superscript j by 2￾j=2 . Thus, in the example from the previous section, the unnormalized coefficients [6 2 1￾1] become the normalized coefficients 6 2 p1 2 p￾1 2 As an alternative to first computing the unnormalized coefficients and then normalizing them, we can include normalization in the de￾composition algorithm. The following two pseudocode procedures accomplish this normalized decomposition: procedure DecompositionStep(C: array [1. . h] of reals) for i 1 to h=2 do C0 [i] (C[2i ￾ 1] + C[2i])=p 2 C0 [h=2 + i] (C[2i ￾ 1] ￾ C[2i])=p 2 end for C C0 end procedure procedure Decomposition(C: array [1. . h] of reals) C C=p h (normalize input coefficients) while h > 1 do DecompositionStep(C[1. . h]) h h=2 end while end procedure Now we can work with an orthonormal basis, meaning one that is both orthogonal and normalized. Using an orthonormal basis turns 3