The Story of Wavelets Theory and Engineering Applications 2D-DWT using MATLAB (review) Implementation issues Advanced Topics: Wavelet Packets Other Applications Density Estimation

Recall: 2D-DWT Just like in 1D we generated an approximation of the 2D function f(x,y). Now, how do we compute the detail lost in approximating this function? Unlike 1D case there will be three functions representing the details lost: Details lost along the horizontal direction Details lost along the vertical direction Details lost along the diagonal direction 1D  Two sets of coeff.; a(k,n) & d (k,n) 2D Four sets of coefficients: a(k,n), b(k, n), c(k, n) & d(k,n)

Implementation of 2D-DWT INPUT IMAGE …… ROWS COLUMNS H ~ 2 1 G 1 2 LL LH HL HH INPUT IMAGE LH HL HH LHH LLH LHL LLL LLH LL LH LH LL LHL LHH HH HL HH HL

Up and Down … Up and Down Downsample columns along the rows: For each row, keep the even indexed columns, discard the odd indexed columns 2 1 Downsample rows along the columns: For each column, keep the even indexed rows, discard the odd indexed rows 1 2 Upsample columns along the rows: For each row, insert zeros at between every other sample (column) 2 1 Upsample rows along the columns: For each column, insert zeros at between every other sample (row) 1 2

Implementing 2D-DWT Decomposition COLUMN j ROW i

Reconstruction LL H 1 2 2 1 H LH G 1 2 HL H 1 2 G 2 1 HH G 1 2 1 2 H 2 1 H LH 1 2 G ORIGINAL IMAGE HL 1 2 H 2 1 G HH 1 2 G

2-D DWT ON MATLAB Load Image Choose (must be wavelet type .mat file) Hit Analyze Choose display options

Recall 1-D DWT g[n] h[n] x[n] B: 0 ~  g[n] h[n] 2 d1: Level 1 DWT Coeff. B: 0 ~ /2 Hz d2: Level 2 DWT d3: Level 3 DWT ……. B: 0 ~ /4 Hz B: 0 ~ /8 Hz In DWT, only approximation coefficients are decomposed. Each decomposition allows dyadic dichotomization of the frequency spectrum What if we were decompose the detail coefficients as well…? Frequency Time

Wavelet Packets : H H H G G H G H H G H G H G H G 2 x[n] A(1) D(1) B: 0 ~  A(1) D(1) H G 0 ~ /2 G /4 ~ /2 H 0 ~ /4 G 3/4 ~  H /2 ~ 3/4 /2 ~  AA(2) DA(2) AD(2) DD(2) H 0 ~ /8 G /8 ~ /4 H /4 ~ 3/8 G 3/8 ~ /2 H /2 ~ 5/8 G 5/8 ~ 3/4 H 3/4 ~ 7/8 G 7/8 ~  AAA(3) DAA(3) ADA(3) DDA(3) AAD(3) DAD(3) ADD(3) DDD(3)

Wavelet Packets Frequency Time

Wavelet Packets on MATLAB

What About Scaling and Wavelet Functions ??? You Ask… In DWT, we used scaling functions to generate lowpass filters, and wavelet functions to generate highpass filters. In WP analysis, filters are generated by related, but different analysis functions. Two-scale (dilation) equations where 2N: Filter length

How Many Decompositions Are Too Many??? For a signal of length N=2L, we can have L levels of 1D-DWT. For the same signal, we can have a maximum of 2N levels of decompositions For a 512 sample signal  x10123 13407807929942597099574024998206

Choosing the Best Tree The best tree is the one that gives the most information. What is “the most information”…you ask…. Entropy based definitions Normalized Shannon entropy Norm based entropy Energy based entropy Threshold based entropy If at any level, splitting a branch results in less sntropy, the splitting provides more information. Matlab Demo: noisychirp

Density Estimation Density:??? 3 5 4 5 3 7 9 8 7 3 4 5 10 7 3 4 5 7 14 12 10 3 5 4 7 9 9 3 4 5 5 4 5 4 3 7 3 4 4 5 6 5 6 7 5 5 4 3 3 4 5 6 8 10 10 2 3 1 1 0 0 3 3 4 5 6 3 2 HISTOGRAM Density function

Density Estimation Density estimation allows us to infer statistical characteristics of data From what distribution is the data coming Reliability, life cycle Average quality, etc. mean Number of “60W” bulbs Watts 60

Density Estimation How do we estimate density??? Matlab demo….Load ex1cusp2.mat from wavelet toolbox Plot the data…What do you observe? Can you infer any information from this data? Plot as “points”… What can you say now? Plot the histogram of the data …>>hist(ex1cusp2) Histogram can be used as a rough estimate of the density Too noisy Takes “every sample” into account, regardless how irrelevant (noisy) it may be Better way? What else, but of course,…wavelets

Wavelets to the Rescue (again) If the histogram is a noisy rough estimate of the density Denoise histogram using wavelet shrinkage denoising Select wavelet Choose denoising thresholds Select number of bins Select thresholding scheme