1、Wavelet Transform第x 章 小波變換小波變換簡介小波及小波變換離散小波變換1DWTDyadic wavelet transformDyadic Wavelet Transform (DWT)Discrete Wavelet Transform (DWT)For easier calculation we can to discrete continuous signal.We have a grid of discrete values that called dyadic grid. Important that wavelet functions compact (e.g.

2、 no over-calculating) .2DWTDyadic wavelet transformDyadic Wavelet Transform (DWT)Haar Transform3DWTDyadic wavelet transformStandard Two Band Dyadic Wavelet TransformHigh PassLow PassLow passHigh passDecimated Wavelet Transform-Analysis StageHigh Freq DetailsLow Freq Approx Signal2 24DWTDyadic wavele

3、t transformStandard Two Band Dyadic Wavelet TransformTime-Freq. Tiling5DWTDyadic wavelet transformDyadic Wavelet Transform (DWT)Multiresolution as a subband coding of a signal, Signal components at different subbands are extractedDifferent analyzing wavelets results in different components at differ

4、ent bands 6DWTSummery of wavelet transformPrimary Features of Wavelets used in Signal ProcessingTime- frequency Localization (useful for transient data analysis)Sparsity: Coefficients are often grouped into high and low amplitudeFunction Approximation: near Optimal Function Approximation Noise Reduc

5、tion, No wavelet can model white noise leading to small amplitude coefficientsDecorrelation: Signal components( coefficients) in wavelet domain have a lower degree of correlation 7Wavelet Transform第x 章 小波變換小波變換簡介小波及小波變換離散小波變換Harr小波圖像編碼8Image encodingHarr wavelet based image encodingIntroduction to i

6、mage encodingDone by compressing the imageUse a process called averaging and differencing to develop a new matrix representing the same image in a more concise manner.Eliminate some of unnecessary information, and arrive at an approximation of our original image9Image encodingHarr wavelet based imag

7、e encodingDyadic Haar Transform10Dyadic Haar Transform8X8 matrix, process involve three steps (2=8)Step 13 5 4 8 13 7 5 3 4 6 10 4 -1 -2 3 1ex. (3 + 5)/2 = 4 3 - 4 = -1Image encodingHarr wavelet based image encodingAveragingDifferencingdetail coefficient11Image encodingHarr wavelet based image encod

8、ingDyadic Haar Transform12Dyadic Haar TransformStep 24 6 10 4 -1 -2 3 1 5 7 -1 3 -1 -2 3 1ex. (4 + 6)/2 = 5 4 - 5 = -1Image encodingHarr wavelet based image encodingAveragingDifferencingdetail coefficient13Image encodingHarr wavelet based image encodingDyadic Haar Transform14Dyadic Harr TransformSte

9、p 25 7 -1 3 -1 -2 3 1 6 -1 -1 3 -1 -2 3 1Image encodingHarr wavelet based image encodingAveragingDifferencingdetail coefficientrow average15Image encodingHarr wavelet based image encodingDyadic Harr TransformNoting is done at this scale!16Harr Transform for image encodingImage encodingHarr wavelet b

10、ased image encodingAn image with 8x8 pixels17Harr Transform for image encodingThe first row 64 2 3 61 60 6 7 57Step1 33 32 33 32 31 -29 27 -25Step2 32.5 32.5 0.5 0.5 31 -29 27 -25Step3 32.5 0 0.5 0.5 31 -29 27 -25Image encodingHarr wavelet based image encoding18Harr Transform for image encodingThe r

11、esult for all the rowsImage encodingHarr wavelet based image encodingrow averagedetail coefficients19Harr Transform for image encodingcolumns the resultschoose some number () and set equal to zeroImage encodingHarr wavelet based image encoding = 520Harr Transform for image encodingapply the inverse

12、of the averaging the differencing operationsImage encodingHarr wavelet based image encoding21Harr Transform for image encodingapply the inverse of the averaging the differencing operationsImage encodingHarr wavelet based image encodingDecompressed ImageOriginal Image22Image encoding1 level HaarOrigi

13、nal2 level Haar23Harr Transform for image encodingWavelet coefficient histogramImage encodingHarr wavelet based image encoding24Harr Transform for image encodingWavelet coefficient entropiesImage encodingHarr wavelet based image encodingEntropyOriginal image7.221-level Haar wavelet5.961-level linear spline wavelet5.532-level Haar wavelet5.022-level linear spline wavelet4.5725Harr Transform for image encodingWavelet coefficient dependenciesImage encodingHarr wav