The Music Signal Approximation Based On The Frame And The Image Compression Based On Second-generation Bandelet Transform

The work of the first part of this article is based on the framework forapproaching the music signal.At present article has a lot of research on automatic classification of musicsignals, and some e?ective methods, such as Hidden Markov Model-based auto-matic classification of audio, but they may have some ?aws, can not be completelysatisfactory results.We listen to music, it will clearly"hear"the voice"frequency"changeover time. These partial frequency is not the single tone but frequency packageswhich closes to each other, use the basic function with better time-frequencylocalization to transform and decomposite the signal can explaine the natureof the sound . Therefore measurement of transient changes in the frequency ofthe time is very important. Window Fourier transform and wavelet transform aretwo basic category of local time-frequency decomposition , but they are subject tothe restriction which Heisenberg uncertainty principle reveals the time-frequencylocalization s. In this paper, the method breaked through the limitations thatcan get the instantaneous frequency of the signal, thereby achieved a precisetime-frequency analysis.This article from the perspective of the framework gives an approximationmethod of the music signals . Construct a framework which used spline function based on the framework theory: {φn,k(t)}n∈Z,k∈Λ.whichφn,k(t) = N(t ? n)exp(iωkt),N(t ? n)is spline function.Gives the following theorem:theorem:{φn,k(t)}n∈Z,k∈ΛMeet the needs of the entire shift linearly inde-pendent.so an,k = 0,n∈Z,k∈Λ.theorem:{φn,k(t)}n∈Z,k∈Λis Riesz base.In this paper, the second part of the job is image compression based onsecond-generation Bandelet transform.Traditional strict sampling tensor product wavelet does not have transla-tional invariance, and only fit that the singular characteristics of isotropic. Todigital images as an example, that the image at the edge of the outline and theboundary will be as isolated points , fail to use the continuity of the boundary di-rection or smoothness; some di?erent image di?erence of few pixel may have manydi?erence the sub-band energy by the wavelet transform,. In order to overcomethis defect, French scholars Pennec and Mallat t provide the Bandelet transformin 2000. Bandelet transform provides a new express way of image based on theto the edge can be adaptive to track the geometry regular direction of the image.Theory to prove: For the regular geometry images it can achieve best sparse ge-ometric image representations used Bandelet basis functions . And compared tothe classical wavelet transform, Bandelet Transform in Image Compression andDenoising areas must re?ect the strengths and potential.Wavelet transform achieve a significant image compression on ?at areas,high-frequency sub-band quantization will have many zero coe?cients; but fornon-?at areas, the high-frequency wavelet sub-band will be more substantialresidual value factor. The regular geometry images, a significant value of high- frequency sub-band coe?cients mainly distribut along the geometric ?ow of theimage. Although the regular geometry image does not have the overall regular-ity, but have regularity along the geometric ?ow direction. This is the importantpremise that wavelet coe?cients can be further compressed.Because of the application of wavelet transform does not use the geometricproperties of images, so wavelet transform can not give sparse geometric imagerepresentations .the second generation transform Bandelet just the application ofthe geometric characteristics of the image, which image can be the best approach,the paper combine the second-generation Bandelet transform and the informationentropy theory describes the Bandelet transform can be better than the waveletcompression e?ect.An image f could be approximated by part sum fM = m∈IM f,gm gm ina set of standard orthogonal basisB = {gm}m.IM correspond M vectors which haslargest amplitude coe?cient f,gm .Its amplitude is greater than the thresholdTM,IM = {m∈N : | f,gm | > TM}. The approximation error isIf the image f has the contours of curve Cαwhich intersect in the corneror cross-point,And f which removed these curves are the Cα.Althoughαis theunknown parameters, the theory shows that structure-based Bandelet Functioncan be the best sparse images that meet the optimal decay rate.If the contours of f is ambiguity by an unknown regular smooth nuclear,the results are still valid.The first generation Bandelet Transform, although could use the regular-ity of the geometric image , but also has the following two shortcomings: (1) Bandelet orthogonal basis functions are not the overall situation and the recon-struction image has the edge e?ect. (2) There are too much computation. Inorder to overcome these shortcomings, the second-generation Bandelet For im-age compression, directly from the discrete form, the orthogonal basis functionwith the overall situation, the algorithm is simple, there is no edge e?ect on thereconstructed image.Bandelet Transform algorithm ?ow:(1)Input: the original image , the threshold value T;(2)Decomposite image use two-dimensional wavelet transform . Can beorthogonal or biorthogonal wavelet transform;(3)Quadtree partition on subband,get the best The direction of the geo-metric ?ow in the partition region ;(4)Bandelet transform on each Bandelet Block,get the Bandelet coe?cients.(5)Arrange the Bandelet coe?cients in matrix form.(6)Output:Quadtree,the best direction of the geometric ?ow,Bandelet co-e?cients.Reconstruct image is inverse process of the above algorithm.Of any section of the sample data B,? is the state-space,b∈? is the statevalue, P(b) is the probability of state value. The entropy of random variable isdefined as:By entropy coding theory ,we know that the compression e?ect will be bet-ter as the entropy is smaller after the transform. In this paper , do numericalexperiments use Bandelet transform and information entropy theory.The resultsshowed that the entropy after Bandelet transform is smaller than the entropy af- ter wavelet transform, and numerical s showed that changes Bandelet For wavelettransform can be smaller than the entropy value, experimental results show that,Bandelet transform natural map As the compression is e?ective, image compres-sion and its e?ect is superior to wavelet transform. Experimental results showthat, Bandelet transform is e?ective compressing natural images.And the e?ectof image compression is better than the wavelet transform.