Dalgacık dönüşümündeki yerel tepe değerlerinden görüntü geriçatma ve sıkıştırma
Image reconstruction from the wavelet maxima and coding
- Tez No: 66634
- Danışmanlar: DOÇ. DR. MUHİTTİN GÖKMEN
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Elektronik ve Haberleşme Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Biyomedikal Mühendisliği Bilim Dalı
- Sayfa Sayısı: 67
Özet
ÖZET Günümüze kadar, ayrıt saptamanın işaretlerin gösteriminde veri miktarını düşürdüğü fakat işaret bilgisinin tümünü içermediği düşünülmüştür. Dalgacık dönüşümü, çok ölçekli işaret ayrıştırma yöntemlerine benzer şekilde işaretlerin farklı ölçeklerde analizine olanak sağlamaktadır. Bu çalışmada, dalgacık dönüşümünün yerel tepe değerlerinin, görüntünün farklı ölçeklerdeki ayrıtlarına karşı geldiği ve bu bilginin görüntüyü tam olarak yeniden elde etmek için yeterli olduğu gösterilmektedir. Dalgacık dönüşümünün tam geriçajma özelliğinden yararlanarak, yerel tepe değerlerinden geriçatma algoritması oluşturulmuştur. Dalgacık dönüşümü yerel tepelerinin bu özelliği nedeni ile", yerel tepelerin konum ve genlik değerlerinin saklanması geriçatma için yeterli olmaktadır. Tezin konuya giriş amaçlı ilk bölümünden sonra, 2. bölümde dalgacık dönüşümünün ayrıntılı açıklaması yapılmış ve sonlu ölçekli dalgacık dönüşümü tanımlanmıştır. 3. bölümde, sonlu ölçekte dalgacık dönüşümünün yerel tepelerinden geriçatma ele alınmıştır. 4. bölümde, 2. ve 3. bölümde anlatılan konular iki boyuta genişletilmiştir. 5. bölümde, yerel tepe değerlerinin sıkıştırılması için kullanılan zincir kodlama, Huffman kodlama ve polinom uydurma yöntemleri anlatılmıştır. 6. bölümde dalgacık dönüşümünün yerel tepelerinden geriçatma ve yerel tepelerin sıkıştırılması ile ilgili çalışmalar anlatılmış ve sayısal sonuçlar verilmiştir. Son bölüm, tez konusu ile ilgili olarak yapılan çalışmaların genel bir değerlendirmesidir.
Özet (Çeviri)
SUMMARY IMAGE RECONSTRUCTION FROM THE WAVELET MAXIMA AND CODING Points of sharp variations are often among the most important features for analyzing the properties of transient signals ör images. in images, they provide the locations of the edges which are generally used for describing the structural information of object boundaries. Many studies have been developed in image processing and computer vision literatures about this subject. Edge detection is generally viewed as a process which reduces the amount of data representing the images, but which does not preserve the whole information. in this study, it is shown that signals can be reconstructed from the wavelet transform maxima and than these maxima is compressed with efficient algorithms. This is achieved by implementing multiscale edge representation of images with wavelet transform. Wavelet transform is a multiscale time-frequency representation of signals. This representation was foraıalized by J. Morlet, A. Grossman and S.G. Mallat in early 1980. They have been found that a signal can be reconstructed with convolutions of some scalable wavelet functions. These wavelets must satisfy energy conservation equation with respect to Parseval theorem. For each scale, the wavelet function is dilated by a scale parameter s and convolved with original signal. Fourier transform of the wavelet functions covers ali frequency domain. it can be shown that local maxima of the wavelet transforms correspond to points of sharp variations of signals. For this purpose, wavelets can be taken as a derivative of some smoothing function. in this study, cubic spline wavelet is used. Cubic spline wavelet is compact and has öne zero in frequency domain. For some scale, the wavelet transform maxima of cubic spline wavelet correspond to the signal local changes of that scale. The scale characterizes the neighborhood size where theses variations are computed. At large scales, the signal variations are measured över large neighborhoods whereas at fine scales the variations due to the finner structures of the signal is obtained. This multiscale maxima representation is essentially equivalent to Canny edge detection. The wavelet transform maxima representation is implemented for both öne and xtwo dimensions. in two dimension, the local maxima representation is taken for horizontal and vertical spatial directions. For images, these maximums are happen to be local maxima curves along both directions, With keeping only local maxima position and amplitude for ali scales, the original image can be reconstructed completely. The compression is made for the wavelet transform maxima points, because they are sufficient for reconstruction of original image. Both maxima positions and amplitudes are compressed for ali scales. For efficient compression, the great amount of maxima curves are removed and local maxima positions are coded with chain coding. Huffman coding is applied to chain codes. The local maxima curve amplitudes are coded with least squares method which is applied for convergence of maxima amplitudes to a polynomial. These polynomials are choosen second ör third degrees. The coeefficient of polynomials are saved for each curve. The first section is an introduction for the study. in the second section, the mathematical results of the wavelet model is summarized briefly for öne dimension in continious case. The wavelet transform is computed with follovving equation where s is scale parameter; Wsf(x) = f*x|/s(x) the basis function \|/s(x) s L2 (R) is called wavelet function which is dilated by s; v-OO-Kl) For digital applications, the wavelet transform parameters s and x must be discretized. s parameter varies över the set of real numbers. For a particular class of wavelets, the scale parameter can be sampled along the dyadic sequence [2J]., without modifying the overall properties of the transform. At each scale 2“, the wavelet transform is continious since it is equal to the convolution of two functions in Hilbert space. x parameter can also be discretized, but the wavelet transform maxima position of discretized signal must be the same with the continious case. By using cubic spline wavelet and assuming the discretized signal is the smoothing signal at the scale l, this is achieved. After ali, the discrete dyadic wavelet transform is defined. This transform can be implemented with a fast pyramidal algorithm. The lowpass filter H and highpass filter G, is used for this algorithm. For scale s, the discrete dyadic xiwavelet transform of S jf(x) is found by following equations: S2J+If(n) = S2Jf*Hj(n) and W2J+1f(n) = S2Jf*Gj(n) The results of wavelet transforms and wavelet transform maxima for 5 finite dyadic scales are shown at the end of the section. in the third section, the vvavelet transform maxima across scales are obtained. The amplitude of the vvavelet transform is related to the singularities appearing in the original signal. This can be achieved by choosing a wavelet vvhose Fourier transform has öne zero at co=0. Cubic spline wavelet has this property. The H and G filters, given in section l is used for fast vvavelet transform. The vvavelet transform using cubic spline vvavelet provides energy conversion equation and the original signal can be reconstructed from the vvavelet transform. But also, keeping the vvavelet transform maxima position and amplitude is enough for reconstruction. This is achieved by projections on F and V spaces iteratively: P = Pr o Pv V space is the dyadic vvavelet transforms of Hilbert space functions. For any function, the projection on the V space is obtained by; Pv= WoW”' equation. W is the dyadic vvavelet transform operatör, W“1 is the in verse dyadic vvavelet transform operatör. F is the set of ali sequences of functions such that for ali scales these functions have the same maxima than the given dyadic vvavelet transform. Clearly, for any scale, there exist an infinite number of functions vvhich have the same local maxima as the dyadic vvavelet transform of some function. Hovvever, these functions are not necessarily the dyadic vvavelet transform of some function. For being a dyadic vvavelet transform, Py projection must be satisfied. The intersection of F vvith V space is öne element and this element can be found vvith an iterative algorithm. This algorithm reconstructs the dyadic vvavelet transform of desired function from the vvavelet transform maxima. The results of the reconstruction algorithm from the vvavelet transform maxima in öne dimension is given section 3.3. This algorithm has been tested on the diracs, sinusoids, square functions and an image scan line of Lenna image. For each scale 2”, signal to noise ratio converges asimptotically a constant value. When the scale increases, signal to noise ratio is decreases. it proves that the xiierror is concentrated in high frequencies. This error can be explained by the spurious high frequencies introduced by the maxima detection procedure. In the fourth section, the wavelet transform is extended in two dimensions. The properties of a two dimensional wavelet transform are essetially the same as in one dimension. Transform is achived by two wavelet functions which are separable and are partial derivative of some two dimensional function along horizontal and vertical directions. The local maxima of two dimensional wavelet transform along x, for y constant, are the points of sharper horizontal variation of original two dimensional signal. Similarly, the local maxima of two dimensional wavelet transform along y, for x constant, are the points of sharper vertical variation. The local maxima belong to curves in the (x,y) plane which are the edges of the image along each direction. The algorithm reconstructing one dimensional signals from the maxima of their wavelet transform can be extended in two dimensions. In this case, V is a space of dyadic wavelet functions in two dimensional Hubert space. I" is the set of all sequences of two dimenional functions such that for all scales these functions have the same maxima along the x and y directions than the given two dimensional dyadic wavelet transform. Then the reconstruction algorithm is implemented as in one dimensional case. In the fifth section, an efficient image compression algorithm is given for the dyadic wavelet transform maxima. Keeping the dyadic wavelet local maxima positions and amplitudes are sufficient for reconstruction of images. This maxima representation have less information than the original image. But, this representation is made along x and y directions for 5 scale. Compression is made on two types of information. At the first, the wavelet transform maxima positions are coded along the x and y directions. These maxima forms curves. The position of these curves are coded with binary image chain coding algorithm. This algorithm provides efficient compression by keeping only the direction of the neighbour maxima position on the 8 neighbourhood. Thus, a maxima position is coding with 3 bits. Additionally, at the beginning of the curve, the coordinates of the first maxima of the curve must be kept. Chain coding is a very effective coding algorithm, but in the first scale, there are many maxima curves of wavelet transform. Even these curves are coded with chain coding algorithm the compression ratio is very less. A maxima removing algorithm is made for this purpose. The 60, 70 and 80 percent of wavelet transform curves are removed and the image is reconstructed with curves left. The reconstruction signal to noise ratio is large enough. As a result of, only the curves that left from removing process are chain coded. Huffman coding is applied to chain coded maxima curves. This process decreases the average bit rate. XlllThe wavelet maxima curve amplitude can be thought as one dimensional discrete function. This function can be coded with some degree of polynomial coefficients. The degree of the polynomial is important, since increasing of the degree causes better convergence. However, larger degree has larger bit rate of coding. 2., 3. and 4. degree of polynomials are used to code the wavelet maxima curves. Signal to ratio of the reconstructed images are large enough. In the last section, all numerical results of the study is discussed. The wavelet maxima representation is good enough for reconstruction of images. However, the iteration on the operator P is not a fast process. 8 iteration is enough for reconstruction of images. Compression ratio of tested images are given in section six. XIV
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