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UVA MATH 3354 - Homework4

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Yang Hong HW4 Due10/17/2014 2. 11.905+K=11.905+0.095=12=B(m,n), In the case 1, F(m,n)=3822 Fq(m,n)=NINT[(211-1)*1]=2047 11.905+K=11.905-0.905=11=B(m,n), In the case 2, F(m,n)=3822 Fq(m,n)=NINT[(210-1)]=1023 3. Block: Advantages: The energy in the transform domain is concentrated in a small region, so the forward DCT method can yield larger compression ratios and maintain image quality . It is not computationally tedious and time-consuming to carry out the 2-D DCT when the image is large as FFBA. And also, the allocation table in the case of block compression may be smaller than in FFBA. It is convenient to use for medical images especially in the DICOM environment. Size is small and easy to store. Disadvantages: Loses information and introduces an approximation to the compressed image file. Full-frame: Advantages: It is developed primarily for large size radiological images. It is different from the block method, it does not produce blocky artifacts and preserves higher image quality in the compressed image. Special for radiological exams. Disadvantages: The size is big and difficult to store. It takes long time. The zigzag sequence is not good to rearrange DCT coefficients. 4. In the case of wavelet transform, the basis functions are derived from a mother wavelet function by dilation and translation operations. We use 1-D case to explain the concept of analysis. Consider there is a signal fm at level m, which can be decomposed into m+1 level by convoluting it with the h(low-pass) filter to form a smooth signal fm+1 and the g(high-pass) filter to form a detailed signal f’m+1. Thenboth fm+1 and f’m+1 are needed to be sampled. The same process can be further applied to fm+1, creating the detailed and smooth signal at the next resolution level until the desired level is reached. At the resolution level m=3, the signal is composed of the detailed signal of the resolution levels f’1,f’2, and f’3 plus one smooth signal f3. And the original signal f0 can be reconstructed by f0=f’1+f’2+f’3 +f3. Each of f’1, f’2, f’3 and f3 can be compressed by different quantization and encoding methods to achieve the required compression ratio. In the case of 2-D wavelet transform, the first level results in four components, the x direction and the y direction. 3-D wavelet transform is a very effective method for compressing data. The 3-D method can be extended from the 1-D and 2-D pyramidal algorithm. Each line in the x direction of the 3-D image data set is first convoluted with filter h(low) and g(high), followed by subsampling every other voxel in the x direction to form the smooth and detailed data line. The resulting voxels are convoluted with h and g in the y direction, followed with subsampling in the y direction. Finally, the same procedure is applied to the z direction. The resulting signal has 8 components. Since h is a low-pass filter, only one components contains all low-frequency information. The seven remaining components are convoluted at least once with the high-pass filter g, so they contain some detailed signals in 3 different directions. 5.Comments: Comparing to the 2-D fft transform of the image, we find that in the 2-D cosine transform image, the data are almost in the low frequency and are concentrated in a small area. However, the 2-D fft transform is still spread. As a result, 2-D cosine transform compresses the image better than 2-D fft does. As 2-D cosine transform keeps information at the low frequency part, it loses high frequency information. However, in the 2-D fft transform, there are still some data in the high frequency. So 2-D cosine transform is a lossy compression method. code: I = dicomread('image12.dcm'); K=mat2gray(I); imshow(K); title('original image'); J = dct2(K); figure(2); imshow(abs(J)); title('2D cosine transform'); J2=fft2(K); figure(3); imshow(abs(J2)); title('2D fft


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UVA MATH 3354 - Homework4

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