HP Inc. The method of claim 1, wherein k is determined from the following relationship: EQU The method of claim 1, wherein k is determined by: a. The method of claim 1, wherein each pixel value is a member of a first alphabet of values, further comprising the step of: mapping each pixel value of said image to a value in a second alphabet, wherein said second alphabet is a subset of said first alphabet. The method of claim 6, wherein after decoding said mapping introduces an error of uniform bound E and each pixel value xi is mapped to a value yi according to the relationship: EQU

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We investigate the lossless and lossy p Table 3: Comparison of the lossless compression performances of di erent integer wavelet transforms on other images in bpp. Image Wavelet Barbara Goldhill Lenna 4,4 4. Although the lossy performance using integer transforms is not quite as good as that for oating point transforms, integer transforms are less complex and o er excellent progressive decoding performance for the many applications that demand the capability to losslessly recover the original image.

Improved lossy performance when using integer transforms is a pursuit of our on-going work. Zandi, J. Allen, E. Schwartz, and M. Said and W. Calderbank, I. Daubechies, W. Sweldens, and B. Dewitte and J. Memon, X. Wu, and B. Antonini, M. Barlaud, P. Mathieu and I. Woods and T. Taubman and A. Li and S. Ordentlich, M. Weinberger, and G. Sementilli, A. Bilgin, J. Kasner, and M. Weinberger, G. Seroussi, and G. Wu and N. Table 1: Comparison of the lossless compression performances of di erent integer wavelet transforms on JPEG images in bpp.

However, the reversible integer transforms used in this work are not orthonormal transforms. For these transforms, the MSE can often be computed by weighting the jC x; y? These scaling factors for each subband can be computed using the method described in 8]. The scaling factors computed using the method in 8] are usually oating point numbers.

They can not be used on the integer wavelet coe cients, since scaling these integers by a oating point number would create a oating point number. We normalize the scaling factors so that the minimum scaling factor is 1, the round each scaling factor to the nearest power of two. The scaling factors for the 2,4 transform for 3 levels of dyadic decomposition are shown in Figure 1.

The default encoding order is to encode one subband at a time, in order from lowest to highest resolution. Within each subband, bitplanes are encoded from most- to least-signi cant.

Within each bitplane, the bits are de-interleaved and encoded by an adaptive binary arithmetic coder in order 1 , 2 , 3. The image is always encoded losslessly. During decoding of a bitplane at any given resolution layer i. We have found this progressive transmission ordering to be nearly optimal when using oating point wavelet transforms. This decoding process can stop at any point to produce a lossy result.

The translation between the encoder ordering and the ordering used by the decoder is handled by a parser. An important feature of the bitplane coder is that the encoding has no inter-sequence dependencies e. This enables the encoded bitstream to be parsed into any arbitrary order, depending on the application. For example, re-synch markers can be inserted after each encoded sequence if the compressed bitstream is ordered by sequence.

This enables a degree of error detection and concealment when transmitting encoded images over noisy channels. Subbands can be emphasized in arbitrary ways, e. In the literature, several image compression methods that bene t from this property have been proposed 1, 2, 9, 10, 11, 12]. Table 1 presents the lossless compression results obtained using di erent integer wavelet transforms on a set of JPEG test images, and Table 2 presents a comparison of these results with other lossless compresion techniques in the literature.

Similarly, Table 3 presents the lossless compression results of di erent integer wavelet transforms on other standard images used in the literature, and Table 4 compares these with other lossless compression methods in the literature. Tables 5, and 6 compare the progressive compression performances of the integer wavelet transforms on Barbara and Goldhill images, respectively.

The results in these tables were obtained using scaling factors as discussed in the previous sections. Note that scaling does not change the lossless performance of the algorithm.

The inverse transform is obtained by reversing the steps of the forward transform. The reader is referred to 3] for details. The notation m; n] represents a transform with m coe cients in the low-pass analysis lter and n coe cients in the highpass analysis lter.

Using mean-squared error MSE as the distortion measure, the information that will produce a larger decrease in MSE can be considered to be more important.

Let I be the original image, and T be an orthonormal transform. Sementilliy, Michael W. SAIC , N. Wilmot Rd. The lossy performance is quite competitive with other e cient lossy compression methods. We also compare the progressively decoded lossy performance of several reversible integer wavelet transforms, and compare these with the popular 7x9 wavelet lter of 6] that produces oating-point coe cients. Although these coe cients can be used to reconstruct the original image perfectly in theory, the use of nite precision arithmetic and quantization results in a lossy scheme.

Recently, reversible integer wavelet transforms, i. In 3], Calderbank et al. Here, the input is rst split into even and odd indexed samples.

Let x n be the input signal. Then, 1. However, until recently, its use has been limited to lossy compression applications. This is due to the fact that most wavelet transforms produce oating-point coe cients which are not well-suited for lossless coding applications.

With the introduction of wavelet transforms that map integers to integers, there has been interest in using wavelet transforms for lossless image coding 1, 2, 3, 4, 5]. Using reversible integer wavelet transforms for compression of images has several advantages.

Perhaps the most important one is that, through the use of appropriate techniques, a fully embedded bitstream can be generated. In other words, the decoder can extract a lossy version of the image, possibly at reduced resolution, at a desired rate from the bitstream, and continue to decode at higher and higher rates, until the image is perfectly reconstructed.

This rate scalability is valuable in many applications. By integrating lossy and lossless compression in a natural fashion, a single image compression method provides excellent lossy performance as well as supporting the many applications that require the ability to exactly recover the original image. In this paper, we investigate both lossless and lossy performance of reversible integer wavelet transforms.

An image coder consisting of a reversible integer wavelet transform and a bit plane coder is presented. We compare the lossless compression performance of the presented scheme with that of other state-of-the-art lossless compression schemes in the literature, including wavelet-based, and non-wavelet based coders.

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## CALIC – A lossless image compression

Variability gtdh, dv Influence Error distribution Group pixels Previous prediction error gt Each group has its new prediction why? How to adapt — past? Comparative study of various still image coding techniques. PowerPoint PPT presentation free to view. It measures the amount of noise introduced through a lossy compression of the image, however, the subjective judgment of the viewer also is regarded as an important measure, perhaps, being the most important measure.

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