• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.765-800
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• 2022

Quantization for probability distributions concerns the best approximation of a d-dimensional probability distribution P by a discrete probability with a given number n of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on ℝ. For such a probability measure P, an induction formula to determine the optimal sets of n-means and the nth quantization error for every natural number n is given. In addition, using the induction formula we give some results and observations about the optimal sets of n-means for all n ≥ 2.

• Proceedings of the IEEK Conference
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• 2003.11a
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• pp.461-464
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• 2003

This paper proposes the multi-level vector error diffusion for smear artifact reduction in the boundary regions. Smear artifact mainly results from a large accumulation of quantization error. Accordingly, to reduce these artifacts, the proposed method excludes the large quantization error in the error diffusion process by comparing the magnitude of the error vector with predetermined first threshold. In addition, if the vector norm of the difference between the error adjusted input vector and the primary co]or that has minimum vector norm for the error adjusted input vector is larger than second threshold, the error is excluded. As a result, the proposed method reduce smear artifact in the boundary region and produces visually pleasing halftone pattern.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.26 no.11A
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• pp.1957-1962
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• 2001

Image and video compression requires quantization error model of DCT coefficients for post processing, restoration or transcoding. Once DCT coefficients are quantized, it is impossible to recover the original distribution. We assume that the original probability density function (pdf) is the Laplacian function. We calculate the variance of the quantized variable, and estimate the variance of the DCT coefficients. We can confirm that the proposed method enhances the accuracy of the quantization error estimation.

• Journal of Electrical Engineering and information Science
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• v.2 no.6
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• pp.138-145
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• 1997

This paper proposes an adaptive algorithm for the quantization step size control of MPEG-2, using the information obtained from the previously encoded picture. Before quantizing the DCT coefficients, the properties of reconstruction error of each macro block (MB) is predicted from the previous frame. For the prediction of the error of current MB, a block with the size of MB in the previous frame are chosen by use of the motion vector. Since the original and reconstructed images of the previous frame are available in the encoder, we can calculate the reconstruction error of this block. This error is considered as the expected error of the current MB if it is quantized with the same step size and bit rate. Comparing the error of the MB with the average of overall MBs, if it is larger than the average, small step size is given for this MB, and vice versa. As a result, the error distribution of the MB is more concentrated to the average, giving low variance and improved image quality. Especially for the low bit application, the proposed algorithm gives much smaller error variance and higher PSNR compared to TM5 (test model 5).

• Journal of Broadcast Engineering
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• v.25 no.2
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• pp.157-165
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• 2020

In this paper. we propose an effective compensation scheme to the quantization error arisen from quantized learning in a machine learning on an embedded system. In the machine learning based on a gradient descent or nonlinear signal processing, the quantization error generates early vanishing of a gradient and occurs the degradation of learning performance. To compensate such quantization error, we derive an orthogonal compensation vector with respect to a maximum component of the gradient vector. Moreover, instead of the conventional constant learning rate, we propose the adaptive learning rate algorithm without any inner loop to select the step size, based on a nonlinear optimization technique. The simulation results show that the optimization solver based on the proposed quantized method represents sufficient learning performance.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.30 no.8C
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• pp.832-837
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• 2005

This paper presents a design method for fixed-width squarer that receives a W-bit input and produces a W-bit squared product. To efficiently compensate for the quantization error, modified Booth encoder signals (not multiplier coefficients) are used for the generation of error compensation bias. The truncated bits are divided into two groups (major/minor group) depending upon their effects on the quantization error. Then, different error compensation methods are applied to each group. By simulations, it is shown that the performance of the proposed method is close to that of the rounding method and much better than that of the truncation method and conventional method. It is also shown that the proposed method leads to up to $28\%\;and\;27\%$ reduction in area and power consumption compared with the ideal squarers, respectively.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.21 no.7
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• pp.1685-1695
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• 1996

The postprocessing of compressed images based on the projections onto convex sets and the constrained minimization imposes several constraints on the procesed data. The quantization constraint has been commonly used in various algorithms. Quantization is many-to-one mapping, by which all the dat in a quantization region are mapped to the corresponding representative level. The basic idea behind the projection onto the QCS(quantization constraint set) is to prevent the processed data from diverging from the original quantization region in order to redue the artifacts caused by filtering in postprocessing. However, there have been few efforts to analye the POQCS(projection onto the QCS). This paper analyzed mathematically the POQCS of filtered data from the viewpoint of minimizing the mean square error. Our analysis shows that a proper filtering technique followed by the POQCS can reduce the quantization distortion. In the conventional POQCS, the outside data of each quantization region are mapped into the corresponding boundary. Our analysis also shows that mappingthe outside data to the boundary of a subregion of the quantization region yields lower distortion than does the mapping to the boundary of the original region. In addition, several examples and discussions on the theory are introduced.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.26 no.12A
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• pp.2085-2091
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• 2001

In lossy compression method, the transformed coefficients are quantized. This results in the quantization noise. The video image quality and bit rate is closely related with the quantization step. In this paper, we proposed a new quantization function for the improved performance. The DC value of each macroblock is compensated depending on the magnitude of DC quantization error. It is implemented very low bit-rate video coding, i.e., H.26L. The experimental result is useful when the object motion is not severe.

• Journal of the Korean Institute of Telematics and Electronics A
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• v.28A no.1
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• pp.1-7
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• 1991

Degradation of communication quality is due to transmission error rather than quantization noise when DPCM signal is transmitted over a heavily noise channel. The communication quality can be improved by employing an error correcting code to the DPCM signal transmission over such a channel. We considered both quantization noise and transmission error simultaneously in evaluating the signal to noise ratio. To efficiently improve the signal to noise ratio, we analyze the unequal symbol error probability of convolutional code, and encoded to more protect significant symbols from channel errors than least ones, so that the signal to noise ratio is improved. We derived related formulas and also made computer simulations.

• Journal of the Korean Institute of Telematics and Electronics B
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• v.28B no.5
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• pp.318-326
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• 1991

A Quantizer reconstruction level control method for block artifact reduction in DCT image coding is described. In our scheme, quantizer reconstruction level control is obtained by adding quantization level step size to the optimum quantization level in the direction of reducing the block artifact by minimizing the mean square error(MSE) and error difference(EDF) distribution in boundary without the other additional bits. In simulation results, although the performance in terms of signal to noise ratio is degraded by a little amount, mean square of error difference at block boundary and mean square error having relation block artifact is greatly reduced. Subjective image qualities are improved compared with other block artifact reduction method such as postprocessing by filtering and trasform coding by block overlapping. But the addition calculations of 1-dimensional DCT become to be more necessary to coding process for determining the reconstruction level.