5/16/2023 0 Comments Noise reduction on express scribeReconstruction can be further improved by alternating projections of F ˜ f r e c onto C i and Ran F ˜ until a signal in their intersection is reached. However, F ˜ f and F ˜ f r e c do not lie in the same quantization cell. As the result of linear reconstruction we obtain a signal f rec Expansion coefficients of f rec, F ˜ f r e c, are obviously in the range of frame expansions, and are closer to the original ones than c is, | | F ˜ f r e c − F ˜ f | | < | | c − F ˜ f | |. Let c denote the set of quantized coefficients of a signal F, c = Q F ˜ f, , and let c be in the partition cell C i. Hence, linear reconstruction does not fully utilize information which is contained in the quantized set of coefficients, so there is room for improvement. However, when dealing with quantization of overcomplete expansions, linear reconstruction is not optimal in the Sense that it does not necessarily yield a signal which lies in the same quantization cell with the original. The reason for this discrepancy is that the error estimates in the previous subsection were derived under the assumption of linear reconstruction given by (3.1.2). Is this always true, or are we underestimating the potential of frame redundancy for error reduction? Let us look at a simple example for the sake of providing some intuition about this effect. According to formulae (3.1.3) and (3.1.5), it seems that the quantization step refinement is more effective than the increase in frame redundancy for the error reduction, since the error decreases proportionally to q 2 in the former case, and proportionally to 1∕ r in the latter case. If the white noise model for the quantization error is accepted, the error variance σ 2 is proportional to the square of the error maximum value, which is half of the quantization step. Effectiveness of the two approaches to partition refinement, that is error reduction, can be assessed based on the results reviewed in the previous subsection. This gives another explanation of the error reduction property in frames, this time for the quantization error. Alternatively, for a given quantization step, more constraints can be added, which corresponds to an increase in redundancy of the frame. One way to refine the partitioning is to tighten the constraints which define cells (3.2.1) by decreasing the quantization step. Roughly speaking, the expected value of quantization error reflects the fineness of the partition. For each of the cells, the quantization maps all the signals in the cell to a single signal in its interior, usually its centroid. It does not store any personal data.(3.2.1) C i = f : n i j − 1 / 2 q ≤ f φ ˜ j < n i j + 1 / 2 q, j ∈ J , The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. The cookie is used to store the user consent for the cookies in the category "Performance". This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other. The cookies is used to store the user consent for the cookies in the category "Necessary". The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". The cookie is used to store the user consent for the cookies in the category "Analytics". These cookies ensure basic functionalities and security features of the website, anonymously. Necessary cookies are absolutely essential for the website to function properly.
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