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| | There are many articles on fat reduction that have been composed. We have read a amount of them that suggest a diet for those wishing to lose weight. Others have spoken of exercise. We have several whom suggest we hiring a personal trainer. The question, yet, nonetheless remains. What is healthy weight loss? What is healthy weight for females?<br><br>BMI will be calculated in metric or imperial measurements. There is a slight variation between the two equations, however, the outcome is accurate enough for wellness assessment reasons. BMI is calculated using a [http://safedietplans.com/bmi-chart bmi chart], an online BMI calculator, or manually utilizing the BMI equation. The results, which are taken to two decimal places, are the same for each. The BMI equation is obtainable in metric or imperial measurements, and there is a extremely slight variation in the 2 figures.<br><br>Now let's have a peep into childhood weight details. Children usually inherit this disease from their parents. The amount of obese children is increasing at an alarming rate. As compared to the last 3 years bmi chart men, this number has doubled. Sedentary escapades that are preferred by kids make them dull and lazy. They tend to take foods which lack in dietary values. As there is not any physical exertion associated, the fats get deposited inside body and create them lethargic. The general capability of overweight kids is very low as compared to others. One of the other childhood obesity details is that such children suffer from many different problems like lack of concentration, inability to sleep properly, plus humiliation faced from peers and neighbors.<br><br>Rest: This has become the most significant parts of my training. If I don't receive enough rest, my body begins to break down. Listen (really closely) to the body.<br><br>The perfect weight for females of a medium frame measuring 64-67 inches is between 124-147 pounds. This leads to a lot of difference inside what will be considered an ideal weight. Women must be accorded for each inch over 5 feet (1.52 m). So a female whom is 52 (1.57m) has an perfect weight of 110 pounds (49.89 kg).<br><br>To a certain extent this really is true, however excess puppy fat is because risky to a child because excess fat is to an adult. It is estimated that over 15% of UK youngsters bmi chart women are overweight or fat, and this figure is increasing fast. The Journal of the American Medical Association reported found on the 4th April which the amount of overweight American youngsters was 33.6%. Obese children grow into fat adults. They never lose this thus called puppy fat unless positive steps are taken. They have a significantly high risk of developing severe wellness problems , both today plus because an adult, including potentially life threatening conditions like bowel cancer, diabetes, strokes, heart conditions plus significant blood pressure. The more overweight the child, the better the risk.<br><br>Nonetheless, it utilizes the U.S. Navy Circumference Method that need the input of information for it to be able to resolve for the body fat percentage. The readings can be put into a body fat chart for the individual to be able to monitor the decrease or increase of body fat percentage.<br><br>If you are thinking what the healthy fat is for a girl, then the BMI is the path to take. You must additionally check the circumference of your waist. If you believe you're obese, then you need to start eating healthy plus exercise frequently so that you can shed those pounds plus become the healthy, fresh we. |
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| '''Rate–distortion theory''' is a major branch of [[information theory]] which provides the theoretical foundations for [[lossy data compression]]; it addresses the problem of determining the minimal number of bits per symbol, as measured by the rate ''R'', that should be communicated over a channel, so that the source (input signal) can be approximately reconstructed at the receiver (output signal) without exceeding a given distortion ''D''.
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| == Introduction ==
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| Rate–distortion theory gives an analytical expression for how much compression can be achieved using lossy compression methods. Many of the existing audio, speech, image, and video compression techniques have transforms, quantization, and bit-rate allocation procedures that capitalize on the general shape of rate–distortion functions.
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| Rate–distortion theory was created by [[Claude Shannon]] in his foundational work on information theory.
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| In rate–distortion theory, the ''rate'' is usually understood as the number of [[bit]]s per data sample to be stored or transmitted. The notion of ''distortion'' is a subject of on-going discussion. In the most simple case (which is actually used in most cases), the distortion is defined as the expected value of the square of the difference between input and output signal (i.e., the [[mean squared error]] ). However, since we know that most [[lossy compression]] techniques operate on data that will be perceived by human consumers (listening to music, watching pictures and video) the distortion measure should preferably be modeled on human [[perception]] and perhaps [[aesthetics]]: much like the use of [[probability]] in [[lossless compression]], distortion measures can ultimately be identified with [[loss function]]s as used in Bayesian [[estimation theory|estimation]] and [[decision theory]]. In audio compression, perceptual models (and therefore perceptual distortion measures) are relatively well developed and routinely used in compression techniques such as [[MP3]] or [[Vorbis]], but are often not easy to include in rate–distortion theory. In image and video compression, the human perception models are less well developed and inclusion is mostly limited to the [[JPEG]] and [[MPEG]] weighting ([[quantization (signal processing)|quantization]], [[Normalization (image processing)|normalization]]) matrix.
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| == Rate–distortion functions ==
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| The functions that relate the rate and distortion are found as the solution of the following minimization problem: | |
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| :<math>\inf_{Q_{Y|X}(y|x)} I_Q(Y;X)\ \mbox{subject to}\ D_Q \le D^*.</math><p>
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| Here ''Q''<sub>''Y'' | ''X''</sub>(''y'' | ''x''), sometimes called a test channel, is the [[conditional probability|conditional]] [[probability density function]] (PDF) of the communication channel output (compressed signal) ''Y'' for a given input (original signal) ''X'', and ''I''<sub>''Q''</sub>(''Y'' ; ''X'') is the '''[[mutual information]]''' between ''Y'' and ''X'' defined as
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| :<math>I(Y;X) = H(Y) - H(Y|X) \, </math>
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| where ''H''(''Y'') and ''H''(''Y'' | ''X'') are the entropy of the output signal ''Y'' and the [[conditional entropy]] of the output signal given the input signal, respectively:
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| :<math> H(Y) = - \int_{-\infty}^\infty P_Y (y) \log_{2} (P_Y (y))\,dy </math>
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| :<math> H(Y|X) =
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| - \int_{-\infty}^{\infty} \int_{-\infty}^\infty Q_{Y|X}(y|x) P_X (x) \log_{2} (Q_{Y|X} (y|x))\, dx\, dy. </math>
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| The problem can also be formulated as a distortion–rate function, where we find the infimum over achievable distortions for given rate constraint. The relevant expression is:
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| :<math>\inf_{Q_{Y|X}(y|x)} E[D_Q[X,Y]] \mbox{subject to}\ I_Q(Y;X)\leq R. </math>
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| The two formulations lead to functions which are inverses of each other.
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| The mutual information can be understood as a measure for ''prior'' uncertainty the receiver has about the sender's signal (''H(Y)''), diminished by the uncertainty that is left after receiving information about the sender's signal (''H''(''Y'' | ''X'')). Of course the decrease in uncertainty is due to the communicated amount of information, which is ''I''(''Y''; ''X'').
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| As an example, in case there is ''no'' communication at all, then ''H''(''Y'' |''X'') = ''H''(''Y'') and ''I''(''Y''; ''X'') = 0. Alternatively, if the communication channel is perfect and the received signal ''Y'' is identical to the signal ''X'' at the sender, then ''H''(''Y'' | ''X'') = 0 and ''I''(''Y''; ''X'') = ''H''(''Y'') = ''H''(''X''). | |
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| In the definition of the rate–distortion function, ''D''<sub>Q</sub> and ''D''<sup>*</sup> are the distortion between ''X'' and ''Y'' for a given ''Q''<sub>''Y'' | ''X''</sub>(''y'' | ''x'') and the prescribed maximum distortion, respectively. When we use the [[mean squared error]] as distortion measure, we have (for [[amplitude-continuous signal]]s):
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| :<math>D_Q = \int_{-\infty}^\infty \int_{-\infty}^\infty
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| P_{X,Y}(x,y) (x-y)^2\, dx\, dy = \int_{-\infty}^\infty \int_{-\infty}^\infty
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| Q_{Y|X}(y|x)P_{X}(x) (x-y)^2\, dx\, dy. </math> <p>
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| As the above equations show, calculating a rate–distortion function requires the stochastic description of the input ''X'' in terms of the PDF ''P''<sub>''X''</sub>(''x''), and then aims at finding the conditional PDF ''Q''<sub>''Y'' | ''X''</sub>(''y'' | ''x'') that minimize rate for a given distortion ''D''<sup>*</sup>. These definitions can be formulated measure-theoretically to account for discrete and mixed random variables as well.
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| An [[Analytical expression|analytical]] solution to this [[Optimization problem|minimization problem]] is often difficult to obtain except in some instances for which we next offer two of the best known examples. The rate–distortion function of any source is known to obey several fundamental properties, the most important ones being that it is a [[Continuous function|continuous]], [[monotonically decreasing]] [[Convex function|convex]] (U) [[function (mathematics)|function]] and thus the shape for the function in the examples is typical (even measured rate–distortion functions in real life tend to have very similar forms).
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| Although analytical solutions to this problem are scarce, there are upper and lower bounds to these functions including the famous [[Shannon lower bound]] (SLB), which in the case of squared error and memoryless sources, states that for arbitrary sources with finite differential entropy,
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| :<math> R(D) \ge h(X) - h(D) \, </math>
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| where ''h(D)'' is the differential entropy of a Gaussian random variable with variance D. This lower bound is extensible to sources with memory and other distortion measures. One important feature of the SLB is that it is asymptotically tight in the low distortion regime for a wide class of sources and in some occasions, it actually coincides with the rate–distortion function. Shannon Lower Bounds can generally be found if the distortion between any two numbers can be expressed as a function of the difference between the value of these two numbers.
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| The [[Blahut–Arimoto algorithm]], co-invented by [[Richard Blahut]], is an elegant iterative technique for numerically obtaining rate–distortion functions of arbitrary finite input/output alphabet sources and much work has been done to extend it to more general problem instances.
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| When working with stationary sources with memory, it is necessary to modify the definition of the rate distortion function and it must be understood in the sense of a limit taken over sequences of increasing lengths.
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| :<math>
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| R(D) = \lim_{n \rightarrow \infty} R_n(D)
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| </math>
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| where
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| :<math>
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| R_n(D) = \frac{1}{n} \inf_{Q_{Y^n|X^n} \in \mathcal{Q}} I(Y^n, X^n)
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| </math> | |
| and
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| :<math>
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| \mathcal{Q} = \{ Q_{Y^n|X^n}(Y^n|X^n,X_0): E[d(X^n,Y^n)] \leq D \}
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| </math>
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| where superscripts denote a complete sequence up to that time and the subscript ''0'' indicates initial state.
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| ===Memoryless (independent) Gaussian source===
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| If we assume that ''P''<sub>''X''</sub>(''x'') is [[normal distribution|Gaussian]] with [[variance]] σ<sup>2</sup>, and if we assume that successive samples of the signal ''X'' are [[stochastically independent]] (or equivalently, the source is ''[[memorylessness|memoryless]]'', or the signal is ''uncorrelated''), we find the following [[analytical expression]] for the rate–distortion function:
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| :<math> R(D) = \left\{ \begin{matrix}
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| \frac{1}{2}\log_2(\sigma_x^2/D ), & \mbox{if } D \le \sigma_x^2 \\ \\
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| 0, & \mbox{if } D > \sigma_x^2.
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| \end{matrix} \right.
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| </math>
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| The following figure shows what this function looks like:
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| [[Image:Rate distortion function.png]]
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| Rate–distortion theory tell us that ''no compression system exists that performs outside the gray area''. The closer a practical compression system is to the red (lower) bound, the better it performs. As a general rule, this bound can only be attained by increasing the coding block length parameter. Nevertheless, even at unit blocklengths one can often find good (scalar) quantizers that operate at distances from the rate–distortion function that are practically relevant.
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| This rate–distortion function holds only for Gaussian memoryless sources. It is known that the Gaussian source is the most "difficult" source to encode: for a given mean square error, it requires the greatest number of bits. The performance of a practical compression system working on—say—images, may well be below the ''R(D)'' lower bound shown.
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| == Connecting rate-distortion theory to channel capacity <ref name="BergerRateDistortion">{{cite book |title=Rate Distortion Theory: A Mathematical Basis for Data Compression |publisher=Prentice Hall |author=Toby Berger |year=1971}}</ref> ==
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| Suppose we want to transmit information about a source to the user with a distortion not exceeding ''D''. Rate–distortion theory tells us that at least ''R''(''D'') bits/symbol of information from the source must reach the user. We also know from Shannon's channel coding theorem that if the source entropy is ''H'' bits/symbol, and the [[channel capacity]] is ''C'' (where ''C'' < ''H''), then ''H'' − ''C'' bits/symbol will be lost when transmitting this information over the given channel. For the user to have any hope of reconstructing with a maximum distortion ''D'', we must impose the requirement that the information lost in transmission does not exceed the maximum tolerable loss of ''H'' − ''R''(''D'') bits/symbol. This means that the channel capacity must be at least as large as ''R''(''D'').
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| == See also ==
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| * [[Decorrelation]]
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| * [[Rate–distortion optimization]]
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| * [[Source coding]]
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| * [[Sphere-packing]]
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| * [[White noise|Whitening]]
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| == References ==
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| {{Reflist}}
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| ==External links==
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| *[http://www-ict.its.tudelft.nl/vcdemo VcDemo Image and Video Compression Learning Tool]
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| {{Compression Methods}}
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| {{DEFAULTSORT:Rate-distortion theory}}
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| [[Category:Data compression]]
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| [[Category:Information theory]]
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There are many articles on fat reduction that have been composed. We have read a amount of them that suggest a diet for those wishing to lose weight. Others have spoken of exercise. We have several whom suggest we hiring a personal trainer. The question, yet, nonetheless remains. What is healthy weight loss? What is healthy weight for females?
BMI will be calculated in metric or imperial measurements. There is a slight variation between the two equations, however, the outcome is accurate enough for wellness assessment reasons. BMI is calculated using a bmi chart, an online BMI calculator, or manually utilizing the BMI equation. The results, which are taken to two decimal places, are the same for each. The BMI equation is obtainable in metric or imperial measurements, and there is a extremely slight variation in the 2 figures.
Now let's have a peep into childhood weight details. Children usually inherit this disease from their parents. The amount of obese children is increasing at an alarming rate. As compared to the last 3 years bmi chart men, this number has doubled. Sedentary escapades that are preferred by kids make them dull and lazy. They tend to take foods which lack in dietary values. As there is not any physical exertion associated, the fats get deposited inside body and create them lethargic. The general capability of overweight kids is very low as compared to others. One of the other childhood obesity details is that such children suffer from many different problems like lack of concentration, inability to sleep properly, plus humiliation faced from peers and neighbors.
Rest: This has become the most significant parts of my training. If I don't receive enough rest, my body begins to break down. Listen (really closely) to the body.
The perfect weight for females of a medium frame measuring 64-67 inches is between 124-147 pounds. This leads to a lot of difference inside what will be considered an ideal weight. Women must be accorded for each inch over 5 feet (1.52 m). So a female whom is 52 (1.57m) has an perfect weight of 110 pounds (49.89 kg).
To a certain extent this really is true, however excess puppy fat is because risky to a child because excess fat is to an adult. It is estimated that over 15% of UK youngsters bmi chart women are overweight or fat, and this figure is increasing fast. The Journal of the American Medical Association reported found on the 4th April which the amount of overweight American youngsters was 33.6%. Obese children grow into fat adults. They never lose this thus called puppy fat unless positive steps are taken. They have a significantly high risk of developing severe wellness problems , both today plus because an adult, including potentially life threatening conditions like bowel cancer, diabetes, strokes, heart conditions plus significant blood pressure. The more overweight the child, the better the risk.
Nonetheless, it utilizes the U.S. Navy Circumference Method that need the input of information for it to be able to resolve for the body fat percentage. The readings can be put into a body fat chart for the individual to be able to monitor the decrease or increase of body fat percentage.
If you are thinking what the healthy fat is for a girl, then the BMI is the path to take. You must additionally check the circumference of your waist. If you believe you're obese, then you need to start eating healthy plus exercise frequently so that you can shed those pounds plus become the healthy, fresh we.