Misplaced Pages

#730269

46-414: Lossless JPEG was developed as a late addition to JPEG in 1993, using a completely different technique from the lossy JPEG standard. It uses a predictive scheme based on the three nearest (causal) neighbors (upper, left, and upper-left), and entropy coding is used on the prediction error. The standard Independent JPEG Group libraries cannot encode or decode it, but Ken Murchison of Oceana Matrix Ltd. wrote

92-406: A confidence interval for  μ . This t-statistic can be interpreted as "the number of standard errors away from the regression line." In regression analysis , the distinction between errors and residuals is subtle and important, and leads to the concept of studentized residuals . Given an unobservable function that relates the independent variable to the dependent variable – say, a line –

138-475: A sample mean ). The distinction is most important in regression analysis , where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals . In econometrics , "errors" are also called disturbances . Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model ). In this case,

184-410: A count of context occurrences. In LOCO-I algorithm, this procedure is modified and improved such that the number of subtractions and additions are reduced. The division-free bias computation procedure is demonstrated in [2] . Prediction refinement can then be done by applying these estimates in a feedback mechanism which eliminates prediction biases in different contexts. In the regular mode of JPEG-LS,

230-429: A dataset. For example, a large residual may be expected in the middle of the domain, but considered an outlier at the end of the domain. The use of the term "error" as discussed in the sections above is in the sense of a deviation of a value from a hypothetical unobserved value. At least two other uses also occur in statistics, both referring to observable prediction errors : The mean squared error (MSE) refers to

276-449: A normally distributed population with mean μ and standard deviation σ, and choose individuals independently, then we have and the sample mean is a random variable distributed such that: The statistical errors are then with expected values of zero, whereas the residuals are The sum of squares of the statistical errors , divided by σ , has a chi-squared distribution with n degrees of freedom : However, this quantity

322-491: A patch that extends the IJG library to handle lossless JPEG. Lossless JPEG has some popularity in medical imaging, and is used in DNG and some digital cameras to compress raw images, but otherwise was never widely adopted. Adobe 's DNG SDK provides a software library for encoding and decoding lossless JPEG with up to 16 bits per sample. ISO/IEC Joint Photography Experts Group maintains

368-402: A population of 21-year-old men is 1.75 meters, and one randomly chosen man is 1.80 meters tall, then the "error" is 0.05 meters; if the randomly chosen man is 1.70 meters tall, then the "error" is −0.05 meters. The expected value, being the mean of the entire population, is typically unobservable, and hence the statistical error cannot be observed either. A residual (or fitting deviation), on

414-445: A prediction of the sample value at the position labeled by X. The three neighboring samples must be already encoded samples. Any one of the predictors shown in the table below can be used to estimate the sample located at X. Any one of the eight predictors listed in the table can be used. Note that selections 1, 2, and 3 are one-dimensional predictors and selections 4, 5, 6, and 7 are two-dimensional predictors. The first selection value in

460-405: A reference software implementation which can encode both base JPEG (ISO/IEC 10918-1 and 18477-1) and JPEG XT extensions (ISO/IEC 18477 Parts 2 and 6-9), as well as JPEG-LS (ISO/IEC 14495). Lossless JPEG is actually a mode of operation of JPEG. This mode exists because the discrete cosine transform (DCT) based form cannot guarantee that encoder input would exactly match decoder output. Unlike

506-455: A regression is a number computed from the sum of squares of the computed residuals , and not of the unobservable errors . If that sum of squares is divided by n , the number of observations, the result is the mean of the squared residuals. Since this is a biased estimate of the variance of the unobserved errors, the bias is removed by dividing the sum of the squared residuals by df = n  −  p  − 1, instead of n , where df

SECTION 10

#1732859119731

552-585: Is a simple and efficient baseline algorithm which consists of two independent and distinct stages called modeling and encoding. JPEG-LS was developed with the aim of providing a low-complexity lossless and near-lossless image compression standard that could offer better compression efficiency than lossless JPEG. It was developed because at the time, the Huffman coding -based JPEG lossless standard and other standards were limited in their compression performance. Total decorrelation cannot be achieved by first order entropy of

598-415: Is a type of regression), the sum of squares of the residuals (aka sum of squares of the error) is divided by the degrees of freedom (where the degrees of freedom equal n  −  p  − 1, where p is the number of parameters estimated in the model (one for each variable in the regression equation, not including the intercept)). One can then also calculate the mean square of the model by dividing

644-669: Is also scalable, progressive, and more widely implemented. JPEG XT includes a lossless integer-to-integer DCT transform mode based on wavelet compression from JPEG 2000. JPEG XL includes a lossless/near-lossless/responsive mode called Modular which optionally uses a modified Haar transform (called "squeeze") and which is also used to encode the DC (1:8 scale) image in VarDCT mode as well as various auxiliary images such as adaptive quantization fields or additional channels like alpha . Entropy coding Too Many Requests If you report this error to

690-549: Is called the Median Edge Detection (MED) predictor or LOCO-I predictor. The pixel X is predicted by the LOCO-I predictor according to the following guesses: The three simple predictors are selected according to the following conditions: (1) it tends to pick B in cases where a vertical edge exists left of the X, (2) A in cases of an horizontal edge above X, or (3) A + B – C if no edge is detected. The JPEG-LS algorithm estimates

736-494: Is necessary if the population mean is known. It is remarkable that the sum of squares of the residuals and the sample mean can be shown to be independent of each other, using, e.g. Basu's theorem . That fact, and the normal and chi-squared distributions given above form the basis of calculations involving the t-statistic: where X ¯ n − μ 0 {\displaystyle {\overline {X}}_{n}-\mu _{0}} represents

782-497: Is not observable as the population mean is unknown. The sum of squares of the residuals , on the other hand, is observable. The quotient of that sum by σ has a chi-squared distribution with only n  − 1 degrees of freedom: This difference between n and n  − 1 degrees of freedom results in Bessel's correction for the estimation of sample variance of a population with unknown mean and unknown variance. No correction

828-470: Is reached. The total run of length is encoded and the encoder would return to the “regular” mode. JPEG 2000 includes a lossless mode based on a special integer wavelet filter (biorthogonal 3/5). JPEG 2000's lossless mode runs more slowly and has often worse compression ratios than JPEG-LS on artificial and compound images but fares better than the UBC implementation of JPEG-LS on digital camera pictures. JPEG 2000

874-464: Is the number of degrees of freedom ( n minus the number of parameters (excluding the intercept) p being estimated - 1). This forms an unbiased estimate of the variance of the unobserved errors, and is called the mean squared error. Another method to calculate the mean square of error when analyzing the variance of linear regression using a technique like that used in ANOVA (they are the same because ANOVA

920-417: The deviation of an observed value of an element of a statistical sample from its " true value " (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean ). The residual is the difference between the observed value and the estimated value of the quantity of interest (for example,

966-407: The influence functions of various data points on the regression coefficients : endpoints have more influence. Thus to compare residuals at different inputs, one needs to adjust the residuals by the expected variability of residuals, which is called studentizing . This is particularly important in the case of detecting outliers , where the case in question is somehow different from the others in

SECTION 20

#1732859119731

1012-476: The Wikimedia System Administrators, please include the details below. Request from 172.68.168.151 via cp1112 cp1112, Varnish XID 390318594 Upstream caches: cp1112 int Error: 429, Too Many Requests at Fri, 29 Nov 2024 05:45:19 GMT Errors and residuals in statistics In statistics and optimization , errors and residuals are two closely related and easily confused measures of

1058-466: The amount by which the values predicted by an estimator differ from the quantities being estimated (typically outside the sample from which the model was estimated). The root mean square error (RMSE) is the square-root of MSE. The sum of squares of errors (SSE) is the MSE multiplied by the sample size. Sum of squares of residuals (SSR) is the sum of the squares of the deviations of the actual values from

1104-461: The assumption that prediction residuals follow a two-sided geometric distribution (also called a discrete Laplace distribution ) and from the use of Golomb -like codes, which are known to be approximately optimal for geometric distributions. Besides lossless compression, JPEG-LS also provides a lossy mode ("near-lossless") where the maximum absolute error can be controlled by the encoder. Prior to encoding, there are two essential steps to be done in

1150-436: The conditional expectations of the prediction errors E { e | C t x } {\displaystyle E\left\{e|Ctx\right\}} using corresponding sample means e ¯ ( C ) {\displaystyle {\bar {e}}(C)} within each context Ctx . The purpose of context modeling is that the higher order structures like texture patterns and local activity of

1196-414: The contexts based on the assumption that After merging contexts of both positive and negative signs, the total number of contexts is ( ( 2 × 4 + 1 ) 3 + 1 ) / 2 = 365 {\displaystyle ((2\times 4+1)^{3}+1)/2=365} contexts. A bias estimation could be obtained by dividing cumulative prediction errors within each context by

1242-459: The data exhibit a trend, the regression model is likely incorrect; for example, the true function may be a quadratic or higher order polynomial. If they are random, or have no trend, but "fan out" - they exhibit a phenomenon called heteroscedasticity . If all of the residuals are equal, or do not fan out, they exhibit homoscedasticity . However, a terminological difference arises in the expression mean squared error (MSE). The mean squared error of

1288-412: The deviations of the dependent variable observations from this function are the unobservable errors. If one runs a regression on some data, then the deviations of the dependent variable observations from the fitted function are the residuals. If the linear model is applicable, a scatterplot of residuals plotted against the independent variable should be random about zero with no trend to the residuals. If

1334-546: The differences between the predicted samples instead of encoding each sample independently. The differences from one sample to the next are usually close to zero. A typical DPCM encoder is displayed in Fig.1. The block in the figure acts as a storage of the current sample which will later be a previous sample. The main steps of lossless operation mode are depicted in Fig.2. In the process, the predictor combines up to three neighboring samples at A, B, and C shown in Fig.3 in order to produce

1380-413: The differences found in the above equation is then quantized into roughly equiprobable and connected regions. For JPEG-LS, the differences g1, g2, and g3 are quantized into 9 regions and the region are indexed from −4 to 4. The purpose of the quantization is to maximize the mutual information between the current sample value and its context such that the high-order dependencies can be captured. One can obtain

1426-414: The entropy, one can use alphabet extension which codes blocks of symbols instead of coding individual symbols. This spreads out the excess coding length over many symbols. This is the “run” mode of JPEG-LS and it is executed once a flat or smooth context region characterized by zero gradients is detected. A run of west symbol “a” is expected and the end of run occurs when a new symbol occurs or the end of line

Lossless JPEG - Misplaced Pages Continue

1472-409: The errors are the deviations of the observations from the population mean, while the residuals are the deviations of the observations from the sample mean. A statistical error (or disturbance ) is the amount by which an observation differs from its expected value , the latter being based on the whole population from which the statistical unit was chosen randomly. For example, if the mean height in

1518-619: The errors, S n {\displaystyle S_{n}} represents the sample standard deviation for a sample of size n , and unknown σ , and the denominator term S n / n {\displaystyle S_{n}/{\sqrt {n}}} accounts for the standard deviation of the errors according to: Var ⁡ ( X ¯ n ) = σ 2 n {\displaystyle \operatorname {Var} \left({\overline {X}}_{n}\right)={\frac {\sigma ^{2}}{n}}} The probability distributions of

1564-416: The image can be exploited by context modeling of the prediction error. Contexts are determined by obtaining the differences of the neighboring samples which represents the local gradient : The local gradient reflects the level of activities such as smoothness and edginess of the neighboring samples. Notice that these differences are closely related to the statistical behavior of prediction errors. Each one of

1610-401: The input variable) may vary even if the errors themselves are identically distributed. Concretely, in a linear regression where the errors are identically distributed, the variability of residuals of inputs in the middle of the domain will be higher than the variability of residuals at the ends of the domain: linear regressions fit endpoints better than the middle. This is also reflected in

1656-463: The lossy mode which is based on the DCT, the lossless coding process employs a simple predictive coding model called differential pulse-code modulation (DPCM). This is a model in which predictions of the sample values are estimated from the neighboring samples that are already coded in the image. Most predictors take the average of the samples immediately above and to the left of the target sample. DPCM encodes

1702-445: The medical imaging field, and defined as an option in DNG standard, but otherwise it is not very widely used because of complexity of doing arithmetics on 10, 12, or 14bpp values on typical embedded 32-bit processor and a little resulting gain in space. JPEG-LS is a lossless or near-lossless compression standard for continuous-tone images. Its official designation is ISO-14495-1/ITU-T.87. It

1748-407: The modeling stage: decorrelation (prediction) and error modeling . In the LOCO-I algorithm, primitive edge detection of horizontal or vertical edges is achieved by examining the neighboring pixels of the current pixel X as illustrated in Fig.3. The pixel labeled by B is used in the case of a vertical edge while the pixel located at A is used in the case of a horizontal edge. This simple predictor

1794-444: The numerator and the denominator separately depend on the value of the unobservable population standard deviation σ , but σ appears in both the numerator and the denominator and cancels. That is fortunate because it means that even though we do not know  σ , we know the probability distribution of this quotient: it has a Student's t-distribution with n  − 1 degrees of freedom. We can therefore use this quotient to find

1840-411: The other hand, is an observable estimate of the unobservable statistical error. Consider the previous example with men's heights and suppose we have a random sample of n people. The sample mean could serve as a good estimator of the population mean. Then we have: Note that, because of the definition of the sample mean, the sum of the residuals within a random sample is necessarily zero, and thus

1886-466: The predicted values, within the sample used for estimation. This is the basis for the least squares estimate, where the regression coefficients are chosen such that the SSR is minimal (i.e. its derivative is zero). Likewise, the sum of absolute errors (SAE) is the sum of the absolute values of the residuals, which is minimized in the least absolute deviations approach to regression. The mean error (ME)

Lossless JPEG - Misplaced Pages Continue

1932-446: The prediction residuals employed by these inferior standards. JPEG-LS, on the other hand, can obtain good decorrelation. Part 1 of this standard was finalized in 1999. Part 2, released in 2003, introduced extensions such as arithmetic coding . The core of JPEG-LS is based on the LOCO-I algorithm, that relies on prediction, residual modeling , and context-based coding of the residuals. Most of the low complexity of this technique comes from

1978-403: The residuals are necessarily not independent . The statistical errors, on the other hand, are independent, and their sum within the random sample is almost surely not zero. One can standardize statistical errors (especially of a normal distribution ) in a z-score (or "standard score"), and standardize residuals in a t -statistic , or more generally studentized residuals . If we assume

2024-483: The standard uses Golomb–Rice codes which are a way to encode non-negative run lengths. Its special case with the optimal encoding value 2 allows simpler encoding procedures. Since Golomb–Rice codes are quite inefficient for encoding low entropy distributions because the coding rate is at least one bit per symbol, significant redundancy may be produced because the smooth regions in an image can be encoded at less than 1 bit per symbol. To avoid having excess code length over

2070-434: The sum of squares of the model minus the degrees of freedom, which is just the number of parameters. Then the F value can be calculated by dividing the mean square of the model by the mean square of the error, and we can then determine significance (which is why you want the mean squares to begin with.). However, because of the behavior of the process of regression, the distributions of residuals at different data points (of

2116-414: The table, zero, is only used for differential coding in the hierarchical mode of operation. Once all the samples are predicted, the differences between the samples can be obtained and entropy-coded in a lossless fashion using Huffman coding or arithmetic coding . Typically, compressions using lossless operation mode can achieve around 2:1 compression ratio for color images. This mode is quite popular in

#730269