In the present paper, the theoretical basis of autoregressive (AR) modelling in spectral analysis is explained in simple terms. Spectral analysis gives information about the frequency content and sources of variation in a time series. The AR method is an alternative to discrete Fourier transform, and the method of choice for high-resolution spectral estimation of a short time series. In biomedical engineering, AR modelling is used especially in the spectral analysis of heart rate variability and electroencephalogram tracings. In AR modelling, each value of a time series is regressed on its past values. The number of past values used is called the model order. An AR model or process may be used in either process synthesis or process analysis, each of which can be regarded as a filter. The AR analysis filter divides the time series into two additive components, the predictable time series and the prediction error sequence. When the prediction error sequence has been separated from the modelled time series, the AR model can be inverted, and the prediction error sequence can be regarded as an input and the measured time series as an output to the AR synthesis filter. When a time series passes through a filter, its amplitudes of frequencies are rescaled. The properties of the AR synthesis filter are used to determine the amplitude and frequency of the different components of a time series. Heart rate variability data are here used to illustrate the method of AR spectral analysis. Some basic definitions of discrete-time signals, necessary for understanding of the content of the paper, are also presented.

In the present paper, the theoretical background of multivariate autoregressive modelling (MAR) is explained. The motivation for MAR modelling is the need to study the linear relationships between signals. In biomedical engineering, MAR modelling is used especially in the analysis of cardiovascular dynamics and electroencephalographic signals, because it allows determination of physiologically relevant connections between the measured signals. In a MAR model, the value of each variable at each time instance is predicted from the values of the same series and those of all other time series. The number of past values used is called the model order. Because of the inter-signal connections, a MAR model can describe causality, delays, closed-loop effects and simultaneous phenomena. To provide a better insight into the subject matter, MAR modelling is here illustrated with a model between systolic blood pressure, RR interval and instantaneous lung volume.

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In the present paper, a 2-dimensional adaptive autoregressive filter is proposed for noise reduction in images degraded with Poisson noise. In autoregressive models, each value of an image is regressed on its neighborhood pixel values, called the prediction region. The autoregressive models are linear prediction models that split an image into 2 additive components, a predictable image and a prediction error image. Methods: In this research, unfiltered images were split into smaller blocks, and best combinations of a prediction region and a block size for the image quality of predictable images were sought by using 3 Poisson noise-corrupted images with different image statistics. The images had dimensions of 128 · 128 pixels. Image quality was assessed by means of the mean squared error of the image. The adaptive autoregressive model was fitted into each block separately. Different degrees of overlapping of the image blocks were tested, and for each pixel the mean predictor coefficient of the different models was determined. The prediction error image was calculated for the entire image, and the filtered image was obtained by subtracting the prediction error image from the original image. The effect of the best adaptive autoregressive filter was illustrated using real scintigraphic data. Results: Generally, a prediction region of 4 orthogonal neighbors of the predicted pixel with a block size of 5 · 5 showed the best results. The use of 75% overlapping of the image blocks and 1 iteration of the filtering was found to improve prediction accuracy. The results were further improved when the 2 error term images were summed and subjected to adaptive autoregressive filtering and the resulting predictable image was added to the iteratively filtered image, allowing both noise reduction and edge preservation. Patient data illustrated effective noise reduction. Conclusion: The proposed method provided a convenient way to reduce Poisson noise in scintigraphic images on a pixel-by-pixel basis. Autoregressi ve modeling uses past values of a 1-dimensional signal (1) or neighborhood values of a 2-dimensional signal (2) to extract important information from a signal. The number of past or neighborhood values used is called the model order. A 2-dimensional autoregressive model can be regarded as a low-pass filter that divides the image into 2 additive components, a predictable image and a prediction error image. Ideally, the prediction errors in a prediction error image should obey gaussian noise. The counting statistics in a scintigraphic image obey Poisson distribution, but with a mean value greater than, say 20, the counting statistics can be approximated by gaussian distribution (3). Therefore, 2-dimensional autoregressive models are, in theory, suitable for noise reduction in scintigraphic images. In the present paper, a new 2-dimensional adaptive autoregressive model for filtering of scintigraphic images is introduced. The adaptive autoregressive filter was tested using an artificial organlike scintigraphic image (4) with ...

A new method to apply spatial two-dimensional power spectral density (2D PSD) analysis to the data measured with Particle Image Velocimetry (PIV) has been introduced. Applying the method to a set of the velocity vector fields characteristic turbulence length scales can be estimated. In this method the computation of 2D PSD has been performed to two kinds of pre-processed data. In the first set, the local average has been spatially subtracted (Spatial decomposition) and in the second set the time-average has been subtracted (Reynolds decomposition). In the computation of 2D PSD the 2D FFT with the variance scaling has been used.Besides 2D spectral analysis this paper uses the distribution analysis of the various turbulence quantities and a structure analysis method to estimate the dimensions of coherent structures in the flow. Another method to analyse turbulence length scales is the estimation of the spatial 2D Auto Correlation Coefficient Function (2D ACCF). All these methods applied side by side to the PIV data increase the understanding ofthe turbulence, its scales and the nature of the coherent structures.