Device and method for processing piecewise smoothed signal

Abstract

A device comprises at least one processor and at least one memory comprising computer program code, wherein the at least one memory and the computer program code are configured to, together with the at least one processor: receive an input signal, the input signal comprising a piecewise smooth signal and a noise signal; determine a priori probability density function of a representation of the piecewise smooth signal, the representation of the piecewise smooth signal comprising a combination of a smooth Gaussian variable and a jump Cauchy variable; determine a likelihood function based on the input signal; determine a posterior probability density function based on the likelihood function and the distribution of the prior probability density function; and estimate the piecewise smooth signal based on an estimate of the posterior probability density function.

Description

Technical Field

[0001] The present invention relates to a device, a method and a computer program for signal processing. The present invention relates in particular to signal processing of piecewise smoothed signals. Background Art

[0002] Several important signals can be classified as piecewise smooth signals. A piecewise smooth signal is a type of signal that includes sharp transitions separating smooth regions. An exemplary 1D piecewise smooth signal is an electrocardiogram (ECG) signal, which may include "jump" regions and smooth regions that repeat for each heartbeat. A high-dimensional example of a piecewise smooth signal includes 2D image data.

[0003] Techniques for denoising piecewise smooth signals are known. For example, in image reconstruction (where the signal is a 2D piecewise smooth image), transitions in the signal (edges in the image) are conventionally processed using this process using total variation (TV) regularization to reduce any noise.

[0004] Another approach is to use the Cauchy difference prior method to perform similar edge-preserving reconstruction. Summary of the invention

[0005] In a first aspect, a device is provided, comprising at least one processor and at least one memory comprising computer program code, wherein the at least one memory and the computer program code are configured to, together with the at least one processor: receive an input signal, the input signal comprising a piecewise smooth signal and a noise signal; determine a priori probability density function of a representation of the piecewise smooth signal, the representation of the piecewise smooth signal comprising a combination of a smooth Gaussian variable and a jump Cauchy variable; determine a likelihood function based on the input signal; determine a posterior probability density function based on the likelihood function and the distribution of the prior probability density function; and estimate the piecewise smooth signal based on an estimate of the posterior probability density function.

[0006] A processor configured to determine a prior probability density function of a representation of a piecewise smooth signal, wherein the representation of the piecewise smooth signal includes a combination of a smooth Gaussian variable and a jump Cauchy variable, can be further configured to: determine a first probability density function representing sample differences of the smooth Gaussian variable; determine a second probability density function representing sample differences of the jump Cauchy variable; and convolve the first probability density function and the second probability density function to generate a prior probability density function.

[0007] A processor configured to convolve a first probability density function and a second probability density function to generate a priori probability density function may be configured to: determine an analytical determination using at least one imaginary error function or a Dawson function; and determine a numerical determination using a grid of sample points and interpolation for points between the grid of sample points.

[0008] The processor configured to estimate the piecewise smooth signal based on an estimate of the a posteriori probability density function may be configured to estimate the piecewise smooth signal based on a maximum a posteriori estimate of the a posteriori probability density function.

[0009] The processor configured to estimate the piecewise smooth signal based on a maximum a posteriori estimate of the posterior probability density function may be further configured to minimize a combination of a logarithm of the likelihood function and a logarithm of the a priori probability density function.

[0010] The processor configured to minimize a combination of the logarithm of the likelihood function and the logarithm of the prior probability density function may be configured to apply one of: Newton's methods; and the Barzilai-Borwein method.

[0011] The processor configured to determine the posterior probability density function based on the likelihood function and the prior probability density function may be further configured to determine the posterior probability density function based on the product of the likelihood function and the prior probability density function according to Bayes' Theorem.

[0012] The processor may be further configured to: window the input signal.

[0013] The smooth Gaussian variables and the jump Cauchy variables can be independent of each other.

[0014] According to a second aspect, a method is provided, which includes: receiving an input signal, the input signal including a piecewise smooth signal and a noise signal; determining a priori probability density function representing the piecewise smooth signal, the representation of the piecewise smooth signal including a combination of a smooth Gaussian variable and a jump Cauchy variable; based on the input signal, determining a likelihood function; based on the likelihood function and the distribution of the prior probability density function, determining a posterior probability density function; and estimating the piecewise smooth signal based on an estimate of the posterior probability density function.

[0015] Determine a priori probability density function of a representation of a piecewise smooth signal, wherein the representation of the piecewise smooth signal includes a combination of a smooth Gaussian variable and a jump Cauchy variable and may also include: determining a first probability density function representing sample differences of the smooth Gaussian variable; determining a second probability density function representing sample differences of the jump Cauchy variable; and convolving the first probability density function and the second probability density function to generate a priori probability density function.

[0016] Convolving the first probability density function and the second probability density function to generate the a priori probability density function may include: determining the analytical determination using at least one imaginary error function or a Dawson function; and determining the numerical determination using a grid of sample points and interpolation for points between the grid of sample points.

[0017] Estimating the piecewise smooth signal based on the estimate of the a posteriori probability density function may include estimating the piecewise smooth signal based on a maximum a posteriori estimate of the a posteriori probability density function.

[0018] Estimating the piecewise smooth signal based on a maximum a posteriori estimate of the posterior probability density function may include minimizing a combination of a logarithm of the likelihood function and a logarithm of the prior probability density function.

[0019] Minimizing the combination of the logarithm of the likelihood function and the logarithm of the prior probability density function may include applying one of: Newton's iterative method; and Barzilai-Borwein method.

[0020] Determining the posterior probability density function based on the likelihood function and the prior probability density function may also include: determining the posterior probability density function based on the product of the likelihood function and the prior probability density function according to Bayes' theorem.

[0021] The method may further include: windowing the input signal.

[0022] The smooth Gaussian variables and the jump Cauchy variables can be independent of each other.

[0023] According to a third aspect, a device is provided, comprising: a component for receiving an input signal, the input signal comprising a piecewise smooth signal and a noise signal; a component for determining a priori probability density function representing the piecewise smooth signal, the representation of the piecewise smooth signal comprising a combination of a smooth Gaussian variable and a jump Cauchy variable; a component for determining a likelihood function based on the input signal; a component for determining a posterior probability density function based on the likelihood function and the distribution of the prior probability density function; and a component for estimating the piecewise smooth signal based on an estimate of the posterior probability density function.

[0024] A component for determining a priori probability density function of a representation of a piecewise smooth signal, wherein the representation of the piecewise smooth signal includes a combination of a smooth Gaussian variable and a jump Cauchy variable and may also include: a component for determining a first probability density function of sample differences representing the smooth Gaussian variable; a component for determining a second probability density function of sample differences representing the jump Cauchy variable; and a component for convolving the first probability density function and the second probability density function to generate a priori probability density function.

[0025] The means for convolving the first probability density function and the second probability density function to generate a priori probability density function may include: means for performing analytical determination using at least one imaginary error function or Dawson function; and means for performing numerical determination using a grid of sample points and interpolation for points between the grid of sample points.

[0026] The means for estimating the piecewise smooth signal based on an estimate of the a posteriori probability density function may include means for estimating the piecewise smooth signal based on a maximum a posteriori estimate of the a posteriori probability density function.

[0027] The means for estimating the piecewise smooth signal based on a maximum a posteriori estimate of the posterior probability density function may include means for minimizing a combination of a logarithm of the likelihood function and a logarithm of the prior probability density function.

[0028] The means for minimizing the combination of the logarithm of the likelihood function and the logarithm of the prior probability density function may include means for applying one of: a Newton iteration method; and a Barzilai-Borwein method.

[0029] The means for determining the posterior probability density function based on the likelihood function and the prior probability density function may also include means for determining the posterior probability density function based on the product of the likelihood function and the prior probability density function according to Bayes' theorem.

[0030] The apparatus may further include: means for windowing the input signal.

[0031] The smooth Gaussian variable and the jump Cauchy variable can be independent of each other.

[0032] In another aspect, there is provided a computer program comprising program code means adapted to perform the steps of the second aspect when the program is run on a data processing apparatus. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] To aid understanding of the present disclosure and to show how embodiments may be put into practice, reference is made by way of example to the accompanying drawings, in which:

[0034] Figure 1a Schematically illustrates an exemplary waveform having transitions and smooth regions;

[0035] Figure 1b Schematically illustrates apparatus suitable for implementing some embodiments;

[0036] Figure 1c Schematically illustrating a method for applying a Figure 1b An exemplary system implemented on the device shown in;

[0037] Figure 2 Schematically illustrating in more detail how the Figure 1c The exemplary prior probability density function determiner shown in ;

[0038] Figure 3 Schematically illustrating how Figure 1cAn exemplary posterior probability density function determiner implemented in;

[0039] Figure 4 Schematically illustrating how Figure 1c An example point estimator implemented in ;

[0040] Figure 5 shows a flow chart illustrating the operation of an exemplary system according to some embodiments;

[0041] Figure 6 shows a flow chart illustrating the operation of an exemplary a priori probability density function determiner according to some embodiments;

[0042] Figure 7 A flow chart illustrating the operation of an exemplary posterior density function determiner and point estimator according to some embodiments is shown. DETAILED DESCRIPTION

[0043] The examples disclosed in this application are applicable to devices, methods, etc. for processing piecewise smooth signals (i.e., signals with transitions and smooth regions). Figure 1a Exemplary signals 1 and 3 are shown. These signals 1, 3 are piecewise smooth signals and include smooth areas 11, 17 and jumps 13, 15. Jumps are prominent signal features that deviate from smooth areas in the signal. Peaks of ECG signals (such as R peaks) are examples of jumps. Jumps can have step function shapes. Other shapes are also possible. Jumps can have unexpected or sudden starting and ending points. More specifically, the concept involves estimating an unknown piecewise smooth signal u(t) based on a measurement m(t), which can be characterized using the observation equation m(t) = A(u(t)) + e(t), where A is a function and e(t) is a signal derived from noise processing. In a linear equation, A can be a matrix (in discrete form).

[0044] An example of an application of signal processing as described herein is to denoise a measured ECG signal. In such an embodiment, A may be chosen as the identity operator, and the measurement is of the form m(t)=u(t)+e(t). In this case, the problem is to estimate the signal u(t) from the noise-corrupted measurement m(t).

[0045] In other examples, when A is selected as the convolution operator, the processing can be deconvolution of a 1D (or 2D or higher dimensional) signal (deblurring / sharpening of an image), or image restoration and painting.

[0046] The concepts discussed in further detail below are characterized by embodiments of processing an input signal using a signal represented as a sum of a (Gaussian) smooth variable and a (Cauchy distributed) variable (which represents the statistically considered jumps) and methods (and apparatus) for determining the combined distribution of the sum without further variable estimation. Thus, these embodiments result in a simple and computationally efficient solution (thus having low CPU and memory requirements in processor-based implementations).

[0047] about Figure 1b An example of an apparatus 100 or system suitable for implementing some embodiments is shown. The apparatus 100 includes a memory 102 and a processor 104. The apparatus 100 also includes a transceiver at 106 for receiving information such as signals to be analyzed from other apparatuses or devices.

[0048] In some embodiments, the transceiver 106 is configured to receive user input, such as input to initiate analysis or change a statistical variable, such as described in further detail herein.

[0049] The transceiver 106 may also be configured to send information, such as processed signal information, to other devices such as remote displays. The transceiver 106 may be configured to send and / or receive information using any suitable protocol and encoding system. In addition, although the transceiver 106 is shown to be connected to a wireless antenna 107 for wireless communication with other devices or equipment, in some embodiments, the transceiver 106 is configured to communicate with other devices or equipment at least in part via a wired connection.

[0050] In some embodiments, the device 100 also includes or is in communication with a display 108 that enables the device 100 to provide output in a visual manner so that it can be understood by a user or viewer. For example, the device 100 can cause visual information to be provided on the display 108 that can be viewed by a user. In some embodiments, the device may also include a signal source or sensor for providing a signal to be directly analyzed, such as a camera or imaging device suitable for generating an image signal to be processed, or an ECG sensor for providing an ECG signal, etc.

[0051] In some examples, the device 100 and the processor 104 and the memory 102 may be configured to execute programs to enable the functions and processes described below to operate.

[0052] Although in the above examples, the apparatus 100 is shown as a single device and thus operates as a single or centralized apparatus in some embodiments, the functions described below can be implemented by a distributed system of apparatuses communicating with each other. For example, the functions can be implemented in a cloud-based system, where the apparatuses or servers are physically or logically separated from each other.

[0053] In some embodiments, processor 104 is configured to implement a posterior probability density function algorithm and a point estimation algorithm based on input signals and other functions, as discussed in further detail below.

[0054] In the following examples, the concept is explained with respect to the specific example of a 1D signal. However, it should be understood that the same process as described below can be applied more generally to high-dimensional signals, whether 2D (e.g., image data) or piecewise smooth signals, as it should be understood that high-dimensional signals can be interpreted as a series of statistically independent 1D signals.

[0055] about Figure 1c , showing an exemplary apparatus for implementing the embodiment.

[0056] The apparatus comprises an input for a digital input signal m(t) 200. The apparatus further comprises a window filter 201, an a priori probability density function determiner 203, an a posteriori probability density function determiner 205, and a point estimator 207.

[0057] The window filter 201 is configured to receive a digital input signal m(t) 200 and generate a series of windowed sample periods, each of which includes a windowed signal m=m=(m0, ..., m n )210, wherein the sample length is n+1. In some embodiments, the window filter 201 is configured to generate a separate sequence, so that the subsequent and ongoing windows do not share samples. In some embodiments, the window filter 201 is configured to generate a sample sequence that at least partially overlaps. The windowed signal 210 sample length can be any suitable sample length. In some embodiments, the window filter 201 is configured to pass the windowed signal 210 to the posterior probability density function determiner 205.

[0058] In some embodiments, the apparatus further comprises a priori probability density function (PDF) determiner 203. The priori probability density function (PDF) determiner 203 may be implemented in hardware or dedicated circuits, software, logic or any combination thereof. In addition, some aspects of the priori probability density function (PDF) determiner 203 may be implemented in hardware, while other aspects may be implemented in firmware or software executable by a controller, microprocessor or other computing device, but the present invention is not limited thereto. However, as a non-limiting example, various aspects of the priori probability density function (PDF) determiner 203 may be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controller or other computing device, software module (e.g., library function), physical circuit (e.g., ASIC), programmed microprocessor or some combination thereof. Therefore, the priori probability density function (PDF) determiner 203 may be implemented by computer software executable by at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The a priori probability density function (PDF) determiner 203 is configured to generate an a priori PDF 220 based on statistical determination, wherein the signal is modeled as the sum of two random signals. The generated a priori PDF 220 is passed to the a posteriori probability density function determiner 205.

[0059] The posterior probability density function determiner 205 can also be realized by hardware or dedicated circuit, software, logic or any combination thereof. In addition, some aspects of the posterior probability density function determiner 205 can be realized by hardware, and other aspects can be realized by firmware or software that can be performed by controller, microprocessor or other computing equipment, but the present invention is not limited thereto. However, as a non-limiting example, various aspects of the posterior probability density function determiner 205 can be realized by hardware, software, firmware, dedicated circuit or logic, general hardware or controller or other computing equipment, software module (for example, library function), physical circuit (for example, ASIC), programming microprocessor or some combination thereof. Therefore, the posterior probability density function determiner 205 can be realized by executable computer software of at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The posterior probability density function determiner 205 is configured to receive the windowed signal 210 and, in addition, the prior PDF, and determine, by using a Bayesian approach, a posterior probability density function 230 for the windowed signal 210, denoted as p(u|m), based on the prior PDF 220. The determined posterior probability density function 230 is then passed to the point estimator 207.

[0060] The point estimator 207 may also be implemented in hardware or dedicated circuits, software, logic, or any combination thereof. In addition, some aspects of the point estimator 207 may be implemented in hardware, while other aspects may be implemented in firmware or software that may be executed by a controller, microprocessor, or other computing device, but the present invention is not limited thereto. However, as a non-limiting example, various aspects of the point estimator 207 may be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controllers or other computing devices, software modules (e.g., library functions), physical circuits (e.g., ASICs), programmed microprocessors, or some combination thereof. Therefore, the point estimator 207 may be implemented by computer software executable by at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The point estimator 207 configured to receive the posterior probability density function 230 is configured to distinguish the piecewise smoothed signal from the noise component. The estimated signal point may be output.

[0061] about Figure 2 , an example of the prior probability density function determiner 203 is shown in more detail. In some embodiments, the prior probability density function determiner 203 is configured to receive lambdaλ and sigmaσ inputs, which control the determination of random variables based on smooth (Gaussian) functions and jump (Cauchy) functions. The prior probability density function determiner 203 may include a smooth (Gaussian) function variable determiner 211, a jump (Cauchy) function variable y determiner 213, and a convolution determiner 215.

[0062] The smooth (Gaussian) function variable determiner 211 can be implemented in hardware or dedicated circuits, software, logic or any combination thereof. In addition, some aspects of the smooth (Gaussian) function variable determiner 211 can be implemented in hardware, while other aspects can be implemented in firmware or software that can be executed by a controller, a microprocessor or other computing device, but the present invention is not limited thereto. However, as a non-limiting example, various aspects of the smooth (Gaussian) function variable determiner 211 can be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controllers or other computing devices, software modules (e.g., library functions), physical circuits (e.g., ASICs), programmed microprocessors or some combination thereof. Therefore, the smooth (Gaussian) function variable determiner 211 can be implemented by computer software executable by at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The smooth (Gaussian) function variable determiner 211 is configured to provide a suitable PDF for the smooth function x based on the sigmaσ input, for example by using the following expression:

[0063]

[0064] Among them, x k (k=0,....,n) is the value of x at the discrete point.

[0065] This PDF p(x) may be passed to the convolution determiner 215 .

[0066] The jump (Cauchy) function variable determiner 213 can be implemented in hardware or dedicated circuits, software, logic, or any combination thereof. In addition, some aspects of the jump (Cauchy) function variable determiner 213 can be implemented in hardware, while other aspects can be implemented in firmware or software that can be executed by a controller, microprocessor or other computing device, but the present invention is not limited to this. However, as a non-limiting example, various aspects of the jump (Cauchy) function variable determiner 213 can be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controllers or other computing devices, software modules (e.g., library functions), physical circuits (e.g., ASICs), programmed microprocessors, or some combination thereof. Therefore, the jump (Cauchy) function variable determiner 213 can be implemented by computer software executable by at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The jump (Cauchy) function variable determiner 213 is configured to provide a suitable PDF for the jump function y based on the lambda λ input, for example, by using the following expression:

[0067]

[0068] Among them, y k (k=0,....,n) is the value of y at the discrete point.

[0069] This PDF p(y) may be passed to the convolution determiner 215 .

[0070] The convolution determiner 215 may be implemented in hardware or dedicated circuits, software, logic, or any combination thereof. In addition, some aspects of the convolution determiner 215 may be implemented in hardware, while other aspects may be implemented in firmware or software that may be executed by a controller, microprocessor, or other computing device, but the present invention is not limited thereto. However, as a non-limiting example, various aspects of the convolution determiner 215 may be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controllers or other computing devices, software modules (e.g., library functions), physical circuits (e.g., ASICs), programmed microprocessors, or some combination thereof. Therefore, the convolution determiner 215 may be implemented by computer software executable by at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The convolution determiner 215 configured to receive the smoothed p(x) and the jump p(y) PDFs is configured to convolve them to determine the PDF of the sum of the smoothed x and jump y variables. The convolution determiner 215 may, for example, calculate the convolution as follows:

[0071]

[0072] Where f is the marginal density convolution of Gaussian p(x) and Cauchy p(y)

[0073] For example, the convolution can be calculated:

[0074]

[0075] Wherein, Erfi(z) is an imaginary error function, which is an entire function defined by Erfi(z)≡-ierf(iz). As another example, in some embodiments, the convolution determiner 215 may use the Dawson function F to express the result of the convolution:

[0076]

[0077] Among them, the Dawson function F is defined as:

[0078]

[0079] The determined convolution, the prior probability density function p(u), may then be output from the prior probability density function determiner 203. In some embodiments, the prior probability density function determiner 203 may include a set of predetermined parameters that are passed to the posterior probability density function determiner 205. In some embodiments, the prior probability density function is implemented as a lookup table of values ​​accessed via preconfigured parameter values ​​λ and σ.

[0080] In other words, in some embodiments, the a priori probability density function may be implemented as a predetermined value in a memory.

[0081] In some embodiments, Figure 3 A posterior probability density function determiner 205 is shown. The posterior probability density function determiner 205 comprises a likelihood function determiner 301 and a posterior likelihood posterior probability density function determiner 303.

[0082] The likelihood function p(m|u) determiner 301 can be implemented by hardware or dedicated circuits, software, logic or any combination thereof. In addition, some aspects of the likelihood function determiner 301 can be implemented by hardware, while other aspects can be implemented by firmware or software that can be executed by a controller, a microprocessor or other computing device, but the present invention is not limited thereto. However, as a non-limiting example, various aspects of the likelihood function determiner 301 can be implemented by hardware, software, firmware, dedicated circuits or logic, general hardware or controllers or other computing devices, software modules (e.g., library functions), physical circuits (e.g., ASICs), programmed microprocessors or some combination thereof. Therefore, the likelihood function determiner 301 can be implemented by computer software executable by at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The likelihood function determiner 301 configured to receive the windowed signal 210 is configured to determine the likelihood function. Likelihood is the assumed probability that an event that has occurred will produce a specific result. The concept is different from the concept of probability in that probability refers to the occurrence of future events, while likelihood refers to past events with known results.

[0083] The likelihood function can be determined based on the model of the input signal m(t). If we model the measurement as m(t) = A(u(t)) + e(t) as described above, and e(t) is modeled as a Gaussian and uniformly distributed variable (e.g., white noise), then:

[0084]

[0085] Here, the norm is the L2 norm (sum of squares).

[0086] The determined likelihood function may be passed to the posterior likelihood a posteriori probability density function determiner 303 .

[0087] In some embodiments, the posterior likelihood a priori probability density function determiner 303 can be implemented in hardware or dedicated circuits, software, logic or any combination thereof. In addition, some aspects of the posterior likelihood a priori probability density function determiner 303 can be implemented in hardware, while other aspects can be implemented in firmware or software that can be executed by a controller, a microprocessor or other computing device, but the present invention is not limited thereto. However, as a non-limiting example, various aspects of the posterior likelihood a priori probability density function determiner 303 can be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controller or other computing devices, software modules (e.g., library functions), physical circuits (e.g., ASIC), programming microprocessors or some combination thereof. Therefore, the posterior likelihood a priori probability density function 303 can be implemented by executable computer software of at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The posterior likelihood a priori probability density function determiner 303 configured to receive a priori PDF and the determined likelihood function is configured to determine the posterior probability density function p(u|m) by using Bayes' theorem so that:

[0088]

[0089] Where p(m|u) is the likelihood function and p(u) is the prior probability density function (the result of the convolution). The posterior probability density function p(u|m) (or the logarithm of p(u|m) in some embodiments) may be output to the point estimator 207.

[0090] Figure 4An exemplary point estimator 207 including a minimizer 401 is shown. The minimizer 401 may be implemented in hardware or dedicated circuits, software, logic, or any combination thereof. In addition, some aspects of the minimizer 401 may be implemented in hardware, while other aspects may be implemented in firmware or software that may be executed by a controller, microprocessor, or other computing device, but the present invention is not limited thereto. However, as a non-limiting example, various aspects of the minimizer 401 may be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controller or other computing device, software module (e.g., library function), physical circuit (e.g., ASIC), programmed microprocessor, or some combination thereof. Therefore, the minimizer 401 may be implemented by computer software executable by at least one data processor (such as in a processor entity), or by hardware, or by a combination of software and hardware. The minimizer 401 configured to receive the posterior probability density function p(u|m) 220 is configured to generate a suitable point estimate by minimizing the posterior probability density function. For example, in some embodiments, the minimizer 401 is configured to determine a maximum a posteriori (MAP) estimate, which is the maximum point of the posterior probability density function. In practice, the MAP estimate can be calculated by minimizing the following function:

[0091] -log p(u|m)=-log p(m|u)-log p(u)

[0092] The minimization may be performed using any suitable method, for example using Newton's iterative method or the Barzilai-Borwein method.

[0093] Figure 5 An exemplary flow chart illustrating the operation of an exemplary system in accordance with some embodiments is shown.

[0094] First, a signal m(t) including a piecewise smoothed signal u(t) and noise e(t) is measured or received, as in Figure 5 As shown in step 501.

[0095] Then, the signal m(t) is windowed to generate a windowed signal 210, as in Figure 5 As shown in step 503.

[0096] In some embodiments, the lambdaλ and sigmaσ parameters for smoothing the x and jumpy variables are selected as in Figure 5 As shown in step 502.

[0097] Furthermore, in these embodiments, the prior PDF is constructed by specifying PDFs for the smooth portion x and the jump y and then convolving the PDFs to generate a priori probability density function p(u)=p(x)*p(y). In other words, the probability density function represents the sum of the smooth and jump variables, as in Figure 5 As shown in step 504.

[0098] After the prior probability density function p(u) and the likelihood function p(m|u) have been determined, the posterior probability density function p(u|m) (or logp(u|m)) is determined, as in Figure 5 As shown in step 505.

[0099] Then, the posterior probability density function is minimized to determine the point estimate, as in Figure 5 As shown in step 507.

[0100] Then, output the point estimate, as in Figure 5 As shown in step 509.

[0101] Figure 6 A flow chart illustrating the operation of an exemplary probability density function determiner in more detail according to some embodiments is shown.

[0102] First, a smoothing function variable x and PDF p(x) are generated (and in some embodiments based on sigma input), as in Figure 6 As shown in step 601.

[0103] Then, the jump function variables and PDF p(y) are generated (and in some embodiments based on the lambda λ input), as in Figure 6 As shown in step 603.

[0104] Then, the prior PDF is generated by applying convolution to the p(x) and p(y) PDFs, as in Figure 6 As shown in step 605 .

[0105] Figure 7 A flow chart illustrating the operation of an exemplary posterior probability density function determiner and point estimator in accordance with some embodiments is shown.

[0106] As in Figure 7 As shown in step 701 , the first operation is to receive the windowed signal 210 .

[0107] Then, the likelihood function PDF p(m|u) can be determined from the windowed signal 210, as in Figure 7 As shown in step 703.

[0108] In addition, the prior probability density function p(u) is also received or obtained, such as in Figure 7 As shown in step 704.

[0109] Then, the method may include determining the posterior probability density function p(u|m) based on the likelihood function p(m|u) and the prior probability density function p(u), as in Figure 7 As shown in step 705.

[0110] The method may then include estimating the point (e.g., by minimizing the posterior probability density function p(u|m) (or -logp(u|m)), as in Figure 7 As shown in step 707.

[0111] In general, various embodiments may be implemented in hardware or dedicated circuits, software, logic, or any combination thereof. Some aspects of the invention may be implemented in hardware, and other aspects may be implemented in firmware or software that may be executed by a controller, microprocessor, or other computing device, but the invention is not limited thereto. Although various aspects of the invention may be illustrated and described as block diagrams, flow charts, or using some other graphical representations, it should be well understood that, as non-limiting examples, the boxes, devices, systems, techniques, or methods described herein may be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controllers or other computing devices, or some combination thereof.

[0112] Embodiments of the present invention may be implemented by computer software executable by a data processor of a mobile device (such as in a processor entity), or by hardware, or by a combination of hardware and software. Computer software or programs are also referred to as program products, including software routines, applets and / or macros, which may be stored in any device-readable data storage medium and which include program instructions for performing specific tasks. A computer program product may include one or more computer executable components configured to execute an embodiment when the program is running. One or more computer executable components may be at least one software code or a portion thereof.

[0113] Further in this regard, it should be noted that any block of the logic flow as in the accompanying drawings may represent program steps, or interconnected logic circuits, blocks and functions, or a combination of program steps and logic circuits, blocks and functions. The software may be stored on physical media such as memory chips, or storage blocks implemented within the processor, magnetic media such as hard disks or floppy disks, and optical media such as, for example, DVDs and data variants thereof, CDs. Physical media are non-transitory media.

[0114] The memory may be of any type suitable for the local technical environment and may be implemented using any suitable data storage technology, such as semiconductor-based memory devices, magnetic memory devices and systems, optical memory devices and systems, fixed memory, and removable memory. The data processor may be of any type suitable for the local technical environment and may include one or more of a general-purpose computer, a special-purpose computer, a microprocessor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), an FPGA, a gate-level circuit based on a multi-core processor architecture, and a processor, as non-limiting examples.

[0115] Embodiments of the present invention may be practiced in various components such as integrated circuit modules. The design of integrated circuits is generally a highly automated process. Complex and powerful software tools are available to convert logic level designs into semiconductor circuit designs ready to be etched and formed on semiconductor substrates.

[0116] The foregoing description has provided a complete and detailed description of exemplary embodiments of the present invention by way of non-limiting examples. However, when read in conjunction with the accompanying drawings and the appended claims, various modifications and changes will become apparent to those skilled in the art in view of the foregoing description. However, the teachings of all such or similar modifications of the present invention will still fall within the scope of protection of the present invention as defined by the appended claims. Further embodiments do exist including combinations of one or more embodiments with any of the other previously discussed embodiments.

Claims

1. An apparatus for signal processing, comprising at least one processor and at least one memory including computer program code, wherein: The at least one memory and the computer program code are configured to, with the at least one processor: Receiving an input signal, wherein the input signal includes a piecewise smoothed signal and a noise signal; determining a priori probability density function of a representation of the piecewise smooth signal, the representation of the piecewise smooth signal comprising a combination of a smooth Gaussian variable and a jump Cauchy variable; Based on the input signal, determining a likelihood function; Determining a posterior probability density function based on the likelihood function and the prior probability density function distribution; as well as estimating the piecewise smoothed signal based on the estimate of the posterior probability density function, wherein the processor configured to determine a priori probability density function of a representation of the piecewise smooth signal, the representation of the piecewise smooth signal comprising a combination of a smooth Gaussian variable and a jump Cauchy variable, is further configured to: determining a first probability density function representing sample differences of the smoothed Gaussian variable; determining a second probability density function representing sample differences of the jump Cauchy variable; and The first probability density function and the second probability density function are convolved to generate the prior probability density function.

2. The device according to claim 1, wherein: The processor configured to convolve the first probability density function and the second probability density function to generate the a priori probability density function is configured to: determining the analytical determination using at least one imaginary error function or a Dawson function; and The numerical determination is determined using a grid of sample points and interpolation for points between the grid of sample points.

3. The device according to claim 1 or 2, wherein: The processor configured to estimate the piecewise smooth signal based on the estimate of the posterior probability density function is configured to: The piecewise smooth signal is estimated based on a maximum a posteriori estimate of the posterior probability density function.

4. The device according to claim 3, wherein: The processor configured to estimate the piecewise smooth signal based on a maximum a posteriori estimate of the posterior probability density function is further configured to: Minimize the combination of the logarithm of the likelihood function and the logarithm of the prior probability density function.

5. The device according to claim 4, wherein: The processor configured to minimize the combination of the logarithm of the likelihood function and the logarithm of the prior probability density function is configured to apply one of: Newton iteration method; as well as Barzilai-Borwein method.

6. The device according to claim 1 or 2, wherein: The processor configured to determine the posterior probability density function based on the likelihood function and the prior probability density function is further configured to: According to Bayes' theorem, the posterior probability density function is determined based on the product of the likelihood function and the prior probability density function.

7. The device according to claim 1 or 2, wherein: The processor is further configured to: The input signal is windowed.

8. The device according to claim 1 or 2, wherein: The smooth Gaussian variable and the jump Cauchy variable are independent of each other.

9. A method for signal processing, comprising: Receiving an input signal, wherein the input signal includes a piecewise smoothed signal and a noise signal; determining a priori probability density function of a representation of the piecewise smooth signal, the representation of the piecewise smooth signal comprising a combination of a smooth Gaussian variable and a jump Cauchy variable; Based on the input signal, determining a likelihood function; Determining a posterior probability density function based on the likelihood function and the prior probability density function distribution; and estimating the piecewise smoothed signal based on the estimate of the posterior probability density function, Wherein, determining the prior probability density function of the representation of the piecewise smooth signal, the representation of the piecewise smooth signal includes a combination of a smooth Gaussian variable and a jump Cauchy variable and further includes: determining a first probability density function representing sample differences of the smoothed Gaussian variable; determining a second probability density function representing sample differences of the jump Cauchy variable; and The first probability density function and the second probability density function are convolved to generate the prior probability density function.

10. The method according to claim 9, wherein: Convolving the first probability density function and the second probability density function to generate the prior probability density function comprises: analytical determination using at least one imaginary error function or Dawson function; and The values ​​are determined using a grid of sample points and interpolation for points between the grid of sample points.

11. The method according to claim 9 or 10, wherein: Estimating the piecewise smooth signal based on the estimate of the posterior probability density function comprises: The piecewise smooth signal is estimated based on a maximum a posteriori estimate of the posterior probability density function.

12. The method according to claim 11, wherein: Estimating the piecewise smooth signal based on the maximum a posteriori estimate of the posterior probability density function comprises: Minimize the combination of the logarithm of the likelihood function and the logarithm of the prior probability density function.

13. The method according to claim 9 or 10, wherein: Determining the posterior probability density function based on the likelihood function and the prior probability density function further comprises: According to Bayes' theorem, the posterior probability density function is determined based on the product of the likelihood function and the prior probability density function.

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