Long-distance water surface target distance measurement filtering method based on monocular vision

Abstract

The invention provides a long-distance water surface target distance measurement filtering method based on monocular vision, and relates to the field of water surface distance measurement. The long-distance water surface target ranging and filtering method based on monocular vision comprises the following steps: S1, carrying out Butterworth high-frequency filtering; s2, detecting a peak value and a valley value; s3, determining midpoints of peaks and valleys; and S4, integrating empirical mode decomposition (EEMD) filtering. According to the method, the system can generate stable distance data through the steps, so that the precision and stability of long-distance monocular distance measurement are greatly improved, and the innovative method not only has important theoretical significance, but also shows practical application value in the field of automatic navigation of ships, and has wide application prospects. Therefore, according to the method, the distance measurement precision is remarkably improved in a dynamic marine environment, meanwhile, dependence on the horizon is eliminated, and the distance measurement requirement of the autonomous ship in collision avoidance is better met.

Description

Technical Field

[0001] The invention relates to the field of unmanned ship surface ranging, and in particular to a long-distance surface target ranging filtering method based on monocular vision. Background Art

[0002] With the continuous development of unmanned ships, autonomous driving and other robotic technologies, machine vision ranging technology has been widely used in many fields, including robot navigation, automated inspection, mapping and industrial applications. These applications usually require accurate target positioning, target tracking and obstacle avoidance. Visual ranging methods can generally be divided into two categories: active ranging and passive ranging. Active ranging methods, such as time of flight (TOF) and structured light ranging, rely on externally emitted light signals to calculate the target distance by measuring the reflection time of the signal or the deformation of the light beam; while passive ranging methods infer depth information by analyzing the scene image. The binocular ranging method uses two cameras to capture the target scene and restores the three-dimensional information of the object by calculating the parallax.

[0003] However, although the binocular ranging method can provide more accurate depth estimation, its implementation complexity is relatively high, requiring a multi-camera system, and is sensitive to factors such as field of view, relative motion of the target, and ambient lighting, which makes it not advantageous in some application scenarios. In contrast, the monocular visual ranging method has the advantages of simple system and low hardware cost because it relies on only one camera. It is especially suitable for some application scenarios with strict requirements on cost and space. In addition, the monocular visual ranging method can also make up for its limitations in depth information recovery by optimizing image processing and depth estimation algorithms, and has certain application potential.

[0004] In the application of surface target ranging, the advantages of monocular vision are particularly prominent due to the relative static nature of long-distance surface targets and the particularity of the environment. Traditional monocular ranging methods based on the horizon usually assume that the horizon must be clearly identified in order to perform reliable ranging. However, in the actual marine environment, affected by sea conditions, weather and visibility, the detection of the horizon is very difficult, or even completely unrecognizable. Furthermore, the influence of the earth's curvature makes the calculation of the horizon not always reliable, which greatly limits the method of monocular ranging that relies on the horizon in practical applications.

[0005] To overcome these challenges, existing ranging and filtering methods usually combine camera parameters with attitude angles provided by inertial measurement units (IMUs) to infer preliminary distance data of targets. However, due to the 6-DOF motion of the ship and measurement errors, these preliminary calculated distance data often show significant fluctuations and noise, and such fluctuations are often asymmetric, resulting in inaccurate direct results.

[0006] To this end, technicians in this field have proposed a long-distance surface target ranging filtering method based on monocular vision, aiming to overcome the shortcomings of existing methods and provide more stable and reliable surface target ranging results by optimizing image processing and filtering algorithms. Summary of the invention

[0007] 1. Technical issues to be resolved

[0008] In view of the shortcomings of the prior art, the present invention provides a long-distance surface target ranging filtering method based on monocular vision, which solves the problem of low ranging accuracy in direct filtering.

[0009] (II) Technical solution

[0010] To achieve the above objectives, the present invention is implemented through the following technical solutions:

[0011] The long-distance surface target ranging filtering method based on monocular vision includes the following steps:

[0012] S1. Buttreworth high frequency filtering;

[0013] S2. Detection of peak and valley values;

[0014] S3. Determination of the midpoint of the peak and valley;

[0015] S4. Integrated empirical mode decomposition (EEMD) filtering.

[0016] Furthermore, the S1. Buttreworth high frequency filtering steps are as follows:

[0017] The filtering process starts with designing a Butterworth filter that has the following transfer function characteristics;

[0018]

[0019] in:

[0020] s is a complex frequency variable;

[0021] ω c is the cut-off angular frequency, defined as, where f c is the cutoff frequency in Hertz;

[0022] n is the order of the filter, which is used to control the attenuation steepness of the filter near the cutoff frequency;

[0023] The Butterworth filter is applied in the digital domain to convert the transfer function of the analog filter into its digital equivalent form through a bilinear transformation; the coefficients a and b of the digital filter are calculated as follows:

[0024] b=[b0,b1,...b n ],a=[a0,a1,...a n ]

[0025] The filtered signal y(t) is calculated by the following difference equation:

[0026]

[0027] in:

[0028] x(t) is the input signal (in this case the raw distance data);

[0029] y(t) is the output signal (filtered distance data);

[0030] b i and a j are the filter coefficients derived from the Butterworth filter design;

[0031] With the above technical solution, the Butterworth filter is used to smooth the original distance data by attenuating high-frequency noise while retaining the basic characteristics of the original signal. The Butterworth filter is designed to have a maximally flat frequency response in the passband, which means that it has no fluctuations and is very suitable for smoothing signals without distorting the original information.

[0032] Furthermore, the steps of detecting the peak and valley values ​​in S2 are as follows:

[0033] For a distance signal D, the peaks and valleys can be identified using the following conditions:

[0034] Peak value: If the following conditions are met at time t, it is considered a peak value:

[0035] D(t i-1 )<D(t i )>D(t i+1 )

[0036] Valley value: If the following conditions are met at time t, it is considered a valley value:

[0037] D(t j-1 )>D(t j )<D(t j+1 )

[0038] Through the above technical solution, by applying the filtering process, the original noisy distance data is smoothed, generating a cleaner signal that more accurately reflects the true underlying distance measurement value.

[0039] Furthermore, the steps of determining the peak-valley midpoint in S3 are as follows:

[0040] Once the peaks and valleys are identified, the midpoint between them is calculated; for p The peak at time t and the subsequent q The valley value at the midpoint t m The calculation formula is:

[0041]

[0042] Similarly, for the q The valley value at time t and the subsequent p The peak value at the midpoint t m The calculation formula is:

[0043]

[0044] Midpoint m The distance value D(t m ) usually interpolate between corresponding peaks and valleys;

[0045] If t m It coincides with a certain time point in the original data:

[0046] D(t m )=D(t m )

[0047] If t m If there is no exact match with any time point, the midpoint distance is interpolated using the following formula:

[0048]

[0049] in:

[0050] t m- is the number of bytes immediately following t in the original data sequence. m Previous time point;

[0051] t m+ is the number of the original data sequence immediately adjacent to t m At a later point in time;

[0052] Through the above technical solution, the method converts the asymmetric distance fluctuations in the original signal into symmetric fluctuations.

[0053] Furthermore, the S4. integrated empirical mode decomposition (EEMD) filtering steps are as follows:

[0054] For a given signal x(t), the EMD process can be summarized as follows:

[0055]

[0056] in:

[0057] IMF i (t) is the i-th eigenmode function;

[0058] r n (t) is the remainder after extracting n IMFs, indicating a non-oscillatory trend;

[0059] In EEMD (Ensemble Empirical Mode Decomposition), white noise is added to the original signal x(t) to generate multiple noisy versions of the signal:

[0060] x k (t) = x(t) + ω k (t)

[0061] in:

[0062] ω k (t) is the kth white noise realization added to the original signal

[0063] Each noisy signal x k (t) After EMD decomposition, a set of corresponding intrinsic mode functions are generated

[0064]

[0065] in:

[0066] IMF i,k (t) is the kth IMF extracted from the i-th noisy signal;

[0067] r n,k (t) is the remaining part of the k-th noisy signal;

[0068] The final IMFs are obtained by averaging the corresponding IMFs of all sets (noisy realizations):

[0069]

[0070] in:

[0071] M is the total number of sets (number of noisy realizations);

[0072] IMF i (t) is the average value of the ith IMF in all sets;

[0073] Signal reconstruction:

[0074]

[0075] in:

[0076] p and q represent the indices of the IMFs selected for reconstruction, and usually IMFs that contain the main modes and exclude noise are selected;

[0077] The filtered signal The original signal is effectively preserved x ( t ) while reducing noise and unnecessary fluctuations;

[0078] Through the above technical solution, the EMD method is used to decompose the signal into multiple intrinsic mode functions (IMFs) and reconstruct the signal. This step further removes random noise in the data and extracts the main signal components.

[0079] (III) Beneficial effects

[0080] 1. The present invention provides a long-distance surface target ranging filtering method based on monocular vision. It has the following beneficial effects:

[0081] 2. The present invention provides a long-distance surface target ranging filtering method based on monocular vision. Through Butterworth high-frequency filtering, peak and valley detection, peak and valley midpoint determination, and integrated empirical mode decomposition (EEMD) filtering, the system can generate stable distance data, thereby greatly improving the accuracy and stability of long-distance monocular ranging. This innovative method not only has important theoretical significance, but also demonstrates practical application value in the field of ship automatic navigation, so that the method significantly improves the ranging accuracy in a dynamic ocean environment, while eliminating the dependence on the horizon, and better meets the distance measurement needs of autonomous ships in collision avoidance.

[0082] 3. The present invention provides a long-distance surface target ranging filtering method based on monocular vision. The ranging process is divided into three main stages: data acquisition and preprocessing, ranging unit, and peak-valley midpoint filtering process. In the initial stage of data acquisition and preprocessing, the system captures raw video data from the camera and performs camera calibration to obtain key parameters such as focal length and optical axis offset. These parameters are crucial to ensure the accuracy of subsequent distance measurements. In addition, the system also collects data from the inertial measurement unit (IMU), including roll, yaw and pitch angles. These IMU data are preprocessed to filter out noise to obtain smooth attitude angle data, which provides basic input for subsequent ranging calculations, thereby improving the accuracy of the ranging and filtering methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] Figure 1 A schematic diagram of monocular ranging for ocean target detection according to the present invention;

[0084] Figure 2 It is a schematic diagram of the two-dimensional projection principle of target detection of the present invention;

[0085] Figure 3 A schematic diagram of asymmetric distance fluctuations observed when performing monocular distance measurement using the principle of similar triangles according to the present invention;

[0086] Figure 4 It is a schematic diagram of the influence of high-frequency noise on the distance measurement curve of the present invention;

[0087] Figure 5 It is the asymmetric fluctuation diagram of the original ranging data of the present invention;

[0088] Figure 6 It is a schematic diagram of the high-frequency noise suppression effect after Butterworth filtering and the preliminary selection of the ranging balance point (blue dots) of the present invention;

[0089] Figure 7 It is a schematic diagram of preliminary data optimization after symmetry of peak-valley median filtering of the present invention;

[0090] Figure 8 It is a schematic diagram of the final distance curve adjustment after EEMD smoothing processing of the present invention;

[0091] Fig. 9 The figure is a flow chart of the monocular distance measurement and filtering process of the present invention. DETAILED DESCRIPTION

[0092] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0093] Embodiment 1:

[0094] The long-distance surface target ranging filtering method based on monocular vision includes the following steps:

[0095] S1. Buttreworth high frequency filtering;

[0096] S2. Detection of peak and valley values;

[0097] S3. Determination of the midpoint of the peak and valley;

[0098] S4. Integrated empirical mode decomposition (EEMD) filtering.

[0099] S1. The steps of Buttreworth high frequency filtering are as follows:

[0100] The filtering process starts with designing a Butterworth filter that has the following transfer function characteristics;

[0101]

[0102] in:

[0103] s is a complex frequency variable;

[0104] ω c is the cut-off angular frequency, defined as, where f c is the cutoff frequency in Hertz;

[0105] n is the order of the filter, which is used to control the attenuation steepness of the filter near the cutoff frequency;

[0106] The Butterworth filter is applied in the digital domain to convert the transfer function of the analog filter into its digital equivalent form through a bilinear transformation; the coefficients a and b of the digital filter are calculated as follows:

[0107] b=[b0,b1,...b n ],a=[a0,a1,...a n ]

[0108] The filtered signal y(t) is calculated by the following difference equation:

[0109]

[0110] in:

[0111] x(t) is the input signal (in this case the raw distance data);

[0112] y(t) is the output signal (filtered distance data);

[0113] b i and a j are the filter coefficients derived from the Butterworth filter design;

[0114] The Butterworth filter is used to smooth the raw distance data by attenuating high frequency noise while retaining the essential features of the original signal. The Butterworth filter is designed to have a maximally flat frequency response in the passband, which means it has no fluctuations and is ideal for smoothing signals without distorting the original information.

[0115] S2. The steps for detecting peak and valley values ​​are as follows:

[0116] For a distance signal D, the peaks and valleys can be identified using the following conditions:

[0117] Peak value: If the following conditions are met at time t, it is considered a peak value:

[0118] D(t i-1 )<D(t i )>D(t i+1 )

[0119] Valley value: If the following conditions are met at time ttt, it is considered a valley value:

[0120] D(t j-1 )>D(t j )<D(t j+1 )

[0121] By applying this filtering process, the raw noisy distance data is smoothed, producing a cleaner signal that more accurately reflects the true underlying distance measurement.

[0122] S3. The steps for determining the midpoint of the peak and valley are as follows:

[0123] Once the peaks and valleys are identified, the midpoint between them is calculated; for p The peak at time t and the subsequent q The valley value at the midpoint t m The calculation formula is:

[0124]

[0125] Similarly, for the q The valley value at time t and the subsequent p The peak value at the midpoint t m The calculation formula is:

[0126]

[0127] Midpoint m The distance value D(t m ) usually interpolate between corresponding peaks and valleys;

[0128] If t m It coincides with a certain time point in the original data:

[0129] D(t m )=D(t m )

[0130] If t m If there is no exact match with any time point, the midpoint distance is interpolated using the following formula:

[0131]

[0132] in:

[0133] t m- is the number of bytes immediately following t in the original data sequence.m Previous time point;

[0134] t m+ is the number of the original data sequence immediately adjacent to t m At a later point in time;

[0135] This method transforms the asymmetric distance fluctuations in the original signal into symmetric fluctuations.

[0136] S4. The steps of integrated empirical mode decomposition (EEMD) filtering are as follows:

[0137] For a given signal x(t), the EMD process can be summarized as follows:

[0138]

[0139] in:

[0140] IMF i (t) is the i-th eigenmode function;

[0141] r n (t) is the remainder after extracting the IMF, indicating a non-oscillatory trend;

[0142] Adding white noise to EEMD

[0143] In EEMD (Ensemble Empirical Mode Decomposition), white noise is added to the original signal x(t) to generate multiple noisy versions of the signal:

[0144] x k (t) = x(t) + ω k (t)

[0145] in:

[0146] ω k (t) is the kth white noise realization added to the original signal

[0147] Decomposition of noisy signals

[0148] Each noisy signal x k (t) After EMD decomposition, a set of corresponding intrinsic mode functions are generated

[0149]

[0150] IMF i,k (t) is the kth IMF extracted from the i-th noisy signal;

[0151] r n,k (t) is the remaining part of the k-th noisy signal;

[0152] The final IMFs are obtained by averaging the corresponding IMFs of all sets (noisy realizations):

[0153]

[0154] in:

[0155] M is the total number of sets (number of noisy realizations);

[0156] IMF i (t) is the average value of the ith IMF in all sets;

[0157] Signal reconstruction

[0158]

[0159] in:

[0160] p and q represent the indices of the IMFs selected for reconstruction, and usually IMFs that contain the main modes and exclude noise are selected;

[0161] The filtered signal The main features of the original signal x(t) are effectively retained while reducing noise and unnecessary fluctuations. The EMD method is used to decompose the signal into multiple intrinsic mode functions (IMFs) and reconstruct the signal. This step further removes random noise in the data and extracts the main signal components.

[0162] Embodiment 2:

[0163] refer to Figure 1 , demonstrating the principle of monocular ranging based on similar triangles. This method involves three main coordinate systems: the image coordinate system (UO1V), the camera coordinate system with O2 as the origin, and the world coordinate system (XO3Y).

[0164] In this principle, points in the world coordinate system are projected through the optical axis, and when the direction and height of the camera are fixed, each point in the world coordinate system will correspond one-to-one to a point in the image coordinate system. Specifically, the target point Q is projected to the image point Q1 through the optical axis QO2Q1, forming a corresponding point in the image coordinate system.

[0165] According to the principle of similar triangles, the ground distance O3Q between the target and the camera can be derived and calculated using the known camera height H, angles α and γ, and the pixel coordinates of the target in the image coordinate system (P1,). The internal and external parameters of the camera (including focal length, optical axis position, etc.) combined with the projection of the target in the image coordinate system jointly determine the final distance measurement result. By integrating these parameters and applying the geometric relationship of similar triangles, the distance between the target object and the camera can be calculated, thereby realizing distance measurement based on monocular vision. Figure 2The position of the target projected onto the photosensitive element is shown.

[0166] The following equation shows the detailed derivation process:

[0167]

[0168] β=α-γ

[0169]

[0170] in:

[0171] H Camera height.

[0172] v0 and u0 are the real axis centers of the image coordinates

[0173] ypix Actual height in pixels

[0174] xpix Actual width in pixels

[0175] fFocal length

[0176] refer to Figure 3 , shows a monocular ranging case of 1800 meters. As shown in the figure, the ranging fluctuation shows obvious asymmetry, which causes the average measured distance to deviate from the true value. In addition, this asymmetry tends to intensify with the increase of distance.

[0177] In the marine environment, ships exhibit a six-degree-of-freedom (6-DOF) motion pattern, including lateral, longitudinal, and vertical translational motions, as well as roll, pitch, and yaw motions. This complex motion pattern causes continuous changes in the camera position and orientation, resulting in continuous changes in the projection position of the target object on the camera sensor, resulting in frequent offsets of the projection point during the measurement process.

[0178] In monocular ranging based on the principle of similar triangles, the six-degree-of-freedom motion of the camera will cause asymmetric distance fluctuations. The calculation of the distance D is based on the camera height H and the angle between the camera and the target, that is, the pitch angle. Since the tangent function changes rapidly near small angles and tends to infinity when approaching ±90°, even a small change in the pitch angle will significantly affect the value of tan(θ).

[0179] refer to Figure 4 In the monocular ranging system, in addition to the six-degree-of-freedom motion of unmanned surface vessels (USVs), the vibration of the equipment on board will also cause high-frequency jitter of the camera. This high-frequency vibration directly affects the angle and position of the camera, resulting in fluctuations in the ranging value. Figure 4 The effect of this vibration on the distance measurement curve is demonstrated, showing the lack of smoothness of the measurement curve.

[0180] refer to Figure 5 , the raw distance data showed significant fluctuations due to the six-degree-of-freedom motion of the ship and the high-frequency vibration of the equipment on board. These fluctuations appeared as sharp peaks and valleys, and some measured distances exceeded 20,000 meters, far beyond the actual target distance range (4,110 meters to 4,247 meters). This high error level comes not only from high-frequency noise, but also includes asymmetric fluctuations caused by dynamic motion, which seriously affects the accuracy of the measurement.

[0181] refer to Figure 6 , in order to reduce the high-frequency noise, a Butterworth filter is used to process the measured data. The green curve shows the effect of this filtering. Compared with the original data, the high-frequency noise is significantly attenuated and the signal becomes smoother. However, due to the continuous asymmetry caused by dynamic motion, although the signal is improved after filtering, it still deviates greatly from the true distance value. After high-frequency filtering, the peaks and valleys in the signal are identified to characterize the fluctuation pattern. The peaks are marked with red crosses and the valleys are marked with green crosses. The midpoint between each peak and valley is then calculated ( Figure 6 The blue dots in the figure effectively achieve the symmetry of the signal and reduce the asymmetric fluctuations caused by the ship's motion.

[0182] refer to Figure 7 , the preliminary midpoint is further refined by removing outliers (shown as purple dots). This step removes extreme deviations caused by transient perturbations and provides a more stable signal. The adjusted midpoint is more consistent with the true distance, proving the effectiveness of the symmetrization process of this method in reducing asymmetric distortion.

[0183] refer to Figure 8 ,In order to further improve the stability and accuracy, EEMD processing is performed on the symmetric ,signal. Figure 8 The final output (red line) is shown, with the smoothed midpoint being very close to the true distance. This final smoothing step ensures that the processed signal is robust to noise and dynamic fluctuations, and has a high degree of fit to the actual target distance.

[0184] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A long-distance surface target ranging filtering method based on monocular vision, characterized by: The following steps are involved: S1. Buttreworth high frequency filtering; S2. Detection of peak and valley values; S3. Determination of the midpoint of the peak and valley; S4. Integrated empirical mode decomposition (EEMD) filtering.

2. The long-distance surface target ranging filtering method based on monocular vision according to claim 1 is characterized in that: The S1. Buttreworth high frequency filtering steps are as follows: The filtering process starts with designing a Butterworth filter that has the following transfer function characteristics; in: s is a complex frequency variable; ω c is the cut-off angular frequency, defined as, where f c is the cutoff frequency in Hertz; n is the order of the filter, which is used to control the attenuation steepness of the filter near the cutoff frequency; The Butterworth filter is applied in the digital domain. The transfer function of the analog filter is converted to its digital equivalent form through a bilinear transformation. The coefficients a and b of the digital filter are calculated as follows: b=[b0,b1,...b n ],a=[a0,a1,...a n ] The filtered signal y(t) is calculated by the following difference equation: in: x(t) is the input signal (in this case the raw distance data); y(t) is the output signal (filtered distance data); b i and a j are the filter coefficients derived from the Butterworth filter design.

3. The long-distance surface target ranging filtering method based on monocular vision according to claim 1 is characterized in that: The steps of detecting the peak and valley values ​​in S2 are as follows: For a distance signal D, the peaks and valleys can be identified using the following conditions: Peak value: If the following conditions are met at time t, it is considered a peak value: D(t i-1 )<D(t i )>D(t i+1 ) Valley value: If the following conditions are met at time t, it is considered a valley value: D(t j-1 )>D(t j )<D(t j+1 )。 4. The long-distance surface target ranging filtering method based on monocular vision according to claim 1 is characterized in that: The steps of determining the peak-to-valley midpoint in S3 are as follows: Once the peaks and valleys are identified, the midpoint between them is calculated, for the p The peak at time t and the subsequent q The valley value at the midpoint t m The calculation formula is: Similarly, for the q The valley value at time t and the subsequent p The peak value at the midpoint t m The calculation formula is: Midpoint m The distance value D(t m ) usually interpolate between corresponding peaks and valleys; If t m It coincides with a certain time point in the original data: D(t m )=D(t m ) If t m If there is no exact match with any time point, the midpoint distance is interpolated using the following formula: in: t m- is the number of bytes immediately following t in the original data sequence. m Previous time point; t m+ is the number of the original data sequence immediately adjacent to t m At a later time point.

5. The long-distance surface target ranging filtering method based on monocular vision according to claim 1 is characterized in that: The S4. integrated empirical mode decomposition (EEMD) filtering steps are as follows: For a given signal x(t), the EMD process can be summarized as follows: in: IMF i (t) is the i-th eigenmode function; r n (t) is the remainder after extracting n IMFs, indicating a non-oscillatory trend; In EEMD (Ensemble Empirical Mode Decomposition), white noise is added to the original signal x(t) to generate multiple noisy versions of the signal: x k (t)=x(t)+ω k (t) in: ω k (t) is the kth white noise realization added to the original signal Each noisy signal x k (t) After EMD decomposition, a set of corresponding intrinsic mode functions are generated in: IMF i,k (t) is the kth IMF extracted from the i-th noisy signal; r n,k (t) is the remaining part of the k-th noisy signal; The final IMF s By calculating the corresponding IMF for all sets (with noise) s Averaging to obtain: in: M is the total number of sets (number of noisy realizations); IMF i (t) is the average value of the ith IMF in all sets; Signal reconstruction in: p and q represent the IMFs selected for reconstruction s The index is usually selected to include the main mode and exclude the noise IMF s ; The filtered signal The main features of the original signal x(t) are effectively preserved while reducing noise and unnecessary fluctuations.

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