Image edge detection method using adjustable direction Fourier single pixel imaging

Abstract

The present invention belongs to the field of image processing technology and proposes a method for image edge detection using adjustable directional Fourier single-pixel imaging, comprising: S1, constructing an adjustable directional filter; S2, using the adjustable directional filter as an adjustable directional operator to modulate Fourier speckle to obtain edge modulated speckle; S3, selecting a low-frequency sampling region radius and filtering the image spatial domain using the edge modulated speckle corresponding to the low-frequency region to obtain a corresponding fan-shaped Fourier spectrum; S4, performing adaptive sampling; S5, using the adaptive sampling method in step S4, obtaining higher-frequency Fourier spectrum coefficients based on the Fourier spectrum coefficients of the target low-frequency semicircular region; and S6, obtaining image edge information through inverse Fourier transform. The present invention can achieve fast and high-quality edge extraction of target edges in different directions at low sampling rates, thereby improving the quality of edge detection.

Description

Technical Field

[0001] The present invention belongs to the technical field of image processing, and in particular relates to a method for image edge detection using adjustable direction Fourier single-pixel imaging. Background Art

[0002] Unlike traditional imaging methods that record target light intensity information using an array detector, single-pixel imaging technology relies on light intensity fluctuations, leveraging the second-order correlation properties of the light field. This information is recorded using a single-pixel detector with no spatial resolution. This allows spatial information to be encoded in the temporal dimension, thereby reconstructing the target image. Because single-pixel imaging sacrifices time for space, it significantly reduces the spatial resolution requirements of the detector compared to imaging with an array detector. It offers high sensitivity, a wide imaging range, and super-resolution. Therefore, it can produce clearer target images than traditional imaging in extremely harsh environments, making it suitable for low-light conditions, scattered environments (cloud, fog, smoke, and dust interference), and long-range scenes. Furthermore, single-pixel detectors are significantly less expensive and easier to manufacture than array detectors, making them a desirable alternative in special circumstances.

[0003] Based on the unique mechanism of single-pixel detection, the single-pixel detector value containing only the target edge information can be obtained by modulating the speckle pattern. The edge of the image can be directly obtained by single-pixel detection, which can omit the redundant and complex image reconstruction process, enhance the anti-interference ability of edge extraction, and realize "image-free" fast edge feature extraction at low sampling rate.

[0004] Currently, research on single-pixel edge detection methods using image-free methods has achieved some success. However, the development of single-pixel edge detection technology still faces many challenges, such as poor imaging quality, low edge detection accuracy, and excessive measurement cycles. In the field of edge feature extraction using single-pixel detection, the future development trend is to efficiently and accurately obtain edge information of unknown targets at extremely low sampling rates. Only by significantly reducing the computational complexity, increasing the speed of edge feature extraction, and improving the quality of edge extraction can single-pixel edge feature extraction technology become more practical. Summary of the Invention

[0005] In view of the problems that the reconstructed edge results are insensitive to edge responses in different directions and the traditional frequency domain sampling efficiency is low when extracting complex target edges using traditional Fourier single-pixel reconstruction images, the present invention provides a method for image edge detection using adjustable direction Fourier single-pixel imaging, which can complete high-quality and fast target edge feature extraction based on single-pixel detection at a low sampling rate.

[0006] To achieve the above object, the present invention discloses a method for image edge detection using adjustable direction Fourier single pixel imaging, comprising the following steps:

[0007] S1: Construct an adjustable direction filter.

[0008] Adjustable directional filter for:

[0009]

[0010] Where,

[0011]

[0012] S2: Using an adjustable directional filter as an adjustable directional operator, modulating the Fourier speckle pattern to obtain edge modulated speckle;

[0013] S3: Select the radius of the low-frequency sampling area and use the edge modulation speckle corresponding to the low-frequency area to filter the image spatial domain to obtain the corresponding fan-shaped Fourier spectrum;

[0014] S4: Perform adaptive sampling

[0015] The edge modulated speckle obtained in step S2 is used for single pixel detection to obtain the Fourier coefficients in a specific direction area. The average Fourier energy coefficient in each direction is calculated by the following formula:

[0016]

[0017] Where n is the number of Fourier coefficients in the direction area, u and v are Fourier domain coordinates, Ω i is the i-th sector area.

[0018] The average energy of all low-frequency direction regions is calculated by the following formula

[0019]

[0020] Setting the threshold Where c is the set modulation parameter, L is the number of sector areas; M i is the average Fourier energy coefficient in the i-th direction; select the region Ω where the average energy is higher than T i The direction is taken as the effective direction; the Fourier coefficients of the corresponding direction are sampled in each region with higher than average energy;

[0021] S5: Obtaining higher-frequency Fourier spectrum coefficients according to the obtained Fourier spectrum coefficients of the target low-frequency semicircular area through the adaptive sampling method of step S4;

[0022] S6: Obtain the edge information of the image through inverse Fourier transform.

[0023] Preferably, the step S2 is specifically as follows:

[0024] First, a set of Fourier speckles with the same size as the image is obtained, and each Fourier speckle has its own direction in the Fourier domain;

[0025] Secondly, the semicircular area is divided into m equal areas, and the symmetry axis direction of each area is defined as the angle θ of the area direction. i , there are m directions and angles in m regions;

[0026] Next, the Fourier speckles are grouped into corresponding areas according to the determined regions;

[0027] Then, according to the angle θ of the region i Use formula (7) to get the angle θ i Adjustable directional filter

[0028] Finally, the adjustable directional filter As the edge operator h, for the angle θ i The Fourier speckle pattern in the region is modulated to obtain the edge modulated speckle in the region. When the Fourier speckle modulation in all regions is completed, the edge modulated speckle is obtained.

[0029] Preferably, the semicircular area is equally divided into 8 areas, the angle in each area is 22.5°, the symmetry axis of the first area is set to 0°, and the direction of the adjustable direction filter is adjusted at intervals of 22.5°.

[0030] Preferably, the step S5 further comprises applying Butterworth filtering to the edges of each sampling area to reduce the influence of ringing artifacts.

[0031] Preferably, f(x,y) is expressed as the impulse response of the filter in the spatial coordinate system, which is called the filter f(x,y). Then the steerable filter is expressed as:

[0032] In the spatial coordinate system, f(x,y) is expressed as the impulse response of the filter, called the filter f(x,y), then the steerable filter can be expressed as:

[0033]

[0034] Among them, f θ (x,y) is the directional tunable filter after f(x,y) is rotated by θ, is the base filter, θ d is the angle of the dth basis angle, M is the number of basis filters, k d(θ) is the d-th interpolation function, x is the abscissa in the spatial coordinate system, y is the ordinate in the spatial coordinate system, and the filter in any direction can be linearly represented by a finite number of basis filters;

[0035] The steerable filter f θ (x,y) is transformed from the spatial coordinate system to the polar coordinate system, where Where r is the radius in polar coordinates, is the angle in the polar coordinate system; according to Euler's formula Then the steerable filter f θ The Fourier series expression of (x,y) is:

[0036]

[0037] Where j is the imaginary unit; n is the number of terms in the Fourier series, starting from -N; the direction-adjustable filter f θ The number of terms in the (x,y) Fourier series is 2N+1, where N is a positive integer; n (r) is the non-zero coefficient of the Fourier series.

[0038] Compared with the prior art, the present invention has the following beneficial effects:

[0039] (1) The present invention proposes an adjustable-direction filter. Building on existing methods for extracting edge features from non-imaged targets based on single-pixel detection, the adjustable-direction filter is used as an edge detection operator. This solves the problem that the Sobel gradient operator is insensitive to multi-directional edge responses when reconstructing images under undersampling, and that high-frequency information is significantly attenuated, resulting in weakened edges and difficulty in obtaining complete and accurate edges. Compared to existing image edge detection methods, the present invention can accurately extract edge features in different directions and, at the same sampling rate, can obtain a more complete and accurate target edge image than the traditional Sobel gradient operator.

[0040] (2) The present invention also proposes a method in which the direction of the edge detection operator corresponds to the direction of the Fourier speckle partition, and uses an efficient adaptive sampling scheme to collect important high-frequency information within a limited number of sampling times to adapt to the uncertainty of the edge direction of natural targets.

[0041] (3) The present invention adopts an adjustable directional filter to modulate Fourier speckle and an adaptive sampling scheme. Based on single-pixel detection technology, it can obtain more precise multi-directional target edges at the same sampling rate, and at the same time solve the problem of a sudden increase in projected speckle as the size of the gradient operator increases in Fourier single-pixel detection. It is simple, effective and easy to implement.

[0042] (4) The present invention optimizes the boundary of the Fourier high-frequency region and uses a Butterworth filter to process the Fourier high-frequency region, which can suppress the ringing artifact phenomenon and improve the quality of edge feature extraction. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 This is a flow chart of a method for image edge detection using adjustable direction Fourier single-pixel imaging according to the present invention;

[0044] Figure 2 A block diagram of the directional adjustable filter system of the present invention;

[0045] Figure 3a is an initial image according to an embodiment of the present invention;

[0046] Figure 3b For the embodiment of the present invention, Image after edge feature extraction by basis filter;

[0047] Figure 3c For the embodiment of the present invention, Image after edge feature extraction by basis filter;

[0048] Figure 3d For the embodiment of the present invention, Image after edge feature extraction by basis filter;

[0049] Figure 4 Schematic diagram of the edge response effect of the filter in different directions of the present invention;

[0050] Figure 5 Schematic diagram of Fourier speckle modulation by the adjustable directional filter of the present invention;

[0051] Figure 6 Schematic diagram of adaptive sampling of the present invention;

[0052] Figure 7 This is a flow chart of edge detection for "cherries" according to the present invention;

[0053] Figure 8 The following are the simulation results of "cherry" and "banana" of the present invention;

[0054] Figure 9 This is the simulation result diagram of the "pepper" of the present invention. DETAILED DESCRIPTION

[0055] The exemplary embodiments, features, and aspects of the present invention will be described in detail below with reference to the accompanying drawings. The same reference numerals in the accompanying drawings represent elements with the same or similar functions. Although various aspects of the embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless otherwise indicated.

[0056] The present invention proposes a method for image edge detection using adjustable direction Fourier single pixel imaging, such as Figure 1 As shown, it includes the following steps:

[0057] S1: Construct an adjustable direction filter.

[0058] If the function f(x,y) can be expressed as a linear combination of functions rotated by themselves, then the function is said to be directionally adjustable. The basic idea of ​​a directionally adjustable filter is to use a linear combination of a set of "base filters" with different directions (each base filter can be expressed as a form of a directionally adjustable filter rotated to a certain angle) to realize a filter response in any direction. This response can be regarded as a function of the direction angle, and the output of the filter is determined by controlling the value of the direction angle.

[0059] In the spatial coordinate system, f(x,y) is expressed as the impulse response of the filter, called the filter f(x,y), then the steerable filter can be expressed as:

[0060]

[0061] Among them, f θ (x,y) is the directional tunable filter after f(x,y) is rotated by θ, is the base filter, θ d is the angle of the dth basis angle, M is the number of basis filters, k d (θ) is the d-th interpolation function, x is the horizontal coordinate in the spatial coordinate system, and y is the vertical coordinate in the spatial coordinate system. From formula (1), we can see that the filter in any direction can be linearly represented by a finite number of basis filters, such as Figure 2 shown.

[0062] For convenience, the steerable filter f θ (x,y) is transformed from the spatial coordinate system to the polar coordinate system, where Where r is the radius in polar coordinates, is the angle in the polar coordinate system. According to Euler's formula Then the steerable filter f θ The Fourier series expression of (x,y) is:

[0063]

[0064] Where j is the imaginary unit; n is the number of terms in the Fourier series, starting from -N; the direction-adjustable filter f θ The number of terms in the (x,y) Fourier series is 2N+1, where N is a positive integer; n (r) is the non-zero coefficient of the Fourier series,

[0065] Formula (2) satisfies the following three theories:

[0066] ① If and only if the interpolation function k d (θ) satisfies

[0067]

[0068] Then formula (1) can be expressed as formula (2).

[0069] ② Let T be the function in formula (2) The non-zero coefficient a n (r), the minimum number of basis filters in formula (1) is T, that is, M ≥ T.

[0070] ③ Let f(x,y)=W(r)P N (x,y), where W(r) is an arbitrary window function. If P N (x, y) is an N-degree polynomial about x and y, then the linear combination of 2N+1 basis functions can make f(x, y) = W(r)P N (x,y) rotates to any angle; if P N (x,y) contains only even (odd) terms (for terms x n y m , n+m is an even number (odd number), then only N+1 basis functions are needed to rotate f(x,y) to any angle.

[0071] Based on the above theory, the following adjustable direction filter is constructed:

[0072] The two-dimensional Gaussian function is defined as:

[0073]

[0074] Where x is the horizontal coordinate in the spatial coordinate system, y is the vertical coordinate in the spatial coordinate system, and σ is the standard deviation.

[0075] Its inverse second-order partial derivative is:

[0076]

[0077] make but This is the base filter constructed in this application, θ d is the angle of the basis filter.

[0078] is a quadratic polynomial, that is, N = 2; according to theories ② and ③, we know that 3 basis filters are needed. From formula (3), for the interpolation function k d (θ) has:

[0079]

[0080] Assume that when d = 1, 2 or 3, θ1 = 0°, θ2 = 45°, θ3 = 90°, and the solution is: k1(θ) = cos 2 (θ), k2(θ)=2cos(θ)sin(θ), k3(θ)=sin 2 (θ). From formula (1), we can see that and Adjustable direction filter based on a basis filter for:

[0081]

[0082] Where,

[0083]

[0084] The schematic diagram of using the three base filters in different directions in this embodiment to extract edge features of an image is shown in FIG. Figure 3a-3d As shown, Figure 3a is the initial image, Figure 3b To pass The image after edge feature extraction by base filter, Figure 3c To pass The image after edge feature extraction by base filter, Figure 3d To pass Image after edge feature extraction using the basis filter.

[0085] When extracting edge features from an image I(x,y), an edge point in the image can be defined as The local maximum response after applying to the image I(x,y) is By convolution, we can get the impulse response of the filter containing image information:

[0086]

[0087] Where * represents the convolution operation.

[0088] Then calculate the zero point of the second-order directional differential, that is, take the derivative of f(x,y) with respect to θ and set it to zero:

[0089]

[0090] Then there is

[0091]

[0092] By using formula (11), the value of θ relative to the local maximum response and the local minimum response can be obtained:

[0093]

[0094] In the formula

[0095]

[0096] The θ value corresponding to the local maximum response in the image is the global maximum response. This θ value can be obtained by substituting all θ values ​​obtained by formula (11) into formula (9) for comparison. The edge response result is as follows: Figure 4 shown.

[0097] S2: Use the adjustable direction operator to modulate the Fourier speckle to obtain edge modulated speckle

[0098] The edge-modulated speckle is obtained by modulating the Fourier speckle pattern using an adjustable direction operator. The specific steps are as follows:

[0099] First, a set of Fourier speckle patterns with the same size as the image is obtained. For example, if the image to be processed has an image resolution of 256*256, a set of 256*256 Fourier speckle patterns is obtained. Each Fourier speckle has its own direction in the Fourier domain.

[0100] Secondly, the semicircular area is divided into m equal areas, and the direction of the symmetry axis of the i-th area is defined as the angle θ of the direction of the area. i , m regions have a total of m directions and angles, i.e. i = 1, 2, ..., m. Since Fourier speckle has a symmetrical structure, it is sufficient to divide only the semicircular region, and the other half can be obtained by conjugation;

[0101] Next, the Fourier speckle patterns are grouped into corresponding regions according to the determined regions. Then, according to the angle θ of the i-th region i Use formula (7) to get the angle θ i Adjustable directional filter

[0102] Finally, the adjustable directional filter The Fourier speckle pattern in this area is modulated using the edge operator h to obtain the edge modulated speckle in this area. When the Fourier speckle modulation in all areas is completed, the edge modulated speckle is obtained.

[0103] In this embodiment, the Fourier speckle pattern is divided into 8 directions, so the adjustable direction filter adjusts the filter direction at 22.5° intervals, as shown in the schematic diagram. Figure 5 shown. Figure 5The process of generating edge-modulated speckles in eight directions is demonstrated: first, a set of Fourier speckles is generated; second, it is equally divided into eight regions, with a specified angle of 22.5° in each region. Next, the Fourier speckles are assigned to the corresponding regions; then, the axis of symmetry of the first region is set to 0°, and the filter direction of the adjustable directional filter is adjusted at intervals of 22.5°. Finally, the obtained adjustable directional filter is used as the edge operator h to modulate the Fourier speckle in the corresponding region; the final edge-modulated speckle can be obtained.

[0104] By using edge-modulated speckle, the edge of the unknown image I can be directly obtained, thus eliminating the step of reconstructing the image. This is because of the following reasons:

[0105] If the adjustable direction filter obtained in step S1 is As the edge operator h, the image I is spatially filtered, and the edge image I e It can be expressed as:

[0106] I e =h*I (14)

[0107] Single-pixel imaging can image any object, including edge images. e (x,y), the single pixel detection process of the edge image can be expressed as:

[0108] D e,φ (f x ,f y )=∫∫ Ω' I e (x,y)P φ (x,y;f x ,f y )dxdy (15)

[0109] Where, P φ (x,y;f x ,f y ) is the Fourier speckle pattern before modulation; D e,φ (f x ,f y ) is the single pixel detection value.

[0110] Combined with formula (14), the single-pixel detection process of edge image can be further expressed as:

[0111]

[0112] Where, P e,φ (f x ,f y ) is the Fourier speckle pattern modulated by the edge operator h, I e (x,y) is the edge image.

[0113] According to formula (16), it can be seen that by using edge modulated speckle, the edge of the unknown image I can be directly obtained.

[0114] S3: Select an appropriate low-frequency sampling area radius and use the edge modulation speckle corresponding to the low-frequency area to filter the image spatial domain to obtain the corresponding fan-shaped Fourier spectrum

[0115] S4: Perform adaptive sampling

[0116] Due to radial correlation in the Fourier domain, if the low-frequency coefficients in certain directions have higher energy, the high-frequency coefficients in the same directions will also have higher energy. Based on this correlation, we can estimate the location of important high-frequency components by selecting low-frequency components, adaptively sample the key Fourier coefficients, and form an image.

[0117] like Figure 6 As shown, the edge modulated speckle obtained in step S2 is used for single pixel detection to obtain the Fourier coefficients in a specific direction area. The average Fourier energy coefficient in each direction is calculated by the following formula:

[0118]

[0119] Where n is the number of Fourier coefficients in the direction area, u and v are Fourier domain coordinates, Ω i is the fan-shaped area in the i-th direction.

[0120] The average energy of all low-frequency direction regions is calculated by the following formula

[0121]

[0122] Setting the threshold Where c is the set modulation parameter, which is 0.1 in this embodiment. L is the total number of sector areas; M i is the average Fourier energy coefficient in the i-th direction; select the region Ω where the average energy is higher than T i The direction is taken as the effective direction; the Fourier coefficients of the corresponding direction are sampled in each region with higher than average energy;

[0123] S5: Using the adaptive sampling method of step S4, higher-frequency Fourier spectrum coefficients are obtained based on the Fourier spectrum coefficients of the target low-frequency semicircular area, and Butterworth filtering is applied to the edges of each sampling area to reduce the impact of ringing artifacts.

[0124] S6: Obtain the edge information of the image through inverse Fourier transform.

[0125] The following will use experimental data,

[0126] The grayscale images with a resolution of 256×256, "banana" and "cherry", were tested at 5% and 10% sampling rates. Figure 7 This is a flow chart of edge detection for "cherries". This method is compared with the traditional Fourier single-pixel edge detection algorithm and the single-pixel edge detection algorithm based on frequency domain filtering. The peak signal-to-noise ratio (SNR) is used as the evaluation index. The results are shown in Figure 2. Figure 8 shown.

[0127] It can be seen that at sampling rates of 5% and 10%, the proposed method achieves higher SNR than the other two methods. This indicates that at the same sampling rate, the proposed method has stronger edge response and reveals more edge details than the other two methods. Furthermore, the proposed method suffers less from ringing artifacts and has lower background noise. Benefiting from the arbitrary-direction Sobel operator, the proposed method is able to preserve the most edges in all directions at low sampling rates.

[0128] Furthermore, the same test was conducted on the 256×256 color image "Pepper" at 5%, 10%, and 25% sampling rates. The results are as follows: Figure 9 shown.

[0129] It can be seen that the quality of all edge images deteriorates as the sampling rate decreases. However, the SNR parameter shows that when the number of sampling times is the same, the SNR of the proposed method is higher than that of the other two methods. At low sampling rates, the proposed method shows the most accurate and most edge details, and is least affected by ringing artifacts.

[0130] The embodiments described above are merely descriptions of preferred implementations of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should fall within the scope of protection determined by the claims of the present invention.

Claims

1. A method for image edge detection using adjustable direction Fourier single pixel imaging, characterized in that: The method comprises the following steps: S1: Construct an adjustable direction filter; Adjustable directional filter for: （7）； Where, θ Angle indicating direction; （8）； Among them, x is the horizontal coordinate in the spatial coordinate system, y is the vertical coordinate in the spatial coordinate system, is the standard deviation; S2: Using an adjustable directional filter as an adjustable directional operator, the Fourier speckle pattern is modulated to obtain edge modulated speckle. The step S2 is specifically as follows: First, a set of Fourier speckles with the same size as the image is obtained, and each Fourier speckle has its own direction in the Fourier domain; Secondly, the semicircular area is divided into m equal areas, and the direction of the symmetry axis of each area is defined as the angle of the direction of the area. ; Next, the Fourier speckles are grouped into corresponding areas according to their directions; Then, according to the angle of the area Use formula (7) to get the angle Adjustable directional filter ; Finally, the adjustable directional filter As the edge operator h, the angle The Fourier speckle in the area is modulated to obtain the edge modulated speckle in the area; when the Fourier speckle modulation in all areas is completed, the edge modulated speckle is obtained; S3: Select the radius of the low-frequency sampling area and use the edge modulation speckle corresponding to the low-frequency sampling area to filter the image spatial domain to obtain the corresponding fan-shaped Fourier spectrum; S4: Perform adaptive sampling The edge modulated speckle obtained in step S2 is used for single pixel detection to obtain the Fourier coefficients in a specific direction area. ; Then calculate the average Fourier energy coefficient in each direction by the following formula: （17）； Where n is the number of Fourier coefficients in the direction area, and is the Fourier domain coordinate, is the fan-shaped area in the i-th direction; The average energy of all low-frequency direction regions is calculated by the following formula : （18）； Setting the threshold , where c is the set modulation parameter and L is the number of sector areas; is the average Fourier energy coefficient in the i-th direction; select the area where the average energy is higher than T The direction is taken as the effective direction; the Fourier coefficients of the corresponding direction are sampled in each region with higher than average energy; S5: Obtaining higher-frequency Fourier spectrum coefficients according to the obtained Fourier spectrum coefficients of the target low-frequency semicircular area through the adaptive sampling method of step S4; S6: Obtain the edge information of the image through inverse Fourier transform.

2. The method for image edge detection using directional Fourier single-pixel imaging according to claim 1, characterized in that: The semicircular area is equally divided into 8 areas, each with an angle of 22.5°. The symmetry axis of the first area is set to 0°, and the direction of the adjustable directional filter is adjusted at intervals of 22.5°.

3. The method for image edge detection using adjustable direction Fourier single pixel imaging according to claim 1, characterized in that The step S5 further includes applying Butterworth filtering to the edges of each sampling area to reduce the influence of ringing artifacts.

4. The method for image edge detection using directional Fourier single-pixel imaging according to claim 1, characterized in that: The adjustable direction filter in step S1 is defined as follows: In the spatial coordinate system, f(x,y) is expressed as the impulse response of the filter, called the filter f(x,y), then the steerable filter is expressed as: （1）； in, It is a directional filter after f(x,y) is rotated by θ. is the base filter, is the angle of the dth basis angle, M is the number of basis filters, is the d-th interpolation function, x is the abscissa in the spatial coordinate system, y is the ordinate in the spatial coordinate system, and the filter in any direction can be linearly represented by a finite number of basis filters; Directionally adjustable filter Convert from spatial coordinate system to polar coordinate system, where ,in is the radius in polar coordinates, is the angle in the polar coordinate system; according to Euler's formula , then the direction-adjustable filter The Fourier series expression is: （2）； Where j is the imaginary unit; n is the number of terms in the Fourier series, starting from -N; direction-adjustable filter The number of terms in the Fourier series is 2N+1, where N is a positive integer; are the non-zero coefficients of the Fourier series.

Images