TWI851470B - System, method and computer readable medium for restoring binary image - Google Patents

Abstract

The invention discloses a system, method and computer readable medium for restoring binary image. A transmitting device performs a two-dimensional discrete Fourier transform on a binary image to obtain a spectrum including two-dimensional discrete Fourier transform coefficients of N*N points, and then samples the spectrum including the two-dimensional discrete Fourier transform coefficients of the N*N points to obtain a sampled spectrum including the two-dimensional discrete Fourier transform coefficients of N+2 points. Next, the transmitting device transmits the sampling spectrum including the two-dimensional discrete Fourier transform coefficients of the N+2 points to a receiving device with a restoration module, so that the restoration module can quickly restores the binary image input by the transmission device based on the sampling spectrum including the two-dimensional discrete Fourier transform coefficients of the N+2 points.

Description

A system, method and computer-readable medium for restoring binary images

The present invention relates to a binary image restoration technology, and more particularly to a system, method and computer-readable medium for restoring a binary image from two-dimensional discrete Fourier transform coefficients.

At present, when obtaining binary images (such as black and white images), often due to equipment (such as mobile phones) or operation (such as hand shaking), only some blurred images can be obtained. From the perspective of the spectrum of binary images, this phenomenon means that only a part of the spectrum of binary images can be obtained instead of the whole or complete picture.

For example, in a binary image of a computer tomography scan, the Fourier transform coefficient of a projection in a certain direction can be directly obtained through measurement, and then the binary image of the original image can be restored through projection in different directions.

For example, when a user scans a QR code with a mobile phone, the QR code obtained by the user may be a blurred image due to the user's hand shaking, which is equivalent to the Fourier transform coefficient being destroyed.

The current known technology usually uses image enhancement to process the image beforehand, where image enhancement assumes that the image is a grayscale continuous signal and then sharpens the grayscale continuous signal.

However, the conventional technology requires about 90 seconds of restoration time (computation time) to restore a binary image of, for example, 19 by 19 (i.e., 19*19), so the restoration time of the binary image is quite lengthy and costly.

Therefore, how to provide an innovative binary image restoration technology to solve any of the above problems and provide a related system or method has become a major research topic for technical personnel in this field.

The system for restoring a binary image described in the present invention includes: a transmission device, which performs a two-dimensional discrete Fourier transform on an input binary image to obtain a spectrum of two-dimensional discrete Fourier transform coefficients including N*N points, and then the transmission device samples the spectrum of two-dimensional discrete Fourier transform coefficients including N*N points to obtain a sampled spectrum of two-dimensional discrete Fourier transform coefficients including N+2 points, so that the transmission device transmits the sampled spectrum of two-dimensional discrete Fourier transform coefficients including N+2 points, wherein N is a prime number; and a receiving device with a restoration module, which is communicatively connected to the transmission device, so that the receiving device receives the two-dimensional discrete Fourier transform coefficients including N+2 points transmitted by the transmission device. The sampling spectrum of the N+2 points of the two-dimensional discrete Fourier transform coefficients is then restored by the restoration module of the receiving device to restore the binary image input by the transmitting device; wherein the restoration module performs a one-dimensional discrete Fourier inverse transform on the spectrum of the two-dimensional discrete Fourier transform coefficients of the N points in the sampling spectrum to obtain a one-dimensional coefficient of the N points. One-dimensional spectrum, when any one-dimensional coefficient of the N points of the one-dimensional spectrum is non-zero, the restoration module performs a one-dimensional signal restoration operation on the non-zero one-dimensional coefficient to restore a binary image, and when any one-dimensional coefficient of the N points of the one-dimensional spectrum is zero, the restoration module performs a waiting and restoring operation on the zero one-dimensional coefficient to restore a binary image.

The present invention discloses a method for restoring a binary image, comprising: performing a two-dimensional discrete Fourier transform on an input binary image by a transmission device to obtain a spectrum of two-dimensional discrete Fourier transform coefficients including N*N points, and then sampling the spectrum of two-dimensional discrete Fourier transform coefficients including N*N points by the transmission device to obtain a sampled spectrum of two-dimensional discrete Fourier transform coefficients including N+2 points, wherein N is a prime number; and transmitting the sampled spectrum of two-dimensional discrete Fourier transform coefficients including N+2 points to a receiving device having a restoration module by the transmission device, and then the restoration module of the receiving device performs a restoration on the spectrum of two-dimensional discrete Fourier transform coefficients including N+2 points. The sampling spectrum of the two-dimensional discrete Fourier transform coefficient of the sampling spectrum is used to restore the binary image input by the transmission device; wherein the restoration module performs a one-dimensional discrete Fourier inverse transform on the spectrum of the two-dimensional discrete Fourier transform coefficients of N points in the sampling spectrum to obtain a one-dimensional spectrum including one-dimensional coefficients of N points, so that when the one-dimensional coefficient of any one of the N points of the one-dimensional spectrum is non-zero, the restoration module performs a one-dimensional signal restoration operation on the non-zero one-dimensional coefficient to restore the binary image, and when the one-dimensional coefficient of any one of the N points of the one-dimensional spectrum is zero, the restoration module performs a waiting and then restoration operation on the zero one-dimensional coefficient to restore the binary image.

The computer-readable medium of the present invention is applied to a computing device or a computer, and stores instructions to execute the above-mentioned method of restoring a binary image.

Therefore, the present invention provides an innovative system, method and computer-readable medium for restoring binary images. When it is known that the image is a binary image, the transmitting device only needs to select N+2 points or more (approximately or at least 6%) of two-dimensional discrete Fourier transform coefficients from the spectrum of the binary image to transmit to the receiving device, so that the restoration module of the receiving device can quickly restore the binary image input by the transmitting device without any distortion.

In other words, the present invention can restore the binary image quickly when the spectrum of the binary image is severely damaged, by using only two-dimensional discrete Fourier transform coefficients (such as spectrum coefficients or spectrum information) of N+2 points or more (approximately or at least 6%) in the spectrum of the binary image. That is, for a binary image with N*N points, the present invention only needs to extract two-dimensional discrete Fourier transform coefficients of N+2 points or more (approximately or at least 6%) from the spectrum of the binary image, and can restore the binary image quickly and losslessly.

In order to make the above features and advantages of the present invention more clearly understandable, the following examples are given and detailed descriptions are provided in conjunction with the attached drawings. The following description will partially explain the additional features and advantages of the present invention, and these features and advantages will be partially known from the description or can be learned through the practice of the present invention. It should be understood that both the general description above and the detailed description below are exemplary and explanatory, and are not intended to limit the scope of the present invention.

1: System

10: Transmission device

11: Spectrum conversion module

12: Two-dimensional discrete Fourier transform method

13: Sampling module

14: Sampling method

15: Spectrum transmission module

20: Receiving device

21: Spectrum receiving module

22: Restore module

23: Restoration method

24: One-dimensional discrete Fourier inverse transform method

25: One-dimensional signal restoration operation

26: Wait and restore the operation

27: Table lookup method

28: Poor Manners

A: Original signal

B: Two-dimensional discrete Fourier transform coefficient

S11 to S14, S21 to S22, S31 to S36: Steps

X: binary image

Y: spectrum

y: one-dimensional spectrum

y(n): one-dimensional coefficient

Z: sampling spectrum

z: spectrum

Figure 1 is a schematic diagram of the structure of a system for restoring binary images described in the present invention.

Figure 2 is a schematic diagram of the process of a method for restoring a binary image described in the present invention.

FIG3 is a schematic diagram of the implementation process of the restoration module or the restoration method shown in FIG1 and FIG2 in a method for restoring a binary image described in the present invention.

The following describes the implementation of the present invention through a specific concrete implementation form. People familiar with this technology can understand other advantages and effects of the present invention from the content disclosed in this manual, and can also implement or use it through other different specific equivalent implementation forms.

FIG1 is a schematic diagram of the structure of a system 1 for restoring binary images described in the present invention, and FIG2 is a schematic diagram of the process of a method for restoring binary images described in the present invention.

As shown in FIG1 , the binary image restoration system 1 is a system for restoring a binary image from a two-dimensional discrete Fourier transform coefficient, and mainly includes a transmitting device 10 and a receiving device 20 that are communicatively connected to each other. The transmitting device 10 may have a spectrum conversion module 11, a sampling module 13, and a spectrum transmission module 15. The spectrum conversion module 11 may have a two-dimensional discrete Fourier transform method 12, and the sampling module 13 may have a sampling method 14. The receiving device 20 may have a spectrum receiving module 21 and a restoration module 22. The restoration module 22 may have a restoration method (such as a restoration algorithm) 23. The restoration module 22 or the restoration method 23 may also include a one-dimensional discrete Fourier inverse transform method 24, a one-dimensional signal restoration operation 25, a waiting and then restoration operation 26, a table lookup method 27 and/or a poor enumeration method 28, etc.

In one embodiment, the transmission device 10 may be a transmission end or an image transmission device, or an electronic device with a transmission function. The receiving device 20 may be a receiving end, an image receiving device, a restoration end or an image restoration device, or an electronic device with a receiving/restoring function. The electronic device may be a computer, a smart phone, a smart watch, a communicator, a communication device, a server, etc. The computer may be a personal computer, a tablet computer, a laptop computer, a desktop computer, etc., and the server may be a general server, a network server, a cloud server, a remote server, etc.

In one embodiment, the spectrum conversion module 11 of the transmission device 10 can be a spectrum converter (chip/circuit), spectrum conversion software (program), etc., the sampling module 13 can be a sampler (chip/circuit), sampling software (program), etc., the spectrum transmission module 15 can be a spectrum transmitter (chip/circuit), spectrum transmission software (program), etc. The spectrum receiving module 21 of the receiving device 20 can be a spectrum receiver (chip/circuit), spectrum receiving software (program), etc., and the restoration module 22 can be a restorer (chip/circuit), restoration software (program), etc.

In one embodiment, the "binary" described in the present invention can be represented by 1 and 0, and the binary image X can be an image, picture, pattern, picture, photo, code, etc. composed of two colors. For example, the binary image X can be a black and white image, a computer tomography image, a quick response code (QR code), etc. The sampling spectrum Z can be a two-dimensional spectrum, and the one-dimensional spectrum y can also be called an intermediate spectrum or an intermediate vector, etc. The one-dimensional signal restoration operation 15 can be a one-dimensional signal restoration method, a one-dimensional signal restoration function, a one-dimensional signal restoration program, etc., and the waiting and restoring operation can be a waiting and restoring method, a waiting and restoring function, and a waiting and restoring program.

In one embodiment, the "at least one" mentioned in the present invention represents more than one (such as one, two or three), "plurality" represents more than two (such as two, three, four, ten or one hundred), and "communication connection" represents communication or connection through various methods such as data, signal, electrical, wired mode (such as wired network) or wireless mode (such as wireless network). However, the present invention is not limited to those mentioned in each embodiment.

It should be noted that the embodiments of the present invention all refer to the longitudinal direction as "row" and the transverse direction as "column" to avoid confusion. However, in other embodiments, the longitudinal direction may be referred to as "column" and the transverse direction may be referred to as "row". For example, in the embodiments of the present invention, "the first row" represents "the first row in the longitudinal direction", but in other embodiments, the first row may also represent "the first row in the transverse direction". Therefore, the "row" described in the present invention may be one of the longitudinal and transverse directions, and the "column" may be the other of the longitudinal and transverse directions.

The system 1 and method for restoring a binary image described in the present invention is a restoration technology applicable to a binary image X. When it is known that the image is a binary image X (such as a binary image of 1 and 0 or black and white), it is only necessary to select an appropriate sample (called a sampling spectrum Z) from the spectrum Y obtained by the two-dimensional discrete Fourier transform method 12 of the binary image X input by the transmission device 10 to transmit it to the receiving device 20, and the restoration module 22 of the receiving device 20 can restore the binary image X from the sampling spectrum Z (i.e., the sampling result of the spectrum Y) through the restoration method 23 without any distortion.

That is, the system 1 and method for restoring a binary image described in the present invention is a restoration technology applicable to a binary image X, which can quickly restore the binary image X with only N+2 points or more (e.g., at least N+2 points and all non-zero) of two-dimensional discrete Fourier transform coefficients B when the length and width of the binary image X are both the same prime number N (e.g., prime number = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...).

For example, when N=19, the present invention can use 21 points (i.e., at least N+2 points and all non-zero) of two-dimensional discrete Fourier transform coefficients B to quickly restore the binary image X, which is equivalent to using only 21/361=5.8% (approximately 6%) of the two-dimensional discrete Fourier transform coefficients B (such as spectrum coefficients), and can restore the binary image X within 2 seconds.

For a binary image X with N*N points, the present invention only needs to extract N+2 points or more (i.e. 6%) of two-dimensional discrete Fourier transform coefficients B from the spectrum Y of the binary image X, and can quickly and losslessly restore the binary image X. For example, in the restoration time (computation time) of a 19x19 (i.e. 19*19) binary image X, the known technology requires 90 seconds of restoration time, but the present invention only requires less than 2 seconds of restoration time, so the present invention only needs 1/45 of the restoration time of the known technology.

For a binary image X with both length and width being multiples of prime number N (e.g., mN*mN=38*38), where m is a multiple (e.g., m=2) and N is a prime number (e.g., N=19), the present invention can use translation sampling technology to sample (N+2)m ² (e.g., 21*2 ² =84) two-dimensional discrete Fourier transform coefficients B to restore the binary image X, and the sampling rate of the two-dimensional discrete Fourier transform coefficient B of the spectrum Y of the binary image X only needs to be about 6% (i.e., 84/38 ² =5.8%≒6%).

The use scenario of the present invention: For example, on a binary image X of a computer tomography scan, the two-dimensional discrete Fourier transform coefficient B of a projection in a certain direction can be directly obtained by measurement, and then the binary image X of the original image can be restored by projection in different directions. In this regard, the present invention only needs to use two-dimensional discrete Fourier transform coefficients B of N+2 points or more (approximately or at least 6%) to restore the binary image X, so as to reduce the number of measurements of computer tomography scans.

For example, when a user scans a QR code with a mobile phone, the QR code obtained by the user may be a blurred image due to the user's hand shaking, which is equivalent to the Fourier transform coefficient being destroyed. The current known technology usually performs image pre-processing through image enhancement, wherein image enhancement assumes that the image is a grayscale continuous signal and then sharpens the grayscale continuous signal. The difference of the present invention is that it utilizes the characteristics of the binary image X, such as the quick response code (QR code), to reduce the number of two-dimensional discrete Fourier transform coefficients B required by the sampling module 13 (sampling method 14) of the transmission device 10 to sample the binary image X, and even if the binary image X is severely damaged, the restoration module 22 (restoration method 23) of the receiving device 20 can quickly restore the binary image X.

Specifically, as shown in FIG. 1 and FIG. 2 , the system 1 and method for restoring a binary image described in the present invention may include the following technical contents of steps S11 to S14 and steps S21 to S22.

[1] Step S11 of FIG. 2: A binary image X including an original signal A is inputted from the transmission device 10. In this embodiment, a binary image X having a length (e.g., horizontal direction) and a width (e.g., vertical direction) of N=5 points is used, that is, a binary image X including an original signal A of N times N points (e.g., N*N=5*5=25 points). For example, the original signals A of the first to fifth columns of the following binary image X are (1 0 0 1 0), (1 1 1 1 1), (0 1 0 0 0), (1 1 0 0 1), (0 0 0 0 0).

Binary image X including original signal A:

The present invention may also use X(n1, n2) to represent the original signal A of any one of the N points (e.g., 5 points) of the binary image X. For example, n1 is the column position of any one of the N points (e.g., 5 points) of the binary image X, n2 is the row position of any one of the N points (e.g., 5 points) of the binary image X, and n1 and n2 are both values from 0 to N-1 (i.e., 0, 1, 2, 3, ..., N-1). In this embodiment, n1 and n2 are both 0, 1, 2, 3, 4. For example, in the first row (such as the first row in the horizontal direction) of the binary image X, X(0,0)=1, X(0,1)=0, X(0,2)=0, X(0,3)=1, X(0,4)=0; in the second row (such as the second row in the horizontal direction) of the binary image X, X(1,0)=X(1,1)=X(1,2)=X(1,3)=X(1,4)=1, and so on.

[2] Step S12 of FIG. 2: The spectrum conversion module 11 of the transmission device 10 performs a two-dimensional discrete Fourier transform (2DDFT) on the binary image X by using the following two-dimensional discrete Fourier transform method 12 to obtain a spectrum Y of two-dimensional discrete Fourier transform coefficients B including N times N points (e.g., N*N=5*5=25 points).

Two-dimensional discrete Fourier transform method12:

In the above two-dimensional discrete Fourier transform method 12, Y is a spectrum, and N is a prime number. n1 is the column position of any one of the N points (e.g., 5 points) of the binary image X, and n2 is the row position of any one of the N points (e.g., 5 points) of the binary image X. n1, n2, k1, and k2 are all values from 0 to N-1 (i.e., 0, 1, 2, 3, ..., N-1). In this embodiment, n1, n2, k1, and k2 are all 0, 1, 2, 3, 4. _{W N} =e ^2πi/N , e is the natural logarithm, π is the pi, and i is the square root of -1.

For example, the spectrum conversion module 11 can perform a two-dimensional discrete Fourier transform (2DDFT) on the binary image X through the two-dimensional discrete Fourier transform method 12 to obtain the following spectrum Y of the two-dimensional discrete Fourier transform coefficient B including N times N points (such as N*N=5*5=25 points). In this embodiment, the two-dimensional discrete Fourier transform coefficient B may include 11, 0.309-3.5797i, -0.809-4.8410i, -0.809-4.8410i, 0.309+3.5797i, 2.118-0.3633i, -1.9271+2.1266i...etc., a total of 25 points, where i is an imaginary number (i.e., square root of negative 1).

The spectrum Y of the two-dimensional discrete Fourier transform coefficient B consisting of N times N points:

[3] Step S13 of Figure 2: The sampling module 13 of the transmission device 10 uses the sampling method 14 to sample the spectrum Y of the two-dimensional discrete Fourier transform coefficient B including N*N points to obtain a sampling spectrum Z of 0 (zero) including N+2 points (e.g. 5+2=7 points) of two-dimensional discrete Fourier transform coefficient B and N*N-(N+2) points (e.g. 5*5-(5+2)=18 points) (i.e. the sampling result of the spectrum Y).

For example, the sampling method 14 of the sampling module 13 can sample the two-dimensional discrete Fourier transform coefficient B (such as 11) of the first point of the first row of the spectrum Y and the two-dimensional discrete Fourier transform coefficient B (such as 0.309-3.5797i) of one of the second point to the Nth point (such as the second point), and sample the two-dimensional discrete Fourier transform coefficients B (such as 2.118-0.3633i, -1.9271+2.1266i, 1.6910-0.9511i, -0.118+1.5388i, -0.809+0.5878i) of all N points (such as 5 points) of the second row to the Nth row (such as the second row) of the spectrum Y, where i is an imaginary number (i.e., square root of negative 1). That is, the sampling method 14 of the sampling module 13 can sample a total of N+2 points (e.g. 5+2=7 points) of the two-dimensional discrete Fourier transform coefficient B of the spectrum Y.

At the same time, after the sampling method 14 of the sampling module 13 samples the two-dimensional discrete Fourier transform coefficients B of all N points of the first point of the first row and one of the second to Nth points (such as the second point) and one of the second to Nth rows (such as the second row) of the spectrum Y, for the points other than the first point of the first row and one of the second to Nth points (such as the second point) and one of the second to Nth rows (such as the third to the fifth row) of the spectrum Y, it is not necessary to sample the two-dimensional discrete Fourier transform coefficients B and replace them with 0 (zero).

In this embodiment, the sampling result of the sampling method 14 of the sampling module 13 of the transmission device 10 on the spectrum Y can be the following two-dimensional discrete Fourier transform coefficient B including N+2 points (such as 5+2=7 points) and the sampling spectrum Z of 0 (zero) of N*N-(N+2) points (such as 5*5-(5+2)=18 points).

Including the two-dimensional discrete Fourier transform coefficients B of N+2 points and the sampling spectrum Z of 0 (zero) of N*N-(N+2) points:

[4] Step S14 of Figure 2: The spectrum transmission module 15 of the transmission device 10 transmits the above-mentioned two-dimensional discrete Fourier transform coefficients B including N+2 points and the sampling spectrum Z of 0 (zero) at N*N-(N+2) points.

[5] Step S21 of Figure 2: The spectrum receiving module 21 of the receiving device 20 receives the two-dimensional discrete Fourier transform coefficients B of N+2 points and the sampling spectrum Z of 0 (zero) of N*N-(N+2) points transmitted by the spectrum transmitting module 15 of the transmitting device 10.

[6] Step S22 of Figure 2: The restoration method 23 of the restoration module 22 of the receiving device 20 restores the binary image X input by the transmitting device 10 based on the sampling spectrum Z of the two-dimensional discrete Fourier transform coefficients B including N+2 points.

FIG3 is a schematic diagram of the implementation process of the restoration module 22 or the restoration method 23 shown in FIG1 and FIG2 (such as step S22) in the system 1 and method for restoring a binary image described in the present invention. At the same time, in FIG3, the detailed implementation process of the restoration module 22 or the restoration method 23 may include the following steps S31 to S36.

[1] As shown in step S31 of FIG. 3 , when the spectrum receiving module 21 of the receiving device 20 receives a sampling spectrum Z including N+2 points of two-dimensional discrete Fourier transform coefficients B and N*N-(N+2) points of 0 (zero), the restoration method 23 of the restoration module 22 of the receiving device 20 obtains a spectrum z including N points of two-dimensional discrete Fourier transform coefficients B from one of the second to Nth rows of the sampling spectrum Z (such as the second row).

For example, the restoration method 23 of the restoration module 22 of the receiving device 20 can obtain a two-dimensional discrete Fourier transform coefficient B including N points (such as 5 points) from one of the second to Nth rows of the sampling spectrum Z (such as the second row) as the spectrum z, that is, the spectrum z can include a two-dimensional discrete Fourier transform coefficient B including N points (such as 5 points). In this embodiment, the two-dimensional discrete Fourier transform coefficients B of the spectrum z obtained from the second row to one of the Nth rows (such as the second row) of the sampling spectrum Z by the restoration method 23 of the restoration module 22 may include 2.118-0.3633i, -1.9271+2.2166i, 1.6910-0.9511i, -0.118+1.5388i, -0.809+0.5878i, as shown below.

The spectrum z obtained from the second row to one of the Nth rows (such as the second row) of the sampling spectrum Z: includes two-dimensional discrete Fourier transform coefficients B of N points (such as 5 points).

[2] As shown in step S32 of FIG. 3 , the restoration module 22 of the receiving device 20 may regard the spectrum z obtained from the second row to one of the Nth rows (such as the second row) of the sampled spectrum Z as one-dimensional (because there is only one row), so the restoration module 22 may use the following one-dimensional discrete Fourier inverse transform method 24 to perform a one-dimensional discrete Fourier inverse transform on the spectrum z of the two-dimensional discrete Fourier transform coefficient B including N points (such as 5 points) to obtain a one-dimensional spectrum y (such as an intermediate spectrum or intermediate vector) including one-dimensional coefficients y(n) of N points (such as 5 points).

One-dimensional discrete Fourier inverse transform method24:

In the above one-dimensional discrete Fourier inverse transform method 24, y is a one-dimensional spectrum (such as an intermediate spectrum or an intermediate vector), N is a prime number, k and n are all values from 0 to N-1 (i.e., 0, 1, 2, 3, ..., N-1). z is a spectrum of one of the second to Nth rows (such as the second row) of the sampling spectrum Z and includes two-dimensional discrete Fourier transform coefficients B of N points (such as 5 points). W _N =e ^2πi/N , e is the natural logarithm, π is the circumference of pi, and i is the square root of -1.

Therefore, in this embodiment, the restoration module 22 can use the one-dimensional discrete Fourier inverse transform method 24 to obtain the following one-dimensional spectrum y (such as an intermediate spectrum or intermediate vector) including one-dimensional coefficients y(n) of N points (such as 5 points). For example, the one-dimensional coefficients y(n) of N points of the one-dimensional spectrum y can include 0.19+0.5878i, 0, 0.3090-0.9511i, 1.6180-0.0000i, 0.

A one-dimensional spectrum y including N points and one-dimensional coefficients y(n):

[3] As shown in steps S33 to S36 of FIG. 3 , the restoration module 22 (restoration method 23) of the receiving device 20 can determine whether the one-dimensional coefficient y(n) of any of the N points (e.g., 5 points) of the one-dimensional spectrum y (e.g., intermediate spectrum or intermediate vector) is non-zero (non-zero) or 0 (zero), that is, y(n)≠0 or y(n)=0?

When the restoration module 22 (restoration method 23) of the receiving device 20 determines that the one-dimensional coefficient y(n) of any one of the N points (such as the first point/the third point/the fourth point) of the one-dimensional spectrum y is non-zero (i.e., y(n)≠0), the restoration module 22 (restoration method 23) can perform the one-dimensional signal restoration operation 25 as shown in step S34 on the non-zero one-dimensional coefficient y(n) to obtain the restoration result of the original signal A of the binary image X as shown in step S36. On the contrary, when the restoration module 22 (restoration method 23) determines that the one-dimensional coefficient y(n) of any one of the N points of the one-dimensional spectrum (such as the second point/fifth point) is 0 (i.e., y(n)=0), the restoration module 22 (restoration method 23) can execute the waiting and restoring operation 26 as shown in step S35 for the one-dimensional coefficient y(n) being 0, so as to obtain the restoration result of the original signal A of the binary image X as shown in step S36. For example, the above y(n) can be the following y(1), y(2), y(3), y(4) or y(5), but is not limited thereto.

Furthermore, when the restoration module 22 of the receiving device 20 determines that the one-dimensional coefficient y(n) of any one of the N points of the one-dimensional spectrum y (such as the first point/the third point/the fourth point) is non-zero, the restoration module 22 can use the table lookup method 27 to perform a one-dimensional signal restoration operation 25 on the one-dimensional coefficient y(n) that is non-zero in any one of the N points of the one-dimensional spectrum y (such as the first point/the third point/the fourth point), so as to deduce or restore the original signal A of any column (such as the first column/the third column/the fourth column) of the binary image X from the one-dimensional coefficient y(n) that is non-zero in any one of the N points of the one-dimensional spectrum y (such as the first point/the third point/the fourth point). Furthermore, when the restoration module 22 of the receiving device 20 determines that the one-dimensional coefficient y(n) of any one of the N points of the one-dimensional spectrum y (such as the second point/fifth point) is 0 (zero), the restoration module 22 (restoration method 23) can perform a waiting and restoring operation 26 on the one-dimensional coefficient y(2) of any one of the N points of the one-dimensional spectrum y (such as the second point/fifth point) that is 0, so as to correspondingly derive or restore the original signal A of any column (such as the second column/fifth column) of the binary image X. In this regard, the technical contents described in the following [3-1] to [3-5] are illustrated by way of example.

[3-1] In the one-dimensional spectrum y including the one-dimensional coefficients y(n) of N points, the one-dimensional coefficient y(1) of the first point is 0.1910+0.5878i. Since the one-dimensional coefficient y(1) of the first point of the one-dimensional spectrum y is non-zero, the restoration module 22 can perform a one-dimensional signal restoration operation 25 on the one-dimensional coefficient y(1) of the first point of the one-dimensional spectrum y in step S34 to derive or restore the original signal A of the first column of the binary image X from the one-dimensional coefficient y(1) of the first point of the one-dimensional spectrum y. For example, the restoration module 22 can use the table lookup method 27 to perform a one-dimensional signal restoration operation 25 on the one-dimensional coefficient y(1) of the first point of the one-dimensional spectrum y according to the content of Table 1 below, so as to use the table lookup method 27 to successfully correspond to or restore the original signal A of the first column of the binary image X from the one-dimensional coefficient y(1) of the first point of the one-dimensional spectrum y as (1 0 0 1 0).

Table 1: A correspondence table for the lookup table method 27. For example, when N=5, a correspondence table consisting of the original signal A of the binary image X and the one-dimensional coefficient y(n) of the one-dimensional discrete Fourier inverse transform method 24.

[3-2] In the one-dimensional spectrum y including the one-dimensional coefficients y(n) of N points, the one-dimensional coefficient y(2) of the second point is 0. Since the one-dimensional coefficient y(2) of the second point of the one-dimensional spectrum y is 0 (zero), the restoration module 22 (restoration method 23) can perform the subsequent waiting and restoring operation 26 on the one-dimensional coefficient y(2) of the second point of the one-dimensional spectrum y in step S35.

[3-3] In the one-dimensional spectrum y including the one-dimensional coefficients y(n) of N points, the one-dimensional coefficient y(3) of the third point is 0.3090-0.9511i. Since the one-dimensional coefficient y(3) of the third point of the one-dimensional spectrum y is non-zero, the restoration module 22 can perform a one-dimensional signal restoration operation 25 on the one-dimensional coefficient y(3) of the third point of the one-dimensional spectrum y in step S34, so as to derive or restore the original signal A of the third column of the binary image X from the one-dimensional coefficient y(3) of the third point of the one-dimensional spectrum y. For example, the restoration module 22 can use the table lookup method 27 to perform a one-dimensional signal restoration operation 25 on the one-dimensional coefficient y(3) of the third point of the one-dimensional spectrum y according to the content of the above Table 1, so as to use the table lookup method 27 to successfully correspond to or restore the original signal A of the third column of the binary image X from the one-dimensional coefficient y(3) of the third point of the one-dimensional spectrum y to be (0 1 0 0 0).

[3-4] In the one-dimensional spectrum y including the one-dimensional coefficients y(n) of N points, the one-dimensional coefficient y(4) of the fourth point is 1.6180-0.0000i (i.e., 1.6180). Since the one-dimensional coefficient y(4) of the fourth point of the one-dimensional spectrum y is non-zero, the restoration module 22 can perform a one-dimensional signal restoration operation 25 on the one-dimensional coefficient y(4) of the fourth point of the one-dimensional spectrum y in step S34, so as to derive or restore the original signal A of the fourth column of the binary image X from the one-dimensional coefficient y(4) of the fourth point of the one-dimensional spectrum y. For example, the restoration module 22 can use the table lookup method 27 to perform a one-dimensional signal restoration operation 25 on the one-dimensional coefficient y(4) of the fourth point of the one-dimensional spectrum y according to the content of the above Table 1, so as to use the table lookup method 27 to successfully correspond to or restore the original signal A of the fourth column of the binary image X from the one-dimensional coefficient y(4) of the fourth point of the one-dimensional spectrum y as (1 1 0 0 1).

[3-5] In the one-dimensional spectrum y including the one-dimensional coefficients y(n) of N points, the one-dimensional coefficient y(5) of the fifth point is 0. Since the one-dimensional coefficient y(5) of the fifth point of the one-dimensional spectrum y is 0 (zero), the restoration module 22 (restoration method 23) can perform the subsequent waiting and restoring operation 26 on the one-dimensional coefficient y(5) of the fifth point of the one-dimensional spectrum y in step S35.

Then, since the one-dimensional spectrum y including the one-dimensional coefficients y(n) of N points has only 5 points, there is no next point. Therefore, the restoration module 22 does not need to wait any longer and immediately starts to perform the waiting and then restoration operation 26 on the one-dimensional coefficients y(2) of the second point and the one-dimensional coefficients y(5) of the fifth point of the one-dimensional spectrum y.

For example, as shown in step S35 of FIG. 3 , the recovery module 22 (recovery method 23) can execute the following recovery procedures P1 to P3 of the waiting recovery operation 26 for y(2)=0 and y(5)=0.

Restoration procedure P1: The restoration module 22 (restoration method 23) can perform horizontal summation on the original signal A of each column of the restored binary image X (such as the original signal A of the first column is (1 0 0 1 0), the original signal A of the third column is (0 1 0 0 0), and the original signal A of the fourth column is (1 1 0 0 1)) to obtain the horizontal summation results (such as 2, 1, 3) of each column (such as the first column, the third column, and the fourth column) of the restored binary image X. At the same time, the restoration module 22 (restoration method 23) can replace the part (such as the unrestored signal) of each column (such as the second column and the fifth column) of the unrestored binary image X with variables (such as v2, v5). Therefore, the restoration module 22 (restoration method 23) can obtain the following one-dimensional signal V (because there is only one row).

One-dimensional signal V:

Restoration procedure P2: From the above y(2)=y(5)=0 and the contents of the corresponding table for table lookup method 27 in Table 1, it can be known that the original signal A of the second or fifth column of the binary image X must be (0 0 0 0 0) or (1 1 1 1 1), and the value of the variable v2 or variable v5 of the one-dimensional signal V after horizontal summation must be one of 0 and 5. Therefore, there are only four possibilities for the variables v2 and v5 of the one-dimensional signal V, namely: (1) the first possibility is v2=0 and v5=0, (2) the second possibility is v2=5 and v5=0, (3) the third possibility is v2=0 and v5=5, (4) the fourth possibility is v2=5 and v5=5.

Restoration procedure P3: From the properties of the two-dimensional discrete Fourier transform coefficient B of the sampling spectrum Z, it can be known that the two one-dimensional discrete Fourier transform coefficients of the variable v2 and the variable v5 of the one-dimensional signal V are the two two-dimensional discrete Fourier transform coefficients B of the first row of the sampling spectrum Z, so the restoration module 22 can use the exhaustive method 28 to check the following four possibilities one by one.

(1) The first possibility: The restoration module 22 can use the exhaustive method 28 to check that v2=0 and v5=0 of the one-dimensional signal V, that is, the one-dimensional signal V is as follows:

For example, the restoration module 22 uses the exhaustive method 28 to check that when v2=0 and v5=0 of the one-dimensional signal V, the two one-dimensional discrete Fourier transform coefficients of the one-dimensional signal V can be obtained as 6.0000+0i and -1.2361+1.1756i, respectively. Therefore, the restoration module 22 can further compare the two one-dimensional discrete Fourier transform coefficients of the one-dimensional signal V with the two two-dimensional discrete Fourier transform coefficients B (such as 11 and 0.309-3.5797i) of the first row of the above-mentioned sampling spectrum Z.

(2) The second possibility: The restoration module 22 can use the exhaustive method 28 to check the one-dimensional signal V for v2=5 and v5=0, that is, the one-dimensional signal V is as follows:

For example, the restoration module 22 uses the exhaustive method 28 to check that when v2=5 and v5=0 of the one-dimensional signal V, the two one-dimensional discrete Fourier transform coefficients of the one-dimensional signal V can be obtained as 11.0000+0i (i.e. 11) and 0.3090-3.5797i, respectively. Therefore, the restoration module 22 can further compare the two one-dimensional discrete Fourier transform coefficients of the one-dimensional signal V with the two two-dimensional discrete Fourier transform coefficients B (i.e. 11 and 0.309-3.5797i) of the first row of the above-mentioned sampling spectrum Z.

Therefore, the two one-dimensional discrete Fourier transform coefficients of the one-dimensional signal V are respectively 11.0000+0i (i.e. 11) and 0.3090-3.5797i, which are consistent with the two two-dimensional discrete Fourier transform coefficients B (i.e. 11 and 0.309-3.5797i) of the first row of the above-mentioned sampling spectrum Z, so the answer obtained by the restoration module 22 using the exhaustive method 28 is v2=5 and v5=0. In other words, the original signal A of the second column of the binary image X is composed of 5 1s (1 1 1 1 1), and the original signal of the fifth column of the binary image X is composed of 5 0s (0 0 0 0 0).

(3) The third possibility and (4) the fourth possibility: Since the restoration module 22 has obtained the answers v2=5 and v5=0 using the poor example method 28, the restoration module 22 does not need to further check the third possibility and the fourth possibility.

At this point, the restoration module 22 has completely restored the binary image X, as shown below.

The restored binary image X:

In the embodiment of the present invention, a binary image X with both length and width being prime number N points (such as N=5, a total of N*N=5*5=25 points) is used as an example. However, for a binary image X with both length and width being multiples of prime number N points (such as mN=2*5=10, multiple m=2, prime number N=5, a total of mN*mN=10*10=100 points), the present invention can also be extended and implemented by using translation sampling technology.

For example, for a binary image X with a length and width that are both multiples of prime number N (e.g., mN=2*5=10, a total of mN*mN=10*10=100 points), the sampling module 13 of the transmission device 10 can adopt a translation sampling technique to sample the binary image X with a length and width that are both multiples of prime number N (e.g., mN*mN=10*10 points). For example, in the binary image X, S represents sampling (Sample) and H represents no sampling (Hide).

Translational sampling technique: sampling of a binary image X with points whose length and width are both multiples of the prime number N.

From the sampling module 13 of the transmission device 10 adopting the translation sampling technology to sample the binary image X with the length and width of the sides being multiples of the prime number N (such as mN*mN=10*10 points), it can be found that the four groups of N*N points (such as 5*5 points) in the upper left corner, upper right corner, lower left corner and lower right corner of the binary image X with the length and width of the sides being multiples of the prime number N (such as mN*mN=10*10 points) are consistent with the sampling method 14 of the upper sampling module 13 for the binary image X with the length and width of the sides being prime numbers N points (such as 5*5 points). Therefore, the restoration module 22 of the receiving device 20 can adopt the translation sampling technology to transform the restoration method (restoration problem) of the binary image X with the length and width of the sides being multiples of the prime number N (such as mN*mN=10*10 points) into the restoration method (restoration problem) of multiple groups of binary images X with the length and width of the sides being prime numbers N (such as 4 groups with 5*5 points).

The principle of this method of converting the restoration method (restoration problem) of a binary image X with both length and width points that are multiples of prime number N (such as mN* mN=10*10 points) into the restoration method (restoration problem) of multiple sets of binary images X with both length and width points that are prime number N (such as 4 sets of 5*5 points) is as follows. For example, the original sequence (spectrum) of one-dimensional mN points (such as mN=2*5=10 points) is converted into a sequence (spectrum) of two sets of N points (such as 5 points), and the two-dimensional image X can be obtained by the same or similar method.

[1] Original sequence: Assume that the transmission device 10 has a one-dimensional original sequence of mN points (e.g., multiple m=2, prime number N=5, mN=2*5=10 points) and is represented as x(n), where n=0,1,2,3,4,5,6,7,8,9.

[2] Sequence segmentation: The restoration module 22 (restoration method 23) of the receiving device 20 can divide the original sequence of one-dimensional mN points (e.g., m*N=2*5=10 points) of the transmitting device 10 into two sequences of N points (e.g., 5 points), and the two sequences of N points (e.g., N=5 points) are respectively called sequence x1(n) and sequence x2(n). For example, x1(n)=x(0),x(2),x(4),x(6),x(8), and x2(n)=x(1),x(3),x(5),x(7),x(9).

[3] Calculation of one-dimensional coefficients: The reduction module 22 (reduction method 23) can calculate the one-dimensional coefficients of two sets of N points (e.g., 5 points) of sequences x1(n) and x2(n), for example, the one-dimensional coefficients are one-dimensional discrete Fourier transform coefficients (abbreviated as DFT). Assume that the calculation results of the one-dimensional coefficients (e.g., DFT) of the two sets of N points (e.g., 5 points) of sequences x1(n) and x2(n) are y1(k) and y2(k), y1(k)=DFT(x1(n)), y2(k)=DFT(x2(n)), and k=0,1,2,3,4.

[4] Combination results: The reduction module 22 (reduction method 23) can combine the calculation results of the one-dimensional coefficients (such as DFT) of two sets of N points (such as 5 points) sequence x1(n) and sequence x2(n) into a one-dimensional coefficient (such as DFT) of 10 points. This can be achieved in the following way.

For example, for k=0,1,2,3,4, the two calculation formulas are y(k)=y1(k)+W _N ^k y2(k) and y(k+5)=y1(k)-W _N ^k y2(k), where y is the one-dimensional coefficient (such as DFT) of mN points (such as 2*5=10 points) of x. On the contrary, if y(k) and y(k+5) are already available, the two calculation formulas y(k)=y1(k)+W _N ^k y2(k) and y(k+5)=y1(k)-W _N ^k y2(k) can be used to reversely obtain y1(k) and y2(k).

Therefore, the contents of [1] to [4] above are methods for converting an original sequence (spectrum) of one-dimensional mN points (e.g. mN=2*5=10 points) into a sequence (spectrum) of two groups of N points (e.g. 5 points).

In addition, the present invention also provides a computer-readable medium for a method of restoring a binary image, which is applied to a computing device or a computer having a processor and a memory, and the computer-readable medium stores instructions, and the computing device or the computer can execute the computer-readable medium through the processor and the memory to execute the above content when executing the computer-readable medium. In one embodiment, the method of restoring a binary image is a method of restoring a binary image from a two-dimensional discrete Fourier transform coefficient. In another embodiment, the computer-readable medium is a non-transitory computer-readable storage medium.

In one embodiment, the processor may be a central processing unit (CPU), a graphics processing unit (GPU), a microprocessor (MPU), a microcontroller (MCU), etc., the memory may be a random access memory (RAM), a read-only memory (ROM), a flash memory, a memory card, a hard drive (such as a cloud/network/external hard drive), an optical disk, a flash drive, a database, etc., and the computing device or computer may be a computer, a smartphone, a tablet computer, a personal computer, a laptop, a desktop computer, a server (such as a cloud/remote/network server), etc.

In summary, the system, method and computer-readable medium for restoring binary images described in the present invention have at least the following characteristics, advantages or technical effects.

1. The present invention enables, when it is known that the image is a binary image, the transmitting device only needs to select N+2 points or more (approximately or at least 6%) of two-dimensional discrete Fourier transform coefficients from the spectrum of the binary image to transmit to the receiving device, so that the restoration module (restoration method) of the receiving device can quickly restore the binary image input by the transmitting device without any distortion.

2. The present invention can restore the binary image quickly when the spectrum of the binary image is severely damaged. The restoration module (restoration method) of the receiving device only needs to use the two-dimensional discrete Fourier transform coefficients (such as spectrum coefficients or spectrum information) of N+2 points or more (approximately or at least 6%) in the spectrum of the binary image.

3. For a binary image with N*N points, the present invention only needs to extract N+2 points or more (approximately or at least 6%) of two-dimensional discrete Fourier transform coefficients from the spectrum of the binary image to quickly and losslessly restore the binary image. For example, the restoration time (computation time) of a 19x19 (i.e. 19*19) binary image requires 90 seconds of restoration time according to the known technology, but the present invention only requires less than 2 seconds of restoration time, so the present invention only needs 1/45 of the restoration time of the known technology.

4. The restoration module of the receiving device of the present invention can restore the binary image input by the transmitting device within a few seconds (e.g., within 2 seconds), which is significantly better than the restoration time of 90 seconds required by the conventional technology.

5. For a binary image with both long and wide sides being multiples of a prime number N (e.g., mN*mN=38*38), where m is a multiple (e.g., m=2) and N is a prime number (e.g., N=19), the present invention can use a translation sampling technique to sample (N+2)m ² (e.g., 21*2 ² =84) two-dimensional discrete Fourier transform coefficients to restore the binary image, and the sampling rate of the two-dimensional discrete Fourier transform coefficients of the frequency spectrum of the binary image only needs to be about 6%.

6. The present invention can restore the binary image by using only N+2 points or more (approximately or at least 6%) of two-dimensional discrete Fourier transform coefficients on binary images such as computer tomography scans, thereby reducing the number of measurements of computer tomography scans.

7. The present invention can utilize the characteristics of a quick response code (QR code) being a binary image to reduce the number of two-dimensional discrete Fourier transform coefficients required by the sampling module (sampling method) of the transmitting device to sample the binary image, and even if the binary image is severely damaged, the restoration module (restoration method) of the receiving device can quickly restore the binary image.

8. The present invention may be applied to industries such as the medical industry, camera chip modules for mobile phones, etc., and may be applied to products such as tomography scanners, scanning modules for quick response codes (QR codes), etc., but is not limited to these.

The above implementation forms are only illustrative of the principles, features and effects of the present invention, and are not intended to limit the scope of implementation of the present invention. Anyone familiar with this technology can modify and change the above implementation forms without violating the spirit and scope of the present invention. Any equivalent changes and modifications completed using the content disclosed by the present invention should still be covered by the scope of the patent application. Therefore, the scope of protection of the present invention should be as listed in the scope of the patent application.

1: System

10: Transmission device

11: Spectrum conversion module

12: Two-dimensional discrete Fourier transform method

13: Sampling module

14: Sampling method

15: Spectrum transmission module

20: Receiving device

21: Spectrum receiving module

22: Restore module

23: Restoration method

24: One-dimensional discrete Fourier inverse transform method

25: One-dimensional signal restoration operation

26: Wait and restore the operation

27: Table lookup method

28: Poor Manners

A: Original signal

B: Two-dimensional discrete Fourier transform coefficient

X: binary image

Y: spectrum

y: one-dimensional spectrum

y(n): one-dimensional coefficient

Z: sampling spectrum

z: spectrum

Claims (15)

A system for restoring binary images, comprising: A transmission device performs a two-dimensional discrete Fourier transform on the input binary image to obtain a spectrum of two-dimensional discrete Fourier transform coefficients including N*N points, and then the transmission device samples the spectrum of two-dimensional discrete Fourier transform coefficients including N*N points to obtain a sampled spectrum of two-dimensional discrete Fourier transform coefficients including N+2 points, so that the transmission device transmits the sampled spectrum of two-dimensional discrete Fourier transform coefficients including N+2 points, wherein N is a prime number; and A receiving device with a restoration module is communicatively connected to the transmitting device, so that the receiving device receives the sampling spectrum of the two-dimensional discrete Fourier transform coefficients of the N+2 points transmitted by the transmitting device, and then the restoration module of the receiving device restores the binary image input by the transmitting device according to the sampling spectrum of the two-dimensional discrete Fourier transform coefficients of the N+2 points; The restoration module performs a one-dimensional discrete Fourier inverse transformation on the spectrum of two-dimensional discrete Fourier transform coefficients of N points in the sampling spectrum to obtain a one-dimensional spectrum including one-dimensional coefficients of N points, so that when the one-dimensional coefficient of any one of the N points of the one-dimensional spectrum is non-zero, the restoration module performs a one-dimensional signal restoration operation on the non-zero one-dimensional coefficient to restore the binary image, and when the one-dimensional coefficient of any one of the N points of the one-dimensional spectrum is zero, the restoration module performs a waiting and restoring operation on the zero one-dimensional coefficient to restore the binary image. The system as described in claim 1, wherein the transmission device has a spectrum conversion module, and the spectrum conversion module performs a two-dimensional discrete Fourier transform on the binary image through a two-dimensional discrete Fourier transform method to obtain a spectrum including two-dimensional discrete Fourier transform coefficients of the N*N points. The system as described in claim 1, wherein the transmission device has a sampling module, so that the sampling module samples the two-dimensional discrete Fourier transform coefficient of the first point and the two-dimensional discrete Fourier transform coefficient of one of the second point to the Nth point on the first line of the spectrum through a sampling method, and samples the two-dimensional discrete Fourier transform coefficient of N points on one of the second line to the Nth line of the spectrum, so that the sampling method of the sampling module samples the two-dimensional discrete Fourier transform coefficients of the N+2 points in total on the spectrum. The system as described in claim 1, wherein the transmission device has a sampling module and a spectrum transmission module, so that the sampling module samples the spectrum including the N*N points of two-dimensional discrete Fourier transform coefficients through a sampling method to obtain a sampling spectrum including the N+2 points of two-dimensional discrete Fourier transform coefficients and N*N-(N+2) points of 0, and then the spectrum transmission module transmits the sampling spectrum including the N+2 points of two-dimensional discrete Fourier transform coefficients and N*N-(N+2) points of 0 to the receiving device. The system as described in claim 1, wherein the restoration module of the receiving device further obtains a spectrum of two-dimensional discrete Fourier transform coefficients including the N points from one of the second to Nth rows of the sampling spectrum, and then the restoration module uses a one-dimensional discrete Fourier inverse transform method to perform a one-dimensional discrete Fourier inverse transform on the spectrum of two-dimensional discrete Fourier transform coefficients including the N points to obtain a one-dimensional spectrum including the one-dimensional coefficients of the N points. The system as described in claim 1, wherein the restoration module of the receiving device uses a one-dimensional discrete Fourier inverse transform method to perform a one-dimensional discrete Fourier inverse transform on the spectrum of the two-dimensional discrete Fourier transform coefficients of the N points to obtain a one-dimensional spectrum including the one-dimensional coefficients of the N points, and then the restoration module determines whether the one-dimensional coefficient of any one of the N points of the one-dimensional spectrum is non-zero or zero. The system as described in claim 1, wherein, when the binary image has a length and a width that are both multiples of the prime number N, the transmission device adopts a translation sampling technique to sample the binary image having the length and the width that are both multiples of the prime number N, and the restoration module of the receiving device adopts the translation sampling technique to convert the restoration method of the binary image X having the length and the width that are both multiples of the prime number N into a plurality of groups of restoration methods of the binary image X having the length and the width that are both prime number N points. A method for restoring a binary image, comprising: A transmission device performs a two-dimensional discrete Fourier transform on the input binary image to obtain a spectrum of two-dimensional discrete Fourier transform coefficients including N*N points, and then the transmission device samples the spectrum of two-dimensional discrete Fourier transform coefficients including N*N points to obtain a sampled spectrum of two-dimensional discrete Fourier transform coefficients including N+2 points, wherein N is a prime number; and The transmitting device transmits the sampling spectrum including the two-dimensional discrete Fourier transform coefficients of the N+2 points to a receiving device having a restoration module, and then the restoration module of the receiving device restores the binary image input by the transmitting device according to the sampling spectrum including the two-dimensional discrete Fourier transform coefficients of the N+2 points; The restoration module performs a one-dimensional discrete Fourier inverse transformation on the spectrum of two-dimensional discrete Fourier transform coefficients of N points in the sampling spectrum to obtain a one-dimensional spectrum including one-dimensional coefficients of N points, so that when the one-dimensional coefficient of any one of the N points of the one-dimensional spectrum is non-zero, the restoration module performs a one-dimensional signal restoration operation on the non-zero one-dimensional coefficient to restore the binary image, and when the one-dimensional coefficient of any one of the N points of the one-dimensional spectrum is zero, the restoration module performs a waiting and restoring operation on the zero one-dimensional coefficient to restore the binary image. The method as described in claim 8 further includes the transmission device performing a two-dimensional discrete Fourier transform on the binary image through a two-dimensional discrete Fourier transform method to obtain a spectrum including the two-dimensional discrete Fourier transform coefficients of the N*N points. The method as described in claim 8 further includes the transmission device sampling the two-dimensional discrete Fourier transform coefficient of the first point and the two-dimensional discrete Fourier transform coefficient of one of the second point to the Nth point on the first line of the spectrum by a sampling method, and sampling the two-dimensional discrete Fourier transform coefficient of N points on one of the second line to the Nth line of the spectrum, so that the sampling method of the sampling module samples the two-dimensional discrete Fourier transform coefficients of the N+2 points in total on the spectrum. The method as described in claim 8 further includes the transmission device sampling the spectrum including the two-dimensional discrete Fourier transform coefficients of the N*N points by a sampling method to obtain a sampling spectrum including the two-dimensional discrete Fourier transform coefficients of the N+2 points and N*N-(N+2) points of 0, and then the transmission device transmits the sampling spectrum including the two-dimensional discrete Fourier transform coefficients of the N+2 points and N*N-(N+2) points of 0 to the receiving device. The method as described in claim 8 further includes the recovery module of the receiving device obtaining a spectrum of two-dimensional discrete Fourier transform coefficients including the N points from one of the second to Nth rows of the sampling spectrum, and then the recovery module uses a one-dimensional discrete Fourier inverse transform method to perform a one-dimensional discrete Fourier inverse transform on the spectrum of two-dimensional discrete Fourier transform coefficients including the N points to obtain a one-dimensional spectrum including the one-dimensional coefficients of the N points. The method as described in claim 8 further includes the recovery module of the receiving device using a one-dimensional discrete Fourier inverse transform method to perform a one-dimensional discrete Fourier inverse transform on the spectrum including the two-dimensional discrete Fourier transform coefficients of the N points to obtain a one-dimensional spectrum including the one-dimensional coefficients of the N points, and then the recovery module determines whether the one-dimensional coefficient of any one of the N points of the one-dimensional spectrum is non-zero or zero. The method as described in claim 8 further includes when the binary image has a number of points whose lengths of both sides are multiples of the prime number N, the transmitting device adopts a translation sampling technique to sample the binary image having a number of points whose lengths of both sides are multiples of the prime number N, and the restoration module of the receiving device adopts the translation sampling technique to convert the restoration method of the binary image X having a number of points whose lengths of both sides are multiples of the prime number N into a plurality of restoration methods of the binary image X having a number of points whose lengths of both sides are prime numbers N. A computer-readable medium, used in a computing device or a computer, stores instructions for executing a method for restoring a binary image as described in one of claims 8 to 14.

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