KR20060065648A - System and method for image sensing and processing - Google Patents

Abstract

A system and method for image sensing and processing using the Arithmetic Fourier Transform (AFT). An image sensing array has sensors located based on a set of Farey fractions, each multiplied by a unit block size of the array. Similar sampling can be achieved by interpolating the pixel values of a conventional, uniformly spaced array of sensors. The AFT can be determined extremely efficiently by computing weighted sums of the representative pixel values. Corresponding Discrete Cosine Transform (DCT) coefficients can then be computed by scaling the AFT coefficients. As a result, the number of multiplication operations required to compute the DCT is dramatically reduced.

Description

Image detection and processing system and method {SYSTEM AND METHOD FOR IMAGE SENSING AND PROCESSING}

A series of important compression standards for still and video images use Discrete Cosine Transform (DCT). For example, a standard JPEG algorithm for compressing still images is shown in FIG. In the algorithm shown, the image is divided into 8x8 pixel blocks with pixel intensity values (e.g., shown at block 102). For every 8x8 block 102, a two-dimensional (2-D) DCT is calculated (step 104). The DCT coefficients are scaled, quantized, and truncated (eg rounded corners) (step 106) to retain only the most important information for the human eye to perceive correctly. For example, since the eye is relatively concentrated at high spatial frequencies and the largest DCT coefficients typically represent the lowest spatial frequency, many high frequency DCT coefficients may be rounded to zero in quantization step 106. Then entropy encoding the quantized coefficients-typically using Hoffman encoding-the representation appears more compact and leaves non-zero DCT coefficients (step 108). The above-described compression scheme can be applied individually to different spectral components of a color image, for example (red, green, blue pixels of an RGB image or luminance-color difference values of an image). Because DCT is a linear operation, it can be applied individually to any linear combination of RGB pixel values.

2-D, NxN point DCT is defined as follows.

only,

Where A is a sampled image, n and m are spatial sampling indexes, and k and l are spatial frequency indexes. Although some well-known algorithms can reduce the number of multiplications by 50 or more factors, the calculation of 8x8 DCT requires a multiplication degree of (2x8x8) x (8x8) = 8192. Nevertheless, the calculation of DCT typically requires a lot of computation for image compression. In addition, some compression techniques, such as JPEG2000, use wavelet representation rather than DCT, but DCT-based techniques are expected to be widely used in the future.

In addition to the JPEG standards used for still image compression, there are several video compression standards that are commonly used, such as Motion JPEG, MPEG (1, 2, 4), and H.26X. Requires calculation of the DCT.

Currently, in the most commercial applications, image compression is performed by separate digital signal processing circuits that derive DCT coefficients based on digitized image data. However, since the conventional DCT algorithm requires a considerable amount of computational power and consumes a large amount of power, such an image processing method is not very favorable for an apparatus in which power maintenance is important. Such devices include, for example, mobile camera phones, digital cameras, and wireless image sensors that monitor health with a machine.

It is therefore an object of the present invention to provide an image sensing and processing system which reduces the number of calculations, in particular the multiplication necessary to derive the DCT coefficients from the image data.

Another object of the present invention is to provide a system for reducing the amount of power consumed by the derivation of the DCT coefficients.

These and other objects are achieved by a system that calculates the DCT coefficients of an image using an Arithmetic Fourier Transform (AFT). By performing the addition by the AFT method, the Fourier transform can be mainly calculated. In hardware implementations, the higher the computational efficiency of the AFT, the less complexity, size, and power consumption of the circuit, and the faster the processing. Irregular sampling is also possible by interpolating signals from a set of sensors at regular intervals, but images can be well sampled using sensors at irregular intervals. The AFT technology of the present invention, in particular analog performance technology, is very economical in terms of circuit complexity and power consumption.

According to one aspect of the invention, the incoming signal is detected by a sensor array comprising at least first and second sensors having first and second sensor positions respectively. The first sensor position is close to the position of the first extreme value of the basis function of the domain transformation, and the basis function has one or more spatial coordinates defined according to the spatial coordinate system of the sensor array. The second sensor position is close to the position of the second extreme value of the same basis function or of another basis function. The system includes at least one filter that receives signals from the first and second sensors and generates a filtered signal that includes a weighted sum of at least the signals from the first second sensor. Also included in the present invention is a special case where a signal from one sensor may include a filter output.

According to a further aspect of the invention, incoming light is detected by a sensor array comprising a plurality of sensors having at least first and second sensors respectively having first and second sensor positions. The incoming optical signal has a first value at the first sensor position and a second value at the second sensor position. The system includes interpolation circuitry to receive, from the first and second sensors, signals indicative of first and second values of the incoming optical signal, respectively. The interpolation circuit interpolates the signals from the first and second sensors to generate an interpolated signal. The interpolated signal represents an approximation of the incoming optical signal at a position close to a first extreme value of at least one basis function of a domain transform, wherein the at least one basis function is defined by at least one defined according to a spatial coordinate system of the sensor array. Have spatial coordinates.

Other objects, features and advantages of the present invention will become apparent from the following detailed description when read in conjunction with the accompanying drawings which show specific embodiments of the present invention.

1 is a block diagram illustrating a conventional exemplary image processing process.

2 is a diagram illustrating data processed according to the present invention.

3 is a diagram illustrating an exemplary picture sampling space and corresponding domain transform basis function in accordance with the present invention.

4 is a graph illustrating error characteristics of an exemplary system and method for sensing and processing images in accordance with the present invention.

5 is a diagram illustrating an exemplary image sampling space in accordance with the present invention.

6 is a graph illustrating error characteristics of an exemplary system and method for sensing and processing images in accordance with the present invention.

7 is a graph illustrating error characteristics of another exemplary system and method for sensing and processing images in accordance with the present invention.

8 is a graph illustrating error characteristics of another exemplary system and method for sensing and processing images in accordance with the present invention.

9 is a diagram illustrating an exemplary image sampling space in accordance with the present invention.

10 illustrates an exemplary sensor array and filter arrangement in accordance with the present invention.

11 is a flowchart illustrating an exemplary image detection and processing procedure in accordance with the present invention.

12 is a flowchart illustrating an exemplary signal filtering process used in the process illustrated in FIG.

13 is a flowchart illustrating an exemplary image detection and processing procedure in accordance with the present invention.

14 is a flowchart illustrating an exemplary signal filtering process used in the process illustrated in FIG. 13.

15 is a diagram illustrating an exemplary sensor array and filtering circuit in accordance with the present invention.

16 is a flowchart illustrating an exemplary image detection and processing procedure in accordance with the present invention.

FIG. 17 is a timeline diagram associated with FIG. 10 illustrating an exemplary timing sequence generated by upon clock generation to generate filtered signal S (3, 12).

18 illustrates an exemplary sensor array and filter arrangement in accordance with the present invention.

The incoming image signal, such as the incoming light pattern from the scene being photographed, is sampled by a sensor array such as a charge coupled device (CCD). Individual sensors are distributed within the array according to spatial patterns that are particularly suitable for enhancing the effectiveness of the AFT algorithm. A good spatial distribution for a 2-D sensor array can be better understood by first assuming a one-dimensional (1-D) case. For example, to find a 1-D AFT equivalent to a 1-D DCT on an 8-point, space or time unit interval (0 to 1), samples of 12 irregular space intervals may be used. Preferred sampling positions are (0, 1/4, 2/7, 1/3, 2/5, 1/2, 4/7, 2/3, 3/4, 4/5, 6/7, 1) . However, it should be noted that if the entire signal being sampled contains several unit intervals, all adjacent unit intervals will share the first and last sample of each interval. In number theory, k / j-type fractions (where k = 0, 1 ... N-1 and j = 1, 2, ... N) are often referred to as the Nth order "parlay fraction". This is called. Thus, the sampling position described above, which provides a good set of samples for calculating an 8-point DCT based on the corresponding 12-point AFT, is 2k / j (where k-0,1 ... 4 and j = 1). Corresponding to an even subset of the 8th order Parlay fractions defined as

The AFT of the signal can be calculated by using the above-described signal sample in combination with a function known as the Mobius function. 1-D AFTs based on Mobius functions are known and examples of the transformation include DW Tufts, G. Sadasiv, "Arithmetic fourier Transform and Adaptive Delta Modulation: a Symbiosis for High Speed Computation," SPIE Vol. 880 High Speed Computing (1988). The 1-D Mobius function μ ₁ (n) is defined as

μ ₁ (l) = 1

If μ ₁ (n) = (-1) ^S , n = (p ₁ ) (p ₂ ) (p ₃ ) ... (p _S ), then p _i is a different prime number

μ ₁ (n) = 0, where p | n for any prime p

Here, the vertical bar display indicates that m | n is an integer n divided by an integer m without a remainder. If n can be represented as an enemy of different prime numbers s, μ ₁ (n) is (-1) ^S , otherwise the value is zero.

Within a unit interval, it is assumed that signal A (t) is periodic with one period. Further, assuming that signal A (t) is band-limited over the entirety of N harmonics, the AFT coefficient is given as follows.

에 기초하여, 다음의 필터링 함수를 갖는 필터의 출 력을 나타낸다.Where each S (n, t _ref ) is a sample distributed at a position corresponding to each parlay fraction of interval 0 to 1

Based on the following, the output of the filter having the following filtering function is shown.

의 합이며, 여기서 t_ref는 임의의 기준 시간이다. t_ref는 단위 간격의 경우에는 양호하게 1과 동일하다. 각각의 AFT 계수는 뫼비우스 함수 μ₁(m)에 의해 가중된, 선택된 필터들의 필터 출력의 합이다.Each filter output S (n, t _ref ) is each sample multiplied by the scale factor 1 / n

Is the sum of t _ref is an arbitrary reference time. t _ref is preferably equal to 1 for unit intervals. Each AFT coefficient is the sum of the filter outputs of the selected filters, weighted by the Mobius function μ ₁ (m).

To process 2-D input signals such as pictures or picture parts (e.g., unit sub-pictures or blocks), the AFT is 2-D in two dimensions using the Mobius function μ ₂ (n, m) defined as 2-D Is expanded.

μ ₁ (n, m) = μ ₁ (n) μ ₁ (m)

Where n and m are positive integers and μ ₁ (n) is the 1-D Mobius function defined in Equations 2a, 2b, 2c.

The equation of the 2-D AFT of the zero mean 2-D input signal A (p, q), where p and q are continuous spatial coordinates in the unit range (e.g., 0 to 1), is expressed as It can be expressed in terms of arbitrary reference points (p _ref , q _ref ) by the D Fourier series.

Where (p _ref , q _ref ) is any reference position, preferably (1,1) for the unit sub-picture.

Signal A (p, q) is band limited to N harmonics in both spatial order q and q-that is, the Fourier series coefficients higher than N are equal to zero. The image data is processed using a filter bank having N ² filters, and each filter has the following filtering function.

의 공간 위치

가, 도 3과 관련해서 상세히 후술되는 바와 같이, 단위 화상 블록의 각각의 파레이 분수

및

에 의해, 기준 위치(p_ref,q_ref)와 관련해서 정의된다는 것을 알 수 있다.Where n = 1, 2, .... N and m = 1, 2, .... N. From equation (8), the samples processed by the filters

Spatial location of

As shown in detail below with respect to FIG. 3, the respective parlay fractions of the unit image blocks

And

It can be seen that by the reference position is defined in relation to the reference position (p _ref , q _ref ).

Substituting the signal A (p, q) in equation (8) with the Fourier series given in equations (6) and (7), the output of each filter is a special set of Fourier series coefficients of A (p, q). It can be seen that it is equal to the sum of.

Derivation of equation (9) is given in Appendix A attached to this specification.

항 이상은 없으며, 여기서

는 x보다 작거나 같은 최대 정수를 나타낸다. 수학식 (9)가 주어지면, 2-D 푸리에 급수 계수에 대한 다음의 관계가 주어질 수 있다(이에 대한 증명은 첨부된 부록 B에 나와 있다).Assuming that the signal is band limited, non-zero

No more than

Denotes the largest integer less than or equal to x. Given Equation (9), the following relationship to the 2-D Fourier series coefficients can be given (the proof for this is given in the appendix B).

Moreover, because of the similar relationship between DCT and Discrete Fourier Transform (DFT), the above-described output from the 2-D AFT algorithm is used to calculate the DCT coefficients of the unit part-picture divided into N × N uniformly spaced pixels. Can be used. First, the image sensor array is divided into pixels of a unit area block, each block having a size of 1 × 1 by definition. The photosensitive elements inside each unit region are located at points based on a set of parlay fractions of unit block size to provide a sample suitable for the filter defined in Equation (8). To calculate the output of the filter, a suitable reference point (p _ref , q _ref ) is selected. Convenient reference points are p _ref = 1 and q _ref = 1 (the corner of the unit area). Then, Equation 8 becomes as follows:

Where n = 1,2, ..., N and m = 1,2, ..., N.

The output of the 2-D AFT is a 2-D Fourier series set. To obtain the DCT coefficients from the Fourier series, the expanded picture block X (p, q) is obtained by extending the original picture block A (p, q) in both directions by the mirror image of the picture block, as shown in FIG. , This becomes:

If the AFT is calculated from the extended picture block X (p, q) and not from the original picture block A (p, q), then the appropriate filter value is as follows:

If the extended picture block X (p, q) conforms to the Nyquist criterion, the resulting AFT coefficients will be equal to the DCT coefficients in the scale factor, and evidence of this result is provided in Appendix C attached herein. On the other hand, if the expanded picture does not meet the Nyquist criterion, the 2-D AFT coefficient is only an approximation of the 2-D DCT coefficient. This situation is more likely to occur for an image rich in high frequency components. However, it is possible to improve this approximation using an aliasing correction technique, which technique will be described later in more detail.

In any case, from equations (12) and (13), each output S (n, m) of the filter can be expressed as follows:

는 x보다 크거나 동일한 최소의 정수를 나타낸다.Where n and m take the value of 1 to N.

Denotes the smallest integer greater than or equal to x.

From Equation 14, it can be seen that a particular point exists in the repeated sample space. As a result, by calculating the DCT rather than the DFT, the number of independent points in the 2-D AFT is reduced by almost one half. For example, to calculate an 8x8 point DCT inside a unit part-image, a set of 12x12 photosensitive elements per unit area is used. Elements at the edge of the unit area are shared between adjacent sub-pictures, thereby reducing the effective number of points per block to 11 × 11. 3 illustrates an example non-uniform sample space 300. In the illustrated example, non-uniformly distributed sample points 348 are used for 2-D AFT calculation. The corresponding effective DCT sample points 398 are evenly distributed.

Using the sampled picture as illustrated in FIG. 3 and a filter whose filtering function is defined according to equation (14), the 2-D AFT coefficients x _{k , 1} can be calculated as follows:

Where E [A] represents the average value of the image, x _{k , 1} represents the 2-D AFT coefficient of the extended block image X, and x _{k , 0} represents the 1-D AFT of the average value of the rows along the p-axis. The coefficient obtained by the calculation, x _0,1 is the coefficient obtained by calculating the 1-D AFT of the mean value of the column along the q-axis. This DCT coefficient can be calculated as follows:

The above explanation uses the 2-D AFT to calculate the DCT coefficients of the image part, allowing the entire operation to be performed primarily as an additive operation, making the 2-D AFT process very efficient with very little multiplication operation. Prove that you can. Factors that provide this increased efficiency can be further understood with reference to FIG. 3. 3 shows an example 2-D sample region 300 of a sensor array corresponding to the region of a pixel 398 of a conventional 8x8 block arranged in a conventional pattern. However, in accordance with the present invention, the illustrated region 300 has some preferred location 348 for use with the aforementioned 2-D AFT technique. Preferred location 348 corresponds to the maximum (ie, maximum) of the base function of the transformation being performed. For example, it is well known that the basic functions of the Fourier transform are the sine and cosine functions of various different frequencies (in the case of time varying signals) or wavelengths (in the case of spatial change signals such as images). For cosine transforms such as DCT, the basic function is the cosine function of various frequencies (for time varying signals) or wavelengths (for spatial change signals), as provided by equation (1). In the example sample region 300 illustrated in FIG. 3, columns 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342 are cosine basic functions 320, 321, 322. , 323, 324, 325, 326, and 327 corresponding to the positions of the respective maximums 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, and 312. The spatial coordinate q is defined according to the spatial coordinate system of either the sensor array or the illustrated region 300. In particular, in the illustrated example, the spatial coordinates q of the basic functions 320, 321, 322, 323, 324, 325, 326, 327 described above are referenced to the left edge (column “331”) of the illustrated region 300. It is equal to the horizontal coordinate of the sensor array. Similarly, the rows 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392 of the preferred sample positions 348 are cosine basic functions 370, 371, 372, 373, 374, Corresponding to local maxima 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362 of 375, 376, 377, these basic functions are the same as q for arrays of sensors or examples. Have a vertical spatial coordinate p defined according to the spatial coordinate system of one of the regions 300.

The 2-D AFT calculation is used only for the selected sample, so for each selected sample the associated basic function has a value of +1 at the position of the sample. This sampling pattern allows the simplified assumption that the pre-scaled input sensor signal needs to be multiplied by a coefficient of +1, 0 or -1 when calculating the AFT coefficient x _{k , 1} , and according to Equation 10 Uses the 2-D Mobius function μ ₂ (m, n).

10 detects an incoming signal (e.g., a light pattern received from a scene being imaged) and processes the signal to output the respective filter in equation (14). It shows with the filter apparatus 1022 which calculates S (n, m). The sensor array portion 1004 has a sensor 1002 positioned at a preferred position for AFT calculation, which positions are defined to have vertical and horizontal distances relative to the corner pixels 1028, which distance is defined by the array portion 1004. Equivalent to various Parlay fountains multiplied by size 1032. Optionally, such filtering may be performed by analog circuit 1022 as illustrated in FIG. 10 or digital filter 1502 as illustrated in FIG. 15. In any case, column selection operations are preferably performed by column selector 1036 under the control of microprocessor 1018, with each filter output S (n, m) being stored in a memory element such as RAM 1016. do.

Regardless of whether the analog filter 1022 is used or the digital filter 1502 is used to calculate the filter output S (n, m), the illustrated arrangement can be calculated according to the example process shown in FIG. . In the illustrated process, an incoming signal, such as a light pattern from the scene, is received by the sensor array 1004 (step 1102). The incoming signal is detected by each sensor 1002 of the array 1004 to generate a sensor signal (step 1104), which is received by the analog filter device 1022 or the digital filter device 1502 (step 1106). Each weighted sum of each set of sensor signals is obtained to produce each filtered signal (step 1118). For example, the weighted sum of the sensor signal set (e.g., the weighted sum of each pixel value 1028, 1029, 1030, 1031 from the intersection of rows 1024, 1026 and columns 1044, 1046) is filtered. Obtained by 1022 or 1052, produces a filtered signal S (2,3) (step 1118).

For analog filter device 1022, the weighted sum obtained in steps 1108 and 1110 may be generated according to the process shown in FIG. In the filtering process 1108 or 1110 shown, the signal from each sensor is amplified with a suitable gain to generate each amplified signal (step 1208). For example, the signal from the first sensor 1028 in row 1024 and column 1044 is amplified with a first gain to generate a first amplified signal (step 1202), and the second sensor 1029 in row 1024 and column 1046. The signal from is amplified with a second gain to produce a second amplified signal (step 1204), with the remainder equally. The resulting amplified signal is integrated to generate a filtered signal (step 1206).

가 선택된다. 그리고나서, n에 대한 값이 주어지면, 적합한

및

가 선택된다. 필터 S(3,12)를 계산하기 위한 일례의 타이밍 사이클은 다음과 같다:The operation of the analog filtering circuit 1022 shown in FIG. 10 will be described with reference to the timing diagram illustrated in FIG. 17. First, microprocessor 1018 determines which filter is to be calculated. That is, select values for n and m. Given a value for m, we match the appropriate column with

Is selected. Then, given a value for n,

And

Is selected. An example timing cycle for calculating filter S (3,12) is as follows:

1.n = 3, m = 12,

여기서 i=1,2,4,5,6,7,12,2.

Where i = 1,2,4,5,6,7,12,

3. Select column 0

다른

4.

Other

5. transfer the charge to the integrator 1010,

6. Select column 1/6

다른

7.

Other

8. Pass the charge to the integrator 1010,

9. Select column 1/3

다른

10.

Other

11. Transfer charge to integrator 1010,

12. Select Column 1/2

다른

13.

Other

14. Pass the charge to the integrator 1010,

15. Select column 2/3

다른

16.

Other

17. transfer the charge to the integrator 1010,

18. Select column 5/6

다른

19.

Other

20. Pass the charge to the integrator 1010,

21. Sampling the output of the integrator,

, 여기서 i=1,2,3,4,5,6,722.

, Where i = 1,2,3,4,5,6,7

23. Pass the charge to integrator 1012,

24. Perform AD conversion using ADC 1014 and store digital value S (3,2) in RAM 1016

25. Reset Integrator 1010 and Amplifier 1012

After each filter output S (m, n) is obtained, a 2-D AFT coefficient is obtained (step 1112). To find the AFT coefficients (step 1112), the filter output is weighted using the appropriate value of the Mobius function as described with reference to equations 15a-15d above (step 1114), and the resulting weighted signal is Are added / summed according to equations 15a-15d (step 1116).

If digital filter 1502 is used, it is preferred that each signal from sensor 1002 in array 1004 is amplified by amplifier 1006, as shown in FIG. 15, and the resulting amplified signal. Note that is received by the digital filter 1502 and processed. Those skilled in the art will be familiar with a number of commercially available and individually programmable specialty digital filters that can be easily programmed by practitioners of ordinary skill in order to perform the mathematical operations described above. will be. Since the resolution of the analog-to-digital converter (ADC) 1014 in a typical image sensor system is 12 bits or less, a 16-bit digital signal processor is suitable for use as the digital filter 1502.

The 2-D AFT is based on the assumption that the average intensity value (also referred to as the "DC" value) of the complete partial-picture is zero, as is the average value for each row and column. If there is a non-zero DC value for the row, column or the entire partial-picture, then that value is preferably used to find the coefficient value for adjusting the appropriate filter output S (n, m). Suitable correction amounts for the case where the entire partial-picture has a non-zero mean E [A] are as follows:

, 및

, And

In addition, if the input signal has a non-zero mean value in any of the rows or columns (ie, if X _{k , 0} or X _{0 , 1} is non-zero), the amount of correction must be calculated. In the case of non-zero mean values in a column or row, it is sufficient to calibrate only X _{k , 1} , where k = 1,2, ..., N-1, and l = 1,2, ... , N-1. The calibration formula is:

Then, the correction coefficients Δ (k, l) and Δ _local (k, l) are added to the uncorrected 2-D AFT coefficients x _{k , 1} , so that the corrected 2-D AFT coefficients A _c (k , l)

In the example shown, an 8x8 DCT case will be considered. It is not necessary to accurately determine the mean value of each of the entire unit region partial-images and the mean value of each of the sporadic rows and columns. Rather, it is sufficient to use estimates of these mean values. For the mean value E [A] of the entire partial-picture A, the closest estimate via the least mean-square error is provided by the filter output averaging the maximum number of points. In general, in the case of N × N, this is S (N, N). In the case of 8x8 DCT, the best estimate of the average E [A] of the full partial-picture A is

For the 8 × 8 DCT case, the overall DC correction value of the result for each 2-D AFT coefficient based on Equations 16a and 16b is provided in Table 1:

The correction values for each 2-D AFT coefficient when there are non-zero column averages and / or row averages are provided in Table 2:

Given the extended partial-picture X and the 2-D AFT coefficient _{xk , l} of the corrected AFR coefficient A _c (k, l), the DCT coefficient of the partial-picture A can be calculated. The relationship between each 8x8 point DCT coefficient DCT (k, l) and its corresponding corrected 2-D AFT coefficient A _c (k, l) is provided in Table 3:

If the image signal to be sampled has a high spatial frequency component that does not correspond to an integer multiple of the unit spatial frequency, aliasing may occur in which a predetermined error occurs in the DCT coefficient calculated using the AFT algorithm. For example, as shown in FIG. 2, there is a sub picture boundary 204 in the expanded sub picture 202 derived from the original sub picture 102. At each of these boundaries, there may be a discontinuity characteristic that is the first derivative of pixel brightness. This discontinuity tends to increase as the input signal frequency approaches half of the Nyquist sampling frequency. In addition, this discontinuity tends to increase as the phase of the input signal approaches π / 2. If there is a substantial discontinuity characteristic, the enlarged sub picture 202 will have significant Fourier components at frequencies that are more than half of the Nyquist frequency. If the Nyquist criterion is violated due to the undersampling of an image signal or other signal, harmonics, i.e. components that violate the Nyquist criterion, are "fold back" so that they appear at frequencies below half the Nyquist frequency. It is well known to be. With image expansion as shown in Fig. 2, when the input signal is uniformly sampled in the eighth step of the unit period, no aliasing effect occurs. However, due to the inhomogeneous placement of samples having positions based on the Parlay fraction as described above, aliasing errors may occur in the DCT coefficients calculated based on the AFT algorithm. The mean square error between the uniformly sampled input signal value and an approximation of this signal indicates the accuracy of the process based on the AFT algorithm. Here, the approximation value is calculated by taking the inverse DCT of the DCT coefficients based on the AFT algorithm. The amount of error may be important when processing image signals having substantial high frequency components. Exemplary results as a frequency function of the mean squared error are illustrated in FIG. 4, which shows the mean squared error of the approximation signal obtained by taking the inverse DCT of the exemplary DCT coefficients derived by the AFT technique described above as a frequency function. It is shown. The illustrated results indicate that this error is maximum in the high frequency components.

The live error by undersampling not only directly affects the accuracy of the filter output S (n, m) before any DC correction is applied, but also the accuracy of the DC correction itself.

An improved estimate of the mean value of the picture can be obtained from the output of filter S, which output S is an average of a set of points taken at unpredictable spatial frequencies as presented in the spectrum of extended picture X. P. Paparao), 1990 SPIE Volume 2 by A. Ghosh, "An Improved Arithmetic Fourier Transform Algorithm" in Optical Information-Processing Systems and Architectures II . As can be seen from the conclusion of the above-mentioned document, increasing the order of the filter S used to calculate the mean value can improve the mean value estimate. Therefore, the mean squared error should be reduced when a filter with an order of 8 or more is used to estimate the mean value in the above mentioned 8x8 DCT case. Since the averaged density and number of light-sensing elements increase when higher order filters are used, it is necessary to select the filter with the highest degree possible, as limited by manufacturing techniques. Certain manufacturing techniques limit the shortest distance between photosensitive elements, thereby limiting the highest filter order possible. In order not to greatly increase the number of photosensitive elements, the order of the filter should be able to be divided by at least one lower order. If the order of the filter can be divided by the lower order, the number of additional photo-sensing elements does not increase substantially since the parlay fraction of the filter with the lower order matches the subset of the parlay fraction associated with the filter with the higher order. Will not. A typical example is filter S (12, 12), where 12 can be divided by 2, 3, 4, 6. A filter with order 12 does not require a greater number of photosensitive elements than a filter with order 8. However, in the case of a filter of order 12, it is preferable that the light sensing element is more closely filled in a predetermined portion of the sub-picture, as shown in FIG. In general, in a filter of order N, the sample positions may be placed at positions 2j / N, where j = 0, 1, ... N / 2. The estimated mean squared error is shown in Figure 6, where a filter of order 12 is used to estimate the global and local mean values.

Optionally, a photosensitive device at the location of the correct Parlay fraction can be used to obtain sample values for higher order filter calculations used to estimate global and local DC values. Alternatively or in addition, sample values can be obtained by performing interpolation processing of neighboring samples using the interpolation method described in detail below. In addition, a filter with an order of 12 or more can be used to estimate the DC value. However, there is a tradeoff in using higher order filters. That is, such a filter may involve an increase in the number of photosensitive elements and / or a reduction in the spacing between the photosensitive elements. In addition, increasing the order of the filter to 12 or more typically does not provide significant additional gain. For example, FIG. 7 shows the mean squared error of a system using a filter of order 16 or greater to estimate global and local DC values. Comparing Figs. 6 and 7 visually, it can be seen that the errors are approximately the same in both cases. Thus, it is evident that an order 12 filter provides an improved tradeoff between the number of sample points (or pixel density) and the overall accuracy.

Aliasing errors in non-DC corrected filters (non-DC corrected filters) can be reduced by adding pixels to the sensor array if the manufacturing technique allows for a sufficiently dense pixel distribution. To correct this aliasing, an AFT coefficient higher than the equivalent uniform sampling frequency (i.e., an order of 8 or higher in the case of 8x8 DCT) may be used to correct the lower coefficient. The higher order coefficients can be obtained directly from additional fraction-spaced sensors, interpolated from neighboring pixels, or estimated as fractions of the lower order coefficients described in detail below.

By adding pixels at the exact parlay fractional position, the higher order AFT coefficients can be calculated accurately, which can be used to correct the lower order AFT coefficients. For example, M is the number of DCT coefficients, and the highest achievable order of the parlay fractional space is N. Here, N is smaller than M (N> M), and when M = 8, N is 9, 10, 11, 12, ... First, the global and local DC coefficients Δ ( k, l ) and Δ _local ( k, l ) are estimated using the highest order N, as described above, and uncorrected AFT as shown in equations 18a and 18b. Is added to the coefficient x _{k , l} . The uncorrected DC AFT coefficient A _C (k, l) as a result is used to determine the aliasing correction value. Where k, l = 0, 1, 2, ... N-1.

The correction formulas in equations 20a-20d are valid when M is even (typical) and 2M is greater than N. The aliased-corrected 2-D AFT coefficient A _CC (k, l) can be calculated by adding the above-listed aliasing correction values to the DC-corrected AFT value A _C (k, l).

FIG. 8 shows the estimated mean square error when the upper parley fraction sample is used to correct for aliasing. In this example, an order 12 (ie, N = 12) parlay fractional sample space was used to provide pixel values, an order 12 filter was used to estimate global and local DC values, and higher order AFT coefficients. (Coefficients of order 8, 9, 10, 11) were used to correct for aliasing as described above with respect to equations 20a-20d. In this example, the estimated maximum mean square error is a value at frequencies 6.5, 6.5, which is 0.0273.

In Figures 4 and 6-8, the estimated mean square error, the input image X is the frequency _{(f 1/2, f 2} /2) 2-D cosine f _1, each frequency point (corresponding to f with Derived by making assumptions about ₂ ). The 2-D DCT coefficients based on the 2-D AFT were calculated as input and the inverse 2-D DCT was calculated to obtain the image Y. The mean square error between image X and image Y is calculated and assigned to frequency points f ₁ and f ₂ .

The preferred number of image samples used for AFT calculation tends to increase substantially as the order of the parlay fractional space increases. For example, when N = 12, a total of 46 photosensitive devices are used per unit section. Producing an image sensor having such a pixel density is not practical and expensive when the higher order AFT coefficient is preferably estimated by interpolation of adjacent pixels. The parlay sampling point used for filters of order M, M + 1, ... N-1 is determined from the set of available samples or a particular filter, preferably filter N of the highest order (order 12 in the example given above). Interpolation from a set of samples processed by an in filter). In either case, the illustrated interpolation system is described in detail below.

As a further method for calculating the higher order 2-D AFT coefficients (coefficients of orders 8, 9, 10, and 11), the coefficients of the higher order are calculated as fractions of neighboring higher orders. Specifically, one or more higher order coefficients are first calculated using accurate parlay sampling points, and other higher order coefficients can be estimated from these exact values as follows. Assuming that picture A is band-limited and has no frequency component greater than half the Nyquist frequency, the correlation between the Fourier series coefficients of each neighboring higher order is typically very high. It also indicates that the even Fourier series coefficients (in this example, coefficients 8, 10, 12) tend to be highly correlated with each other, and likewise the odd Fourier series coefficients (in this example, coefficients 9, 11) are highly correlated with each other. Simulations indicate that they tend to be. Accordingly, when the Fourier coefficients of one even higher order are known, the coefficients of another even higher order can be estimated. Similarly, if the Fourier coefficients of one higher order are known, the coefficients of the other higher order can be estimated. For example, if an order 7 parley fraction space and an order 12 filter are used, the order 9 filter may be used to estimate the even higher order coefficients. An example of a suitable sample space for this estimation (for aliasing correction) is shown in FIG. 9, where the position of the light sensing element is defined as a parlay fraction 2j / n. Where j = 0, 1, 2, ... n-1, and n = 1, 2, 3, 4, 5, 6, 7, 9, and 12.

In addition, a single system may comprise (a) adding a sensor at a higher order parley fractional position, and (b) interpolating a value from an existing sensor to estimate the value of the signal input at the appropriate higher order position. The techniques mentioned can be combined. For example, as shown in FIG. 9, when the desired higher order pixel position 906 is very close to the lower order pixel position 904, and when there is a sensor at the lower order position 904, the upper order Rather than placing the sensor at the position 906 of the order, it may be desirable to calculate the estimated value for the higher order pixels 906 by interpolation. However, if the desired higher order position 910 is farther away from the nearest lower order pixels 908, 912, it may be desirable to add an extra sensor to the sensor array at the higher order position 910. Can be.

16 schematically illustrates an example of a process for image sensing and processing in accordance with the present invention. The pixel value 1602 is processed to calculate the filter S (n, m) according to Equation 14 above (step 1604). The set of uncorrected AFT coefficients x _{k , l} is calculated based on the filter values S (n, m) (step 1606). If the entire image and each matrix has a non-zero DC component, average value correction is not necessary (step 1608). Thus, the AFT coefficient x _{k , l} is rated for power to derive the DCT coefficient 1618, as illustrated in equations 15e-15h (step 1616). However, if the mean value correction is appropriate (step 1608), the mean value correction amount is calculated (step 1610) and used to correct the AFT coefficient x _{k , l} for the corrected coefficient Ac (k, l) (step 1612). If no aliasing correction is needed (step 1614), the process proceeds to step 1616. However, if aliasing correction is appropriate (step 1614), the aliasing correction is calculated as mentioned above (step 1620), and the DC corrected AFT coefficient A _c to derive the aliased corrected coefficient A _CC (k, l). (k, l) is used to further correct (step 1622). DCT coefficient 1618 is calculated based on the aliased corrected AFT coefficient A _CC (k, l) (step 1616).

As discussed above, interpolation of values measured from neighboring sensors in the sensor array can be used to estimate the value of a pixel adjacent to the sensor's location. For example, referring to the unit area 300 shown in FIG. 3, a Parlay fraction when the AFT method of the present invention can be used in a conventional sensor array with sensors located at evenly spaced locations 398. An interpolation method can be used to estimate the value of the image at location 348 based on. If this calculation method is performed by a digital signal processor such as the digital filter 1502 shown in Fig. 15, the calculation of the value at a particular parlay fractional position 345 is, for example, the nearest and evenly spaced position 394. , 395, 396, 397 can be performed by calculating an average value of each value generated by the sensor located at.

13 shows an example of a process of deriving AFT algorithm using interpolated pixel values. In the illustrated process, the input image signal is received by the sensor array (step 1302). This sensor array can be, for example, a conventional array comprising sensors with uniformly distributed spatial positions. The input signal is detected by the sensors of the array to generate a plurality of sensor signals (step 1304). This sensor signal is received by an interpolation circuit that interpolates the sensor signal (step 1308), for example by averaging the signal (step 1306), thereby interpolating a pixel value at a position defined by the parlay fraction as described above. Will generate a set of signals. The interpolated signal is received by a filter device such as the analog filter 1022 shown in FIG. 10 or the digital filter 1502 shown in FIG. 15 (step 1310). Filter 1022 or 1502 derives each weighted sum of each interpolated signal set to produce each filtered signal (step 1316). For example, the weighted sum of the first set of interpolated signals is derived to produce a first filtered signal (step 1312), and the weighted sum of the second set of interpolated signals generates a second filtered signal. It is directed to (step 1314). When using the analog filter device 1022, the weighted sum derived in steps 1312 and 1314 can be written according to the process shown in FIG. In the illustrated filtering process 1312 or 1314, the interpolated signal from a particular matrix is amplified with an appropriate gain to produce each amplified signal (step 14080. For example, the interpolated first signal is a first gain). Is amplified to generate a first amplified signal (step 1402), and the interpolated second signal is amplified to a second gain to generate a second amplified signal (step 1404), as a result of which the amplified signal is integrated and filtered. Generate a signal (step 1406).

Once each filter output S (m, n) is derived, a 2-D AFT coefficient is derived (step 1112). To derive the AFT coefficients (step 1112), the filter output is weighted using the appropriate values of the Mobius function as described for equations 15a-15d above (step 1114), and the resulting weighted signal is It is added according to 15a-15d (step 1116).

Further improving the computational efficiency can be achieved using analog circuitry to perform the above-mentioned interpolation. 18 illustrates an example for interpolating pixel values from sensor 1806 of sensor array portion 1802 to derive additional pixels 1808, 1810, 1812 (pixels in rows 1814 and columns 1816). Analog interpolation circuit 1804 is shown and used in the AFT calculation according to the present invention. To interpolate pixels 1808 in row 1814, pixels 1826 in rows 1818 and 1820 are used. Similarly, to interpolate pixels 1810 in columns 1816, pixels 1828 in columns 1822 and 1824 are used. The target pixel does not necessarily need to be spaced apart by an equivalent distance from its neighboring pixels, but it can be estimated to have an equivalent distance with an error of 0.5%. Thus, each interpolated pixel value is approximated as the average of two neighboring pixel values. There is a case where the pixel 1812 is located at the intersection of the row 1814 and the column 1816. This pixel value may be interpolated as an average of four neighboring pixels (pixel values 1830 at intersections 1818, 1822, 1818, 1824, 1820, 1822, and 1820, 1824). If the target pixel is at an equivalent distance from neighboring pixels, the maximum number of sampling capacitors can be used.

An exemplary timing cycle for calculating filters S (3, 12) using interpolation circuit 1804 is as follows.

1.n = 3, m = 12,

2.Φ ³ _int = 1, Φ ⁱ _int = 0, where i = 1,2,4,5,6,7,12,

3. Select column 0

4.Φ ² _s2 = 1, Φ ⁸ _s2 = 1, Φ ⁱ _sj = 0

5. Transfer charge to integrator (1832), Φ _t = 1, Φ ⁱ _sj = 0

6. Select column 1/6

7.Φ ¹ _sl = 1, Φ ⁸ _sj = 1, Φ ⁱ _sj = 0

8. Transfer charge to integrator (1832), Φ _t = 1, Φ ⁱ _sj = 0

9. Select column 1/3

10.Φ ¹ _s1 = 1, Φ ⁸ _s1 = 1, Φ ⁱ _sj = 0

11. Transfer charge to integrator (1832), Φ _t = 1, Φ ⁱ _sj = 0

12. Select column 1/2

13.Φ ¹ _sl = 1, Φ ⁸ _sj = 1, Φ ⁱ _sj = 0

14. Transfer charge to integrator (1832), Φ _t = 1, Φ ⁱ _sj = 0

15. Select column 2/3

16.Φ ¹ _s1 = 1, Φ ⁸ _s1 = 1, Φ ⁱ _sj = 0

17. Transfer charge to integrator (1832), Φ _t = 1, Φ ⁱ _sj = 0

18. Select Column 4/5-Interpolation Column

13. Note that the values in Φ ¹ _s2 = 1, Φ ⁸ _s2 = 1, Φ ⁱ _sj = 0, column 4/5 are sampled with gain 2, not gain 4.

20. Transfer charge to integrator (1832), Φ _t = 1, Φ ⁱ _sj = 0

21. Select Column 6/7-Interpolation Column

22. Note that the values in Φ ¹ _s2 = 1, Φ ⁸ _s2 = 1, Φ ⁱ _sj = 0, column 6/7 are sampled with gain 2, not gain 4.

23. Transfer charge to integrator (1832), Φ _t = 1, Φ ⁱ _sj = 0

24. Sampling integrator output, Φ _s3 = 1

25.Φ ¹² _amp = 1, Φ ⁱ _amp = 0, where i = 1,2,4,5,6,7

26. Transfer charge to amplifier 1834, Φ _s3 = 0, Φ _t3 = 1

27. Perform AD conversion using ADC 1836 and store digital values S (3, 12) in RAM 1838

28. Reset the integrator and amplifier

Table 4 shows a comparison of the computational efficiencies of the various methods for calculating 1-D, 8-point DCT, including the AFT method of the present invention. This comparison is expressed in terms of the respective numbers of the various types of operations used to calculate the 1-D DCT.

Conventional DCT High Speed DCT in Chen 1997 Fast DCT at [Narashima 1978] AFT Addition 56 26 NA 33 multiplication 42 16 18 7 Full Computation (8 points) 2-D DCT 1820 282 350 82 (8x8) 2-D DCT Normalized to Full Math AFT 493 12 18 One

As can be seen from Table 4, in terms of the total number of operations, the AFT method of the present invention is approximately 3.4 times more efficient than the most efficient conventional method for calculating 1-D DCT. Also, since the total number of operations in the 2-D DCT is proportional to approximately the square of the number of calculations in the 1-D DCT, the AFT method of the present invention is the most efficient conventional method for calculating the 2-D DCT. It is about 12 times more efficient. In addition, since the multiplication in the AFT calculation includes the prescaling of each pixel intensity (brightness) by an integer value, these multiplications can be easily implemented using an analog circuit such as the filter 1022 shown in FIG. have. By effectively eliminating most digital multiplications, this analog filter 1022 allows the AFT system of the present invention to use 73 times less computation than the most effective conventional system.

While the present invention has been described in connection with specific exemplary embodiments, it will be appreciated that various modifications and variations can be made in accordance with the disclosed embodiments without departing from the spirit and scope of the invention as set forth in the appended claims.

Appendix A

This Appendix A demonstrates the following relationship.

The output of the filter is

And the Fourier series expansion of the image A is given by the equations (6) and (7), which is expressed again as follows.

Therefore, the filter's output formula (Equation (A-2)) can be rewritten as

By rearranging the sum order, equation (A-4) can be represented again as equation (A-5).

Substituting the relation (A-6), the output of the filter is as shown in the formula (A-7).

Appendix B

This appendix demonstrates the following relationship.

The Kronecker function is defined as

The values of m and n are positive integers and the summation is performed on all positive integer values m / n, which correctly divides the positive integers.

In order to prove the relationship in equation (B-1), equations (B-3) and (9) can be used to derive the following relationship.

Appendix C

2D DCT and 2D AFT Coefficient Equivalence

Image X is an expanded version of the unit area sub image A (as shown in FIG. 1).

According to the two-dimensional case of the Nyquist reconstruction formula, the continuous picture X can be represented by the samples as follows.

Without compromising generality, the sampling period T is assumed to be equal in both dimensions and equal to 1/8. As a result, there are 16x16 samples. It is assumed that the image X is periodic in units of 2x2 periods. Therefore, formula (C-1) can be expressed as follows.

Using the dual form of the inverse Fourier transform and the Poisson formula, we can see that the sum of the sine functions is equal to the right side of equation (C-3).

Based on Formula (C-3), Formula (C-2) can be expressed as follows.

Since X (p, q) is an extension of the image A (p, q), as shown in equation (C-5), equation (C-4) can be represented again by equation (C-6).

The product term of the cosine function is

If the order of addition is rearranged by replacing the equator with the formula (C-7), the formula (C-6) becomes as follows.

It can be seen from the formula (C-8) that the (n, m) sum term does not depend on the signs of k and l. Also according to the definition of the two-dimensional DCT given in equation (C-10), equation (C-8) can be expressed as follows.

The definition of the two-dimensional DCT is as follows.

here

Equation (C-9) can therefore be expressed as

Further, the expanded image X (p, q) can be represented as follows by the two-dimensional Fourier series.

Where x _{k , l} (k, l = 1,2..8) are the 2D AFT coefficients of the extended image X. The second and third terms of equation (C-12) are due to the presence of local row and column non-zero mean values. The coefficients in the second term are calculated as the 1D AFT of the row-average and the coefficients in the third term are calculated as the 1D AFT of the column-average.

Substituting the orthogonal cosine functions in the formulas (C-11) and (C-12) and the two formulas of the image X (t, τ), the 2D AFT and DCT coefficients are excluded from the multiplying constant factors in the respective DCT coefficients. We can conclude that is the same.

Claims (30)

In the sensing device, A sensor array having at least a first sensor and a second sensor and associated with a spatial coordinate system, wherein the first sensor has a first sensor location and the second sensor has a second sensor location, the first sensor location being Close to the location of the first extremum of at least one basis function of the domain transform, the at least one basis function having at least one spatial coordinate defined according to the spatial coordinate system, A sensor array, wherein the sensor position is close to a position of a second extreme value of the at least one basis function; And At least one filter coupled to receive a signal from the first sensor, coupled to receive a signal from the second sensor, and configured to generate a first filter output signal, the first filter output At least one filter, the signal comprising a weighted sum of at least a signal from the first sensor and a signal from the second sensor Detection device comprising a. The method of claim 1, And the domain transform comprises at least one of a Fourier transform and a cosine transform. The method of claim 2, The first sensor position has a first distance from a reference position of the unit cell of the sensor array, the unit cell has a unit cell size, the second sensor position has a second distance from the reference position, Wherein the first distance is essentially equal to the product of the unit cell size and the first parlay fraction, and the second distance is essentially the same as the product of the unit cell size and the second parlay fraction. The method of claim 1, The first sensor position has a first distance from a reference position of the unit cell of the sensor array, the unit cell has a unit cell size, the second sensor position has a second distance from the reference position, And the first distance is essentially equal to the unit cell size and the product of the first parlay fraction, and the second distance is essentially the same as the unit cell size and the product of the second parlay fraction. The method of claim 4, wherein The sensor array further includes a third sensor and a fourth sensor, wherein the third sensor has a third sensor position and the fourth sensor has a fourth sensor position, the third sensor position from the reference position. Have a third distance and the fourth sensor position has a fourth distance from the reference position, the third distance is essentially equal to the product of the unit cell size and the third parlay fraction and the fourth distance is the unit cell size Essentially the same as the enemy of and the fourth parlay fraction, The at least one filter, A first filter coupled to receive a signal from the first sensor and coupled to receive a signal from the second sensor and configured to generate the first filter output signal; Coupled to receive a signal from the third sensor and coupled to receive a signal from the fourth sensor, the weighted sum of at least a signal from the third sensor and a signal from the fourth sensor A second filter configured to generate a second filter output signal; And A third filter coupled to receive the first and second filter output signals and configured to generate a third filter output signal Including; Wherein the third filter output signal includes a sum of the product of the first filter output signal and the first value of the Mobius function and the product of the second filter output signal and the second value of the Mobius function. Device. The method of claim 1, The sensor array further includes a third sensor and a fourth sensor, the third sensor has a third sensor position, the fourth sensor has a fourth sensor position, and the third sensor position is the at least one base. Close to the position of the third extreme value of the function and the fourth sensor position is close to the position of the fourth extreme value of the at least one basis function, The at least one filter, A first filter coupled to receive a signal from the first sensor and coupled to receive a signal from the second sensor and configured to generate a first filter output signal; A second filter coupled to receive a signal from the third sensor and coupled to receive a signal from the fourth sensor, the second filter being configured to generate a second filter output signal, the second filter output signal being A second filter comprising a weighted sum of at least a signal from the third sensor and a signal from the fourth sensor; And A third filter coupled to receive the first and second filter output signals and configured to generate a third filter output signal Including; Wherein the third filter output signal includes a sum of the product of the first filter output signal and the first value of the Mobius function and the product of the second filter output signal and the second value of the Mobius function. Device. The method of claim 6, The first, second, third and fourth sensors are included in a plurality of sensors, wherein the plurality of sensors further comprise fifth, sixth, seventh and eighth sensors, wherein the plurality of sensors comprise a plurality of sensors. Configured to generate a sensor signal, the plurality of sensor signals including respective signals from the first, second, third, and fourth sensors, wherein the plurality of sensor signals include fifth, sixth, and fifth signals. Further comprising respective signals from the seventh and eighth sensors, The at least one filter, Are coupled to receive respective signals from the fifth and sixth sensors and are configured to generate a fourth filter output signal comprising a weighted sum of at least respective signals from the fifth and sixth sensors A fourth filter; Coupled to receive respective signals from the seventh and eighth sensors, and configured to generate a fifth filter output signal comprising a weighted sum of at least respective signals from the seventh and eighth sensors A fifth filter; And A sixth filter coupled to receive the fourth and fifth filter output signals and configured to generate a sixth filter output signal More, The sixth filter output signal includes a sum of the product of the fourth filter output signal and the third value of the Mobius function, and the product of the fourth filter output signal and the product of the fourth value of the Mobius function, The sensing device further comprises a first correction circuit configured to generate a first correction signal, Wherein the first correction signal comprises (a) the sum of the values of the Mobius function, (b) (i) the mean of the plurality of signals and (ii) the product of at least one of the sixth filter output signals, And the first correction circuit is further configured to generate a first corrected filter output signal comprising a sum or a difference between the third filter output signal and the first correction signal. The method of claim 7, wherein The plurality of sensors further includes ninth, tenth, eleventh, and twelfth sensors, wherein the plurality of sensor signals further include respective signals from the ninth, tenth, eleventh, and twelfth sensors; , The at least one filter, Are coupled to receive respective signals from the ninth and tenth sensors and are configured to generate a seventh filter output signal comprising a weighted sum of at least respective signals from the ninth and tenth sensors. Seventh filter; Coupled to receive respective signals from the eleventh and twelfth sensors, and configured to generate an eighth filter output signal comprising a weighted sum of at least respective signals from the eleventh and twelfth sensors; An eighth filter; And A ninth filter coupled to receive the seventh and eighth filter output signals and configured to generate a ninth filter output signal More, The ninth filter output signal includes a sum of the product of the seventh filter output signal and the fifth value of the Mobius function, and the product of the eighth filter output signal and the product of the sixth value of the Mobius function, The sensing device further comprises a second correction circuit configured to generate a second correction signal, The second correction signal includes an eighth filter output signal, The second correction circuit is further configured to generate a second corrected filter output signal comprising a sum or a difference between the first corrected filter output signal and the second corrected signal. . The method of claim 6, The first, second, third and fourth sensors are included in a plurality of sensors, wherein the plurality of sensors further comprise fifth, sixth, seventh and eighth sensors, wherein the plurality of sensors comprise a plurality of sensors. Configured to generate a sensor signal, the plurality of sensor signals including respective signals from the first, second, third, and fourth sensors, wherein the plurality of sensor signals include fifth, sixth, and fifth signals. Further comprising respective signals from the seventh and eighth sensors, The at least one filter, Are coupled to receive respective signals from the fifth and sixth sensors and are configured to generate a fourth filter output signal comprising a weighted sum of at least respective signals from the fifth and sixth sensors A fourth filter; Coupled to receive respective signals from the seventh and eighth sensors, and configured to generate a fifth filter output signal comprising a weighted sum of at least respective signals from the seventh and eighth sensors A fifth filter; And A sixth filter coupled to receive the fourth and fifth filter output signals and configured to generate a sixth filter output signal More, The sixth filter output signal includes a sum of the product of the fourth filter output signal and the third value of the Mobius function, and the product of the fourth filter output signal and the product of the fourth value of the Mobius function, The sensing device further comprises a first correction circuit configured to generate a correction signal, The correction signal includes a sixth filter output signal, And the correction circuit is further configured to generate a corrected filter output signal comprising a sum or a difference between the third filter output signal and the correction signal. The method of claim 1, The at least one filter, A first amplifier coupled to receive the signal from the first sensor, the first amplifier generating a first amplifier output signal; A second amplifier coupled to receive the signal from the second sensor, the second amplifier generating a second amplifier output signal; And An integrator coupled to receive and integrate the first and second amplifier output signals to generate an integrated signal Including; And the first filter output signal comprises the integrated signal. The method of claim 1, And the at least one filter comprises a digital filter. In the sensing device, A plurality of sensors having at least a first sensor and a second sensor, the sensor array associated with a spatial coordinate system, wherein the first sensor has a first sensor position and the second sensor has a second sensor position; The sensor array is coupled to receive an incoming signal, the incoming signal having a first incoming signal value at the first sensor location and having a second incoming signal value at the second sensor location; And An interpolation circuit coupled to receive a signal from the first sensor and coupled to receive a signal from the second sensor, wherein the signal from the first sensor represents the first incoming signal value The signal from the two sensors represents a second incoming signal value, and the interpolation circuit is configured to generate a first interpolated signal by interpolating a signal from the first sensor and a signal from the second sensor, and The first interpolated signal represents an approximation of the incoming signal at a position close to the first extreme value of the at least one basis function of the domain transform, wherein the at least one basis function comprises at least one spatial coordinate defined according to the spatial coordinate system. With interpolation circuit Detection device comprising a. The method of claim 12, And the domain transform comprises at least one of a Fourier transform and a cosine transform. The method of claim 12, The interpolation circuit interpolates a first set of at least two signals from the plurality of sensors to generate a second interpolated signal, the second interpolated signal being close to a second extreme value of the at least one basis function. An approximation of the incoming signal at a location, The sensing device further comprises at least one filter configured to receive the first and second interpolated signals, And the at least one filter is configured to generate a first filter output signal comprising a weighted sum of at least the first and second interpolated signals. The method of claim 14, The interpolation circuit interpolates a second set of at least two signals from the plurality of sensors to generate a third interpolated signal, the third interpolated signal being close to a third extreme value of the at least one basis function. Representing an approximation of the incoming signal at a location, wherein the interpolation circuit interpolates a third set of at least two signals from the plurality of sensors to generate a fourth interpolated signal, the fourth interpolated signal being at least An approximation of the incoming signal at a position close to the fourth extreme of one basis function, The at least one filter, A first filter coupled to receive the first interpolated signal from the interpolation circuit and coupled to receive a second interpolated signal from the interpolation circuit, the first filter configured to generate the first filter output signal; Is coupled to receive the third interpolated signal from the interpolation circuit and is coupled to receive the fourth interpolated signal from the interpolation circuit, and wherein at least a weighted sum of the third and fourth interpolated signals A second filter configured to generate a second filter output signal comprising; And A third filter coupled to receive the first and second filter output signals and configured to generate a third filter output signal Including; Wherein the third filter output signal includes a sum of the product of the first filter output signal and the first value of the Mobius function and the product of the second filter output signal and the second value of the Mobius function. Device. The method of claim 14, The at least one filter, A first amplifier coupled to receive the first interpolated signal from the interpolation circuit, the first amplifier generating a first amplifier output signal; A second amplifier coupled to receive the second interpolated signal from the interpolation circuit to generate a second amplifier output signal; And An integrator coupled to receive and integrate the first and second amplifier output signals to generate an integrated signal Including; And the first filter output signal comprises the integrated signal. The method of claim 14, And the at least one filter comprises a digital filter. In the detection method, Receiving an incoming signal by a sensor array comprising at least a first sensor and a second sensor, wherein the first sensor has a first sensor location and the second sensor has a second sensor location, and the first sensor The position is close to the position of the first extreme value of at least one basis function of the domain transformation, wherein the at least one basis function has at least one spatial coordinate defined according to the spatial coordinate system, and the second sensor position is the Receiving an incoming signal, close to the location of the second extreme value of at least one basis function; Detecting the incoming signal by the first sensor to generate a first sensor signal; Detecting the incoming signal by the second sensor to generate a second sensor signal; Receiving the first sensor signal by the at least one filter; Receiving the second sensor signal by the at least one filter; And Generating, by the at least one filter, a first filtered signal comprising a weighted sum of at least the first sensor signal and the second sensor signal Detection method comprising a. The method of claim 18, The sensor array further includes a third sensor and a fourth sensor, the third sensor has a third sensor position, the fourth sensor has a fourth sensor position, and the third sensor position is the at least one base. Close to the position of the third extreme value of the function and the fourth sensor position is close to the position of the fourth extreme value of the at least one basis function, The detection method, Detecting the incoming signal by the third sensor to generate a third sensor signal; Detecting the incoming signal by the fourth sensor to generate a fourth sensor signal; Receiving the third sensor signal by the at least one filter; Receiving the fourth sensor signal by the at least one filter; Generating, by the at least one filter, a second filtered signal comprising a weighted sum of at least the third sensor signal and the fourth sensor signal; And A third filtered signal comprising, by the at least one filter, a sum of the product of the first filtered signal and the first value of the Mobius function and the product of the second filtered signal and the second value of the Mobius function Steps to raise Detection method further comprising. The method of claim 19, The first, second, third and fourth sensors are included in a plurality of sensors, and the plurality of sensors further include fifth, sixth, seventh and eighth sensors, and the first, second, The third and fourth sensor signals are included in the plurality of sensor signals, and the plurality of signals further include fifth, sixth, seventh and eighth sensor signals, The detection method, Detecting the incoming signal by the fifth sensor to generate a fifth sensor signal; Detecting the incoming signal by the sixth sensor to generate a sixth sensor signal; Detecting the incoming signal by the seventh sensor to generate a seventh sensor signal; Detecting the incoming signal by the eighth sensor to generate an eighth sensor signal; Receiving fifth, sixth, seventh and eighth sensor signals by the at least one filter; Generating, by the at least one filter, a fourth filtered signal comprising a weighted sum of at least the fifth and sixth sensor signals; Generating, by the at least one filter, a fifth filtered signal comprising a weighted sum of at least the seventh and eighth sensor signals; Generating a sixth filtered signal by the at least one filter; A sixth filtered by the at least one filter comprising a sum of the product of the fourth filtered signal and the third value of the Mobius function and the product of the fifth filtered signal and the product of the fourth value of the Mobius function Generating a signal; generating a first correction signal comprising (a) the sum of the values of the Mobius function, (b) (iii) an average of the plurality of signals and (ii) an enemy of at least one of the sixth filter output signal ; And Generating a first corrected filter output signal comprising a sum or a difference between the third filtered signal and the first correction signal Detection method further comprising. The method of claim 20, The plurality of sensors further includes ninth, tenth, eleventh, and twelfth sensors, and the plurality of signals further include ninth, tenth, eleventh, and twelfth sensor signals The detection method, Detecting the incoming signal by the ninth sensor to generate a ninth sensor signal; Detecting the incoming signal by the tenth sensor to generate a tenth sensor signal; Detecting the incoming signal by the eleventh sensor to generate an eleventh sensor signal; Detecting the incoming signal by the twelfth sensor to generate a twelfth sensor signal; Receiving ninth, tenth, eleventh, and twelfth sensor signals by the at least one filter; Generating, by the at least one filter, a seventh filtered signal comprising a weighted sum of at least the ninth and tenth sensor signals; Generating, by the at least one filter, an eighth filtered signal comprising a weighted sum of at least the eleventh and twelfth sensor signals; A ninth filtered by the at least one filter comprising a sum of the product of the seventh filtered signal and the fifth value of the Mobius function and the product of the eighth filtered signal and the sixth value of the Mobius function Generating a signal; Generating a second correction signal comprising the eighth filtered signal; And Generating a second corrected filter output signal comprising a sum or a difference between the first corrected filter output signal and the second corrected signal Detection method further comprising. The method of claim 19, The first, second, third and fourth sensors are included in a plurality of sensors, and the plurality of sensors further include fifth, sixth, seventh and eighth sensors, and the first, second, The third and fourth sensor signals are included in the plurality of sensor signals, and the plurality of signals further include fifth, sixth, seventh and eighth sensor signals. The detection method, Detecting the incoming signal by the fifth sensor to generate a fifth sensor signal; Detecting the incoming signal by the sixth sensor to generate a sixth sensor signal; Detecting the incoming signal by the seventh sensor to generate a seventh sensor signal; Detecting the incoming signal by the eighth sensor to generate an eighth sensor signal; Receiving the fifth, sixth, seventh and eighth sensor signals by the at least one filter; Generating, by the at least one filter, a fourth filtered signal comprising a weighted sum of at least the fifth and sixth sensor signals; Generating, by the at least one filter, a fifth filtered signal comprising a weighted sum of at least the seventh and eighth sensor signals; A sixth filtered by the at least one filter comprising a sum of the product of the fourth filtered signal and the third value of the Mobius function and the product of the fifth filtered signal and the product of the fourth value of the Mobius function Generating a signal; Generating a correction signal comprising the sixth filter output signal; And Generating a corrected filter output signal comprising a sum or a difference between the third filtered signal and the correction signal Detection method further comprising. The method of claim 18, Generating the first filtered signal Amplifying the first sensor signal to generate a first amplified signal; Amplifying the second sensor signal to generate a second amplified signal; And Integrating the first and second amplified signals to generate an integrated signal Including; And the first filtered signal comprises the integrated signal. The method of claim 18, The generating of the first filtered signal comprises digitally calculating a weighted sum of at least the first sensor signal and the second sensor signal. In the detection method, Receiving a pull signal by a sensor array associated with a spatial coordinate system, said first sensor having a first sensor position and said second sensor, said sensor comprising a plurality of sensors having at least a first sensor and a second sensor; A sensor having a second sensor location, the incoming signal having a first incoming signal value at the first sensor location, and the incoming signal having a second incoming signal value at the second sensor location; Detecting the incoming signal by the first sensor to generate a first sensor signal indicative of the first incoming signal value; Detecting the incoming signal by the second sensor to generate a second sensor signal indicative of the second incoming signal value; Receiving the first sensor signal by an interpolation circuit; Receiving the second sensor signal by the interpolation circuit; And Interpolating the first and second sensor signals by the interpolation circuit to generate a first interpolated signal Including; The first interpolated signal represents an approximation of the incoming signal at a location close to a first extreme of at least one basis function of a domain transform, And the at least one basis function has at least one spatial coordinate defined according to the spatial coordinate system. The method of claim 25, And the domain transform comprises at least one of a Fourier transform and a cosine transform. The method of claim 25, Interpolating, by the interpolation circuit, a first set of at least two signals from the plurality of sensors to generate a second interpolated signal, the second interpolated signal being the at least one basis function. Said first set interpolation step representing an approximation of said incoming signal at a position near a second extreme value of; Receiving the first and second interpolated signals by at least one filter; And Generating, by the at least one filter, a first filtered signal comprising a weighted sum of at least the first and second interpolated signals Detection method further comprising. The method of claim 27, Interpolating, by the interpolation circuit, a second set of at least two signals from the plurality of sensors to generate a third interpolated signal, the third interpolated signal being the at least one basis function. Said second set interpolation step representing an approximation of said incoming signal at a position near a third extreme value of; Interpolating, by the interpolation circuit, a third set of at least two signals from the plurality of sensors to generate a fourth interpolated signal, the fourth interpolated signal being the at least one basis; Said third set interpolation step representing an approximation of said incoming signal at a position close to a fourth extreme of a function; Receiving the third and fourth interpolated signals by the at least one filter; Generating, by the at least one filter, a second filtered signal comprising a weighted value of at least the third and fourth interpolated signals; And A third filtered signal comprising, by the at least one filter, a sum of the product of the first filtered signal and the first value of the Mobius function and the product of the second filtered signal and the second value of the Mobius function Steps to raise Detection method further comprising. The method of claim 27, The generating of the first filtered signal may include: Amplifying the first interpolated signal to generate a first amplified signal; Amplifying the second interpolated signal to generate a second amplified signal; And Integrating the first and second amplified signals to generate an integrated signal Including; And the first filtered signal comprises the integrated signal. The method of claim 27, The generating of the first filtered signal comprises digitally calculating a weighted sum of at least the first and second interpolated signals.

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