CN107197291B - Low-complexity direct-calculation cubic convolution spline interpolation method - Google Patents

Low-complexity direct-calculation cubic convolution spline interpolation method Download PDF

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CN107197291B
CN107197291B CN201710376446.8A CN201710376446A CN107197291B CN 107197291 B CN107197291 B CN 107197291B CN 201710376446 A CN201710376446 A CN 201710376446A CN 107197291 B CN107197291 B CN 107197291B
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洪少华
叶林飞
王琳
徐位凯
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Xiamen University
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Abstract

A low-complexity direct-calculation cubic convolution spline interpolation method relates to digital image compression. Circularly calculating the extended image and a two-dimensional cubic convolution interpolation function to obtain a coefficient
Figure DDA0001304099700000011
5 re-calculated reconstruction filter coefficients are compared with the known ones
Figure DDA0001304099700000012
Intermediate data obtained by cyclic convolution calculation
Figure DDA0001304099700000013
The 5 new reconstructed filter coefficients are again compared to the known ones

Description

Low-complexity direct-calculation cubic convolution spline interpolation method
Technical Field
The invention relates to digital image compression, in particular to a direct-computation cubic convolution spline interpolation technology which can greatly reduce computation complexity and processing time.
Background
At present, the compression technology of multimedia data information is various, and some of the compression technologies are widely applied to various international compression standards, such as JPEG, h.26x, and the like. However, with the application of these conventional multimedia compression techniques and the intensive research on these techniques, it has been found that these compression techniques are more or less insufficient, for example, in the case of a relatively high compression rate, the image/video recovered by the compression techniques may have severe blocking effect. How to achieve high compression ratio and low complexity and still maintain good subjective and objective quality of images/videos, namely, an efficient compression technology is a problem to be solved urgently.
The cubic Convolution Spline Interpolation (CSI) is a compression method for resampling image data, and based on the difference between a minimized original image and a reconstructed image, a least square method and a cubic convolution interpolation function are combined, so that the image quality is better than that of other interpolation methods through verification, and the algorithm can be used for matching with an image coding standard like JPEG to obtain an improved image compression technology with higher compression ratio and better image quality than that of the original image coding standard. However, the computational complexity is also relatively large, especially in the computational circular convolution part of the algorithm.
In 2000, T.K Truong et al, combined with the least square method and the Cubic Convolution Interpolation kernel function, proposed was the use of the Fast Fourier Transform (FFT) method to implement the Cubic Convolution Spline Interpolation algorithm (T.K Truong, L.J.Wang, I.S. Reed. "ImageData Compression Using the Convolution Spline Interpolation," ImageProcessing, IEEE Transactions on 9.11 (2000): 1988-1995), and thereafter in 2001 Lung-JenWang et al (Lung-J.Wang, Wen-selling Hsieh, T.K Truong. "A Fast Effect Compression of client-Spline Interpolation," Processing ", IEEE Transactions on 49.6 (2001)' J.P.P.P.19. the Convolution algorithm was proposed to implement the Cubic Convolution Spline Interpolation algorithm (I.P.P.19) in combination of the least square method and the Cubic Convolution kernel function, I.P.S. 11. the method was first proposed in 2911. the trade of software Convolution algorithm, IEEE Transactions on 11819. 9. and the method of creating the Convolution of the cube Spline Interpolation algorithm No. 11. in No. 11. the first paragraph of the second application of the fifth application . These three common implementations: the methods based on Fast Fourier Transform (FFT) method, wdtf (winograd discrete Fourier transform) and overlap solution, and direct computation method are distinguished by the computation of the cyclic convolution part. As an optimal direct calculation method, computer simulation shows that 11 reconstructed filter coefficients are still needed to achieve the image quality based on the FFT method, and obviously, the complexity of the direct calculation method is still high, and the result is still not ideal, so that the direct calculation method is still not beneficial to the hardware implementation of a flow structure.
The cubic convolution spline interpolation algorithm combines a least square method and a cubic convolution interpolation kernel function to estimate an original function X (t), and is shown as a formula (1-1):
Figure BDA0001304099680000021
wherein k is an integer of 0 to n-1, XkIs sample point data, i.e. data retained after compression of image data, R (t) is a cubic convolution interpolation kernel, S (t) is based on sample point data XkEstimation of the original function x (t) by a cubic convolution interpolation function. According to the least square method, the following equation is derived from the three documents:
Figure BDA0001304099680000022
wherein, (j-k)nModulo (j-k) with n,
Figure BDA0001304099680000023
autocorrelation coefficient, y, being a cubic convolution interpolation functionjIs the convolution of the x (t) function with the interpolation function which is a cubic convolution.
The idea of the original direct calculation method as the optimal scheme is as follows: is provided with [ X ]]TBeing the transpose of matrix X, the circular convolution formula (1-2) can be expressed in matrix form as follows:
Y＝BX (1-3)
wherein Y ═ Y0,y1,…yn-1]T,X＝[x0,x1,…xn-1]T,B＝[b0,b1,…bn-1]CIs a circulant matrix of size nxn, and when τ is 2, b0＝420/256，b1＝bn-1＝63/256，b2＝bn-2＝-18/256，b3＝bn-3＝1/256，b4＝0，b5＝0，…，bn-40. From the matrix idea, solving equations (1-3) can obtain:
X＝B-1Y＝AY (1-4)
wherein A ═ B-1(A is the inverse of B), since the matrix B is a circulant matrix of n × n, it is readily apparent from the nature of the matrix that the matrix A is also a circulant matrix of size n × n:
A＝[a0,a1,a2,a3,…,an-3,an-2,an-1]C
at a compression ratio τ of 2, a0＝0.64640，a1＝-0.10937，a2＝0.04667，a3＝-0.01398，……，an-3＝-0.01398，an-20.04667, and an-1-0.10937. Considering that the matrix A is a circulant matrix, equation (1-4) calculates the sampling point data xjCan be simplified as follows:
Figure BDA0001304099680000031
it can be found that the coefficient aiThe value of (i is more than or equal to L and less than or equal to n-L) gradually tends to zero value along with the increase of the value of L, wherein L is a fixed positive integer. Obviously, to reduce the computational complexity, these coefficients can be considered as approximately zero values, i.e. coefficient aiIs considered to be of finite length [ -L, L]The data of (1). Since the signal X (t) is a periodic function with a period of n tau and y is a periodic function of n, i.e. yk＝yk+n. Thus, the formula (1-5) can be represented as:
Figure BDA0001304099680000032
according to the above formula, the direct calculation method is implemented as follows, corresponding to the one-dimensional situation: firstly, the original signal or image data is convolved with a cubic convolution interpolation function to obtain a coefficient yjDue to reconstruction of the filter coefficient aiCan be calculated in advance and the reconstructed filter coefficient a of any original signal or imageiAre identical, only a suitable limited length of [ -L, L ] is finally taken]Filter coefficient a ofiWith respect to the precedingResult yjMultiplying and accumulating the 2L +1 products to obtain sampling point data xjI.e. compressed signal or image data. The three documents mentioned above show that y is calculatedj2 τ -1- (-2 τ +1) +1 ═ 4 τ -1 correlation coefficients are required for each datum of (a), and 2 coefficients of 0 are still contained therein. Thus calculating yjM is required1(4 τ -1-2) × n multipliers and s1(4 τ -1-2-1) × n adders, and accordingly, this direct calculation method yields xjTotal required m1+ (2L +1) x n multipliers and s1+2L n adders.
For the six gray-scale images shown in fig. 1, computer simulation is performed by using a cubic convolution spline interpolation algorithm of an original direct calculation method, and the results of fig. 2 and fig. 3 show that when L is 5, that is, based on 11 reconstruction filter coefficients, the subjective and objective quality of the images restored by the cubic convolution spline interpolation method of direct calculation can only reach the subjective and objective quality of the images restored based on FFT (fast fourier transform). In the one-dimensional case, 11n multipliers and 10n adders are still needed for calculating the cyclic convolution part (equations 1-6), and in the two-dimensional case, 22n adders are needed for calculating the cyclic convolution part1n2A multiplier and 20n1n2An adder in which n1、n2Indicating the size of the image. Obviously, the existing CSI algorithm is based on 11 reconstructed filter coefficients akThe complexity of the direct calculation scheme (k is more than or equal to 5 and less than or equal to 5) is still high, and the result is still not ideal, so that the direct calculation scheme is not beneficial to large-scale integrated circuit realization.
Disclosure of Invention
In order to overcome the defect of higher complexity caused by direct calculation based on 11 reconstructed filter coefficients in the prior art, the invention provides a cubic convolution spline interpolation method with low complexity and direct calculation.
The invention comprises the following steps:
1) circularly calculating the extended image and a two-dimensional Cubic Convolution Interpolation (CCI) function to obtain a coefficient
2) 5 weights obtained by recalculationThe filter coefficients are known
Figure BDA0001304099680000042
Intermediate data obtained by cyclic convolution calculation
Figure BDA0001304099680000043
3) The 5 new reconstructed filter coefficients are again compared to the known ones
Figure BDA0001304099680000044
The cyclic convolution yields compressed image data.
In step 1), the expanded image and a two-dimensional Cubic Convolution Interpolation (CCI) function are circularly calculated to obtain a coefficient
Figure BDA0001304099680000045
The specific method of (3) may be: for a standard gray image, the standard gray image is expanded to the image with the size of 517 × 517, namely, 3 rows of pixels are supplemented above the original image, 2 rows of pixels are supplemented below the original image, 3 columns of pixels are supplemented on the left side, and 2 columns of pixels are supplemented on the right side, through mirror symmetry compensation, the gray value of the expanded image is convolved with a two-dimensional Cubic Convolution Interpolation (CCI) function to obtain a middle coefficient with the size of 256 × 256
Figure BDA0001304099680000046
In step 2), the intermediate coefficient calculated in step 1) is used
Figure BDA0001304099680000047
Performing cyclic convolution calculation with 5 new reconstructed filter coefficients to obtain intermediate data
Figure BDA0001304099680000048
Since the 5 filter coefficients can be calculated in advance, and the 5 filter coefficients are the same for any size of image, there is similarity and convenience in operation.
In step 3), the intermediate data obtained by the calculation in step 2) are used
Figure BDA0001304099680000049
By circularly convolving with those 5 new reconstructed filter coefficients again, the compressed image data of size 256 × 256 can be calculated.
The method has the advantage of low computation amount, and is more suitable for realizing a VLSI (very Large Scale integration) circuit by a flow structure provided by Tsung-Ching Lin. This flowing water structure includes: time delay unit D and cyclic shift unit SiAdder, multiplier and gating switch 5. The working process is as follows:
1) in each clock period, the reconstructed filter coefficients enter 5 cells in a cyclic shift manner, and corresponding clock periods are delayed according to the number of delay units D in each Cell.
2) In each clock period, convolving the original signal or image data with the cubic convolution interpolation function to obtain the coefficient yjAnd the cells are driven into 5 cells in sequence.
3) The corresponding delayed filter coefficient aiAnd yjAfter multiplication, the product is stored and accumulated with the product corresponding to the next clock period.
4) When C is detectediWhen the Cell count is 1, the accumulated value of the Cell, i.e., the sampling point data x is outputjI.e. compressed data.
The computer simulation display method only needs 5 filter coefficients, namely, the operation complexity of the cyclic convolution part calculated in the original cubic Convolution Spline Interpolation (CSI) technology is reduced by nearly 55 percent (under the two-dimensional condition), and the subjective and objective quality of the image based on the FFT method can be achieved, so that the calculation complexity and the processing time delay are greatly reduced, and the realization of a large-scale integrated circuit is facilitated.
Drawings
Fig. 1 is a six-set standard grayscale image. In fig. 1, (1) Cameraman (256 × 256), (2) Girl (256 × 256), (3) airplan (512 × 512), (4) Lena (512 × 512), (5) Boat (576 × 720), and (6) Airport (1024 × 1024) are shown in this order.
Fig. 2 shows a method based on the original direct calculation when the compression ratio is 2: 1(τ ═ 2) in the one-dimensional case. In fig. 2, the peak signal-to-noise ratio (PSNR) of the corresponding gray image varies with the reconstructed filter coefficient.
Fig. 3 shows a method based on the original direct calculation when the compression ratio is 4: 1(τ ═ 2) in the two-dimensional case. In fig. 3, the peak signal-to-noise ratio (PSNR) of the corresponding gray image varies with the reconstructed filter coefficient.
FIG. 4 is a flow chart of an image compression method based on a low-complexity cubic convolution spline interpolation technique.
Fig. 5 shows Lena images of original size 512 × 512 at a compression ratio of 4: 1(τ ═ 2) in the two-dimensional case. In fig. 5, the image is compressed to a size of 256 × 256.
FIG. 6 is a schematic diagram of a flow structure proposed by Tsung-Ching Lin.
Detailed Description
The computer simulation display new method only needs 5 filter coefficients, namely, the operation complexity of the cyclic convolution part calculated in the original cubic Convolution Spline Interpolation (CSI) technology is reduced by nearly 55 percent (under the two-dimensional condition), and the subjective and objective quality of the image based on the FFT method can be achieved, so that the calculation complexity and the processing time delay are greatly reduced, and the realization of a large-scale integrated circuit is facilitated.
Fig. 4 shows a flow chart of an image compression method based on a low-complexity cubic convolution spline interpolation technology, which comprises the following steps:
step 1, circularly calculating the expanded image and a two-dimensional Cubic Convolution Interpolation (CCI) function to obtain a coefficient
Figure BDA0001304099680000051
Step 2, 5 reconstructed filter coefficients obtained by recalculation are compared with known ones
Figure BDA0001304099680000052
Intermediate data obtained by cyclic convolution calculation
Figure BDA0001304099680000053
Step 3, finally 5 new reconstructed filter coefficientsAgain in accordance with what is known
Figure BDA0001304099680000054
The cyclic convolution yields compressed image data.
For a standard gray image, such as the Lena chart with the size of 512 × 512 in fig. 1, the process of the present invention is adopted as follows:
step 1, extending the image to a size of 517 × 517, that is, supplementing 3 rows of pixels above, 2 rows of pixels below, 3 columns of pixels on the left side, and 2 columns of pixels on the right side of the original image (all by simple mirror symmetry complement). Convolving the gray value of the expanded image with a two-dimensional Cubic Convolution Interpolation (CCI) function to obtain an intermediate coefficient with the size of 256 multiplied by 256
Figure BDA0001304099680000061
Step 2, calculating the intermediate coefficient obtained in the previous step
Figure BDA0001304099680000062
Performing cyclic convolution calculation with 5 new reconstructed filter coefficients to obtain intermediate dataSince the 5 filter coefficients can be calculated in advance, and the 5 filter coefficients are the same for any size of image, there is similarity and convenience in the operation.
Step 3, the obtained product is calculated in the previous step
Figure BDA0001304099680000064
By circularly convolving with those 5 new reconstruction filter coefficients again, the compressed image data having a size of 256 × 256, i.e., the grayscale image shown in fig. 5, can be calculated.
Because the method has the advantage of lower operation amount, the method is more suitable for realizing a VLSI (very Large Scale integration) circuit by a flow structure (see figure 6) proposed by Tsung-Ching Lin. This flowing water structure includes: time delay unit D and cyclic shift unit SiAn adder,Multiplier, gating switch 5. The working process is as follows:
1. in each clock period, the reconstructed filter coefficients enter 5 cells in a cyclic shift manner, and corresponding clock periods are delayed according to the number of delay units D in each Cell.
2. In each clock period, convolving the original signal or image data with the cubic convolution interpolation function to obtain the coefficient yjAnd the cells are driven into 5 cells in sequence.
3. The corresponding delayed filter coefficient aiAnd yjAfter multiplication, the product is stored and accumulated with the product corresponding to the next clock period.
4. When C is detectediWhen the Cell count is 1, the accumulated value of the Cell, i.e., the sampling point data x is outputjI.e. compressed data.
For the direct calculation method, the following assumptions are made for its reconstructed filter coefficients: a is only when-2. ltoreq. k.ltoreq.2kThe value of (a) is valid, and when | k | ≧ 3kThe value assumes 0. Obviously, this would greatly reduce the computational complexity of computing the cyclic convolution portion (equations 1-6), however, since the filter coefficient a is equal to 3 in | k ≧ 3kNeglecting, the subjective and objective quality of the recovered image is greatly reduced. The performance of the recovered picture is improved if there is a way to compensate for the missing reconstruction filter coefficients. By using the concept of boundary condition extension of Cubic Convolution Interpolation (CCI), the method can be realized by akRecalculating reconstructed filter coefficient (k is more than or equal to 5 and less than or equal to 5)
Figure BDA0001304099680000065
This new set of filter coefficients is used to compute the cyclic convolution components (1-6) to improve the computational efficiency, i.e. the computational complexity is reduced 6/11 ≈ 54.55% with the same image quality.
For cubic convolution interpolation function (CCI), when the boundary is x-1(beyond a point to the left of the given range of the X (t) function) and xn+1(beyond a point to the right of the given range of the X (t) function), the CCI interpolation function provides a third order approximation to X (t): x is the number of-1＝x2-3x1+3x0And xn+1＝xn-2-3xn-1+3xn. Extending these boundary conditions to xj-1Or xj+1Equation x can be obtainedj-1＝3xj-3xj+1+xj+2And xj+1＝3xj-3xj-1+xj-2Likewise, it is also possible to obtain:
yj-1＝3yj-3yj+1+yj+2(1-7)
yj+1＝3yj-3yj-1+yj-2(1-8)
compressing the data x according to the above relationj(equations 1-6) New Filter coefficients can be used
Figure BDA0001304099680000071
Expressed, derived as follows:
wherein the content of the first and second substances,
Figure BDA0001304099680000073
Figure BDA0001304099680000074
and k is less than or equal to L-4 for-L +4
Figure BDA0001304099680000075
Explicit computation of compressed data xjIs reduced from 2L +1 to 2L-1. Continuing to replace the above equations with equations (1-7), (1-8), it can continue to be derived as follows:
new filter coefficientsCan be calculated in advance. In particular, when the compression ratio τ is equal to2, the following can be obtained:
when the compression ratio τ is 3, it is possible to obtain:
Figure BDA0001304099680000079
from the above equation, it can be seen that n compressed data x are calculatedjOnly 5 reconstructed filter coefficients are needed, so in the one-dimensional case this low complexity direct computation method only needs 5n multipliers and 4n adders when computing the cyclic convolution portion. Obviously, 11 reconstructed filter coefficients a are used as beforekCompared with direct calculation methods (k is more than or equal to 5 and less than or equal to 5), the invention can save 11n-5 n-10 n-4 n-6 n multipliers and adders respectively. Considering that multiplication is much more complicated than addition, neglecting the addition amount, i.e. using the method of the present invention in one-dimensional situation can save about 6n/11n ≈ 54.55% arithmetic operation amount. If the coefficient y is calculatedjThe required multiplication is taken into account that the percentage of the arithmetic operation quantity required by the invention and the original direct calculation method is (m)1+5n)/(m1+11n), wherein m1To calculate yjThe number of multiplication operations required. In particular, m is a compression ratio τ of 2 in the one-dimensional case1The calculation amount required by the new low-complexity direct calculation method is only 10n/16n which is about equal to 63 percent of the original 11-point direct calculation method.
TABLE 1
Figure BDA0001304099680000081
Table 1 shows statistics of the CSI technique based on arithmetic operation amounts under different implementation methods when the compression ratio is 2: 1(τ ═ 2) in the one-dimensional case, where the first scheme uses FFT to calculate the cyclic convolution part, and the second scheme uses 11 reconstruction filter coefficients akAnd (k is more than or equal to 5 and less than or equal to 5) calculating a circular convolution part, wherein the third scheme uses the circular convolution part calculated by the method. The latter diagram refers to the schemeThe first, second and third schemes are the same as the above, and no further description is given.
Because the cubic convolution spline interpolation function (CSI) has the characteristic of two-dimensional separability, when two-dimensional signals or image data are processed, time division can be decomposed into two steps for operation, namely, two-dimensional operation is separated into one-dimensional operation which is convenient for calculation. Likewise, the following formula can be derived using a two-dimensional low-complexity direct calculation method based on the CSI technique:
Figure BDA0001304099680000082
Figure BDA0001304099680000083
wherein the content of the first and second substances,
Figure BDA0001304099680000084
in order to calculate the intermediate data of the compressed image,
Figure BDA0001304099680000085
is two-dimensional compressed image data. From the above two equations, in the two-dimensional case, 5 × n is required to calculate the cyclic convolution portion using the method of the present invention2×n1+5×n1×n2＝10n1n2A multiplier and 4 xn2×n1+4×n1×n2＝8n1n2An adder in which n1、n2Is the size of the two-dimensional image. Compared with the original direct calculation method, the invention can simultaneously reduce 22n1n2-10n1n2＝20n1n2-8n1n2＝12n1n2A multiplier and an adder. Similar to the one-dimensional case, considering that multiplication is much more complicated than addition, neglecting the addition, it can be seen that the calculation amount of the cyclic convolution part calculated by the invention is reduced by 12n compared with the calculation amount required by the original direct calculation method1n2/22n1n2And ≈ 54.55%. Further, in the above-mentioned case,consider a calculation
Figure BDA0001304099680000091
The total operation complexity of the method is only 25n of the original direct calculation method1n2/37n1n2≈68％。
TABLE 2
Figure BDA0001304099680000093
Table 2 shows the statistics of the CSI technique based on the arithmetic operation amount under different methods when the compression ratio is 4: 1 (tau ═ 2) under the two-dimensional situation, wherein N is1×N2Indicating the size of the image.
The invention reduces the filter coefficients required for calculating the cyclic convolution part in the cubic Convolution Spline Interpolation (CSI) technology from 11 to 5, and still maintains the good subjective and objective quality of the recovered image from the comparison result of peak signal-to-noise ratios (PSNR) of different schemes in tables 3 and 4. The calculation amount of the cyclic convolution part is more than that of the original 11 reconstruction filter coefficients akThe computation amount brought by (k is more than or equal to 5 and less than or equal to 5) is reduced by about 55 percent, the hardware realization difficulty, the time delay and the resource consumption are greatly reduced, and the method is very suitable for the flow structure provided by Tsung Ching Lin to realize a VLSI (VeryLarge Scale integration) circuit.
TABLE 3
TABLE 4
Figure BDA0001304099680000101
Table 3 shows peak signal-to-noise ratios (PSNR) of corresponding gray images based on different methods for the CSI technology at a compression ratio τ: 1 in the one-dimensional case; TABLE 4 compression ratio τ in the two-dimensional case21-hour CSI technology based on peak signal-to-noise ratio (PSNR) of corresponding gray level images under different methods)。

Claims (4)

1. The low-complexity direct-calculation cubic convolution spline interpolation method is characterized by comprising the following steps of:
1) circularly calculating the extended image and a two-dimensional cubic convolution interpolation function to obtain a coefficient
Figure FDA0002141948270000011
2) 5 newly reconstructed filter coefficients obtained by recalculation are compared with the known filter coefficients
Figure FDA0002141948270000012
Intermediate data obtained by cyclic convolution calculation
Figure FDA0002141948270000013
3) The 5 new reconstructed filter coefficients are again compared to the known ones
Figure FDA0002141948270000014
The cyclic convolution yields compressed image data.
2. The low-complexity direct-computation cubic convolution spline interpolation method of claim 1, wherein in step 1), the coefficients are obtained by circularly computing the extended image and a two-dimensional cubic convolution interpolation function
Figure FDA0002141948270000015
The specific method comprises the following steps: for a standard gray image, the standard gray image is expanded to the image with the size of 517 × 517, namely, 3 rows of pixels are supplemented above the original image, 2 rows of pixels are supplemented below the original image, 3 columns of pixels are supplemented on the left side, and 2 columns of pixels are supplemented on the right side, values are supplemented through a mirror symmetry or a period expansion method, the gray value of the expanded image is convolved with a two-dimensional cubic convolution interpolation function to obtain a middle coefficient with the size of 256 × 256
Figure FDA0002141948270000016
3. The low-complexity direct-computation cubic convolution spline interpolation method of claim 1, wherein in step 2), the intermediate coefficients computed in step 1) are applied
Figure FDA0002141948270000017
Performing cyclic convolution calculation with 5 new reconstructed filter coefficients to obtain intermediate data
Figure FDA0002141948270000018
The 5 filter coefficients are calculated in advance, and for any size image, the 5 filter coefficients are the same.
4. The low-complexity direct-computation cubic convolution spline interpolation method of claim 1, wherein in step 3), the intermediate data computed in step 2) are usedAnd circularly convolved again with those 5 new reconstruction filter coefficients, compressed image data of size 256 × 256 is calculated.
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