Method for compressing and decompressing image signals and apparatus for compressing and decompressing image signals

[0003] 1. Field of the Invention

[0004] The present invention relates to a method for compressing of image signals and decompressing of image signals and to an apparatus to implement of image signals compressing of image signals and decompressing of image signals.

[0005] The present application claims the Convention Priority of Japanese Patent Application No. Hei11-326188 filed on Nov. 16, 1999, which is hereby incorporated by reference.

[0006] 2. Description of the Related Art

[0007] Conventionally, when a digitized image is transmitted, since an amount of image information is enormous, the image information is compressed prior to its transmission. As one of methods for compressing image information, international standard coding systems called MPEG (Moving Picture Experts Group) and JPEG (Joint Photographic Experts Group) are known. In the international standard coding systems, input image data is compressed by DCT (Discrete Cosine Transformation) and compressed image data is returned to its original image data by IDCT (Inverse Discrete Cosine Transformation).

[0008] The DCT used in the international standard coding systems is performed in accordance with the following equation (17) provided for in International Standard (STD•IEC 61834-2-ENGL 1, 998) 8-8 DTC, which prescribes the international standard coding systems. The IDCT used in the international standard coding systems is performed in accordance with the following equation (18) provided for in the International Standard (STD•IEC 61834-2-ENGL 1, 998) 8-8 IDCT, which also prescribes the international standard coding systems. $\begin{array}{cc}F\left(h,v\right)=C\left(v\right)C\left(h\right)\sum _{y=0}^7\sum _{x=0}^7\left\{f\left(x,y\right)\cos \alpha \right\}\cos \beta \mathrm{where}C\left(h\right)=\frac{1}{2\sqrt{2}}\left(h=0\right),C\left(h\right)=\frac{1}{2}\left(h=1,2,\dots ,7\right)C\left(v\right)=\frac{1}{2\sqrt{2}}\left(v=0\right),C\left(v\right)=\frac{1}{2}\left(v=1,2,\dots ,7\right)\alpha \equiv \frac{\pi v\left(2y+1\right)}{16},\beta \equiv \frac{\pi h\left(2x+1\right)}{16}& \left(17\right)\\ f\left(x,y\right)=\sum _{v=0}^7\sum _{h=0}^7\left\{C\left(v\right)C\left(h\right)F\left(h,v\right)\cos \alpha ·\cos \beta \right\}\mathrm{where}C\left(h\right)=\frac{1}{2\sqrt{2}}\left(h=0\right),C\left(h\right)=\frac{1}{2}\left(h=1,2,\dots ,7\right)C\left(v\right)=\frac{1}{2\sqrt{2}}\left(v=0\right),C\left(v\right)=\frac{1}{2}\left(h=1,2,\dots ,7\right)\alpha \equiv \frac{\pi v\left(2y+1\right)}{16},\beta \equiv \frac{\pi h\left(2x+1\right)}{16}.& \left(18\right)\end{array}$

[0009] An example of an 8-8 DCT device (called a “first technology” in this application) is disclosed in the Japanese Patent Application Laid-open No. Hei5-181896, in which the 8-8 DCT is performed on every block composed of “8×8” picture element data constituting image data, which is constructed of a product obtained by multiplying eight pieces of picture element data arranged in a transverse direction by eight pieces of picture element data arranged in a longitudinal direction, in accordance with following equation (19) and equation (20). $\begin{array}{cc}\begin{array}{c}\left[\begin{array}{c}{F}_0\\ F_4\\ F_2\\ F_6\end{array}\right]=\left[\begin{array}{ccc}{a}_0+a_1+a_3+a_2& 0& 0\\ a_0-a_1+a_3-a_2& 0& 0\\ 0& a_0-a_3& a_1-a_2\\ 0& -a_1-a_2& a_0-a_3\end{array}\right]\left[\begin{array}{c}{P}_3\\ P_5\\ P_1\end{array}\right]\\ \left[\begin{array}{c}{F}_1\\ F_3\\ F_5\\ F_7\end{array}\right]=\left[\begin{array}{cccc}{a}_4& a_5& a_6& a_7\\ -a_6& a_4& -a_7& -a_5\\ -a_5& a_7& a_4& a_6\\ a_7& a_6& -a_5& a_4\end{array}\right]\left[\begin{array}{c}{P}_6\\ P_4\\ P_2\\ P_0\end{array}\right]\end{array}\}& \left(19\right)\\ \begin{array}{c}\frac{1}{2}\left[\begin{array}{c}{f}_0+f_4\\ f_1+f_5\\ f_2+f_6\\ f_3+f_7\end{array}\right]=\left[\begin{array}{ccc}{a}_0+a_3& a_2& a_1\\ a_0-a_3& -a_1& a_2\\ a_0-a_3& a_1& -a_2\\ a_0+a_3& -a_2& -a_1\end{array}\right]\left[\begin{array}{c}{P}_3\\ P_5\\ P_1\end{array}\right]\\ \frac{1}{2}\left[\begin{array}{c}{f}_0-f_4\\ f_1-f_5\\ f_2-f_6\\ f_3-f_7\end{array}\right]=\left[\begin{array}{cccc}{a}_5& a_7& -a_6& -a_4\\ a_6& a_5& a_4& -a_7\\ -a_7& -a_4& a_5& -a_6\\ a_4& -a_6& -a_7& a_5\end{array}\right]\left[\begin{array}{c}{P}_6\\ P_4\\ P_2\\ P_0\end{array}\right]\end{array}\}& \left(20\right)\end{array}$

[0010] In the 8-8 DCT devices, the 8-8 DCT and 8-8 IDCT are performed by arithmetic operations according to a determinant equation (19) and determinant equation (20) derived from the equation (17) and the equation (18).

[0011] The first technology disclosed in the above Japanese Patent Application Laid-open No. Hei5-181896 includes two 8-8 DCT devices. One of the two 8-8 DCT, which performs the 8-8 DCT by arithmetic operations according to the determinant equation (19), is composed of eight selecting circuits adapted to select one piece of input data out of 8 pieces of input data (a0 to a7), seven fixed coefficient multipliers connected to the selecting circuits and adapted to multiply each of selected signals output from the selecting circuits by a different coefficient and 5 types of adding/subtracting units connected to outputs of the fixed coefficient multipliers and adapted to perform adding and subtracting operations to each of output signals from the fixed coefficient multipliers in a variety of differently combined ways and is so configured that three types of the adding/subtracting units out of the five types of the adding/subtracting units are used for the cosine transformation and the remaining two types of the adding/subtracting units for the inverse cosine transformation and that eight cosine transformation coefficients are obtained by switching, four times, the way of selecting data in the selecting circuits and by switching, four times, the calculating way from the addition to subtraction or vice versa in the adding and subtracting units. However, input data (a0 to a7), when the cosine transformation is performed, include a0=x0+x7, a1=x1+x6, a2=x2+x5, a3=x3+x4, a4=x0−x7, a5=x1−x6, a6=x2−x5 and a7=x3−x4, which can be obtained from 8 input image data (x0 to x7). Input data (a0 to a7), when the inverse cosine transformation is performed, includes a0=x0, a1=x6, a2=x2, a3=x4, a4=−x7, a5=x1, a6=−x5 and a7=x3, which can be obtained from eight input image data (x0 to x7).

[0012] The other out of the two 8-8 DCT devices is composed of four selecting circuits adapted to select one piece of input data out of eight pieces of input data (a0 to a7), four fixed coefficient multipliers connected to the selecting circuits and adapted to multiply each of selected signals output from the selecting circuits by a different coefficient and five types of adding/subtracting units connected to outputs of the fixed coefficient multipliers and adapted to perform adding or subtracting operations to each of output signals from the fixed coefficient multipliers in a variety of differently combined ways and is so configured that two types of the adding/subtracting units out of the four types of the adding/subtracting units are used for the cosine transformation and the remaining two types of the adding/subtracting units for the inverse cosine transformation and that eight cosine transformation coefficients are obtained by switching, eight times, the way of selecting data in the selecting circuits, and by switching, eight times, the calculating way from the addition to subtraction or vice versa in the adding and subtracting units and by switching, eight times, the selection of two types of coefficients in the fixed coefficient multipliers. Eight pieces of input data (a0 to a7) for this DCT device are the same as those for the above one out of the two DCT devices.

[0013] In both these two DCT devices, DCT coefficients can be calculated by performing the DCT on input data obtained from each line composed of eight pieces of picture element data, which constitutes the block composed of 8×8 picture element data, arranged either in the transverse direction or in the longitudinal direction, that is, 64 DCT coefficients can be obtained by performing the DCT first on input data obtained from a first line composed of the eight picture element data arranged, for example, in the traverse direction and then sequentially on input data obtained from a second line, third line, . . . , eighth line, each being composed of the eight picture element data also arranged in the traverse direction. The DCT is used to transform data signals on a time-axis to signal components in the frequency region.

[0014] Therefore, the 64 DCT coefficients obtained by the DCT device represent frequency components in the frequency region. The 64 DCT coefficients are used for coding only low frequency components being centralized on the upper left of the block composed of 8×8 picture element data, which can provide image data in which information contained in the 8×8 picture element block is compressed. Then, by performing the DCT applied singly to the 8×8 picture element block also on all the 8×8 picture element blocks contained in the image, an image data having information compressed for every 8×8 picture element block can be obtained for transmission.

[0015] Moreover, an example of another DCT device (called a “second technology” in this application) which can use both the 8-8 DCT function and 2-4-8 DCT (described later in detail) function is disclosed in the Japanese Patent Application Laid-open No. Hei6-243160. The second technology device is composed of a factor switching controller, a first sorting circuit to sort data in accordance with control signals fed from the factor switching controller, a first fourth-degree inner product calculating circuit, a second sorting circuit to sort data in accordance with control signals fed from the factor switching controller, an inner product calculating circuit for the 2-4-8 DCT to perform the 2-4-8 DCT in accordance with control signals fed from the factor switching controller, an eighth-degree/fourth-degree inner product circuit to select an inner product calculating circuit for the 8-8 DCT, a fourth-degree inner product circuit to select either of functions of the inner product calculating circuit for 2-4-8 DCT or functions of the inner product calculating circuit for 8-8 DCT in accordance with control signals fed from the factor switching circuit and a third sorting circuit to sort data in accordance with control signals fed from the factor switching circuit.

[0016] However, in the one 8-8 DCT device out of the first technology, an inconvenience occurs in that the 2-4-8 DCT required in operations of a digital video cannot be easily performed. This is due to the following reason. That is, the 2-4-8 DCT is performed in accordance with the following equation (21) and the 2-4-8 IDCT in accordance with the following equation (22). $\begin{array}{cc}\begin{array}{c}F\left(h,v\right)=C\left(v\right)C\left(h\right)\sum _{z=0}^3\sum _{x=0}^7\left\{f\left(x,2z\right)+f\left(x,2z+1\right)\right\}\cos \gamma ·\cos \beta \\ F\left(h,v+4\right)=C\left(v\right)C\left(h\right)\sum _{z=0}^3\sum _{x=0}^7\left\{f\left(x,2z\right)+f\left(x,2z+1\right)\right\}\cos \gamma ·\cos \beta \end{array}\}& \left(21\right)\end{array}$

[0017] where

[0018] v=0,1, . . . , 7

[0019] z=integer of y/2 $C\left(h\right)=\frac{1}{2\sqrt{2}}\left(h=0\right),C\left(h\right)=\frac{1}{2}\left(h=1,2,\dots ,7\right)C\left(v\right)=\frac{1}{2\sqrt{2}}\left(v=0\right),C\left(v\right)=\frac{1}{2}\left(v=1,2,\dots ,7\right)$ $\gamma \equiv \frac{\pi v\left(2z+1\right)}{8},\beta \equiv \frac{\pi h\left(2x+1\right)}{16}$ $\begin{array}{cc}\begin{array}{c}f\left(x,2z\right)=\sum _{v=0}^3\sum _{h=0}^7\left\{C\left(v\right)C\left(h\right)\left(F\left(h,v\right)+F\left(h,v+4\right)\right)\right\}\cos \gamma ·\cos \beta \\ f\left(x,2z+1\right)=\sum _{v=0}^3\sum _{h=0}^7\left\{C\left(v\right)C\left(h\right)\left(F\left(h,v\right)-F\left(h,v+4\right)\right)\right\}\cos \gamma ·\cos \beta \end{array}\}& \left(22\right)\end{array}$

[0020] where

[0021] v=0,1, . . . , 7

[0022] z=integer of y/2 $C\left(h\right)=\frac{1}{2\sqrt{2}}\left(h=0\right),C\left(h\right)=\frac{1}{2}\left(h=1,2,\dots ,7\right)C\left(v\right)=\frac{1}{2\sqrt{2}}\left(v=0\right),C\left(v\right)=\frac{1}{2}\left(v=1,2,\dots ,7\right)$ $\gamma \equiv \frac{\pi v\left(2z+1\right)}{8},\beta \equiv \frac{\pi h\left(2x+1\right)}{16}$

[0023] As is apparent from comparison between the equation (17) and equation (18) being used for the 8-8 DCT and 8-8 IDCT respectively and the equation (21) and equation (22) being used for the 2-4-8 DCT and 2-4-8 IDCT respectively, fixed coefficients {values expressed as cosine values in the equation (17) and equation (18)} of the fixed coefficient multipliers used in the 8-8 DCT device are different from those {values expressed as cosine values in the equation (21) and equation (22)} of the fixed coefficient multiplier used in the 2-4-8 DCT device and, moreover, data to be multiplied by the above fixed coefficients are also different. Therefore, when the 2-4-8 DCT is to be performed on image data, the 2-4-8 DCT device has to be separately fabricated in a manner being similar to the 8-8 DCT device disclosed in the above Japanese Patent Application Laid-open No. Hei6-243160.

[0024] Furthermore, in the DCT device disclosed as the second technology, both the 8-8 DCT and the 2-4-8 DCT can be used. However, the DCT device, while it is operated in a mode to operate as a two-dimensional 4×8 DCT device, is so configured that control signals, for example, signals with logic “1” are fed from the coefficient switching controller to the first sorting circuit, second sorting circuit, eighth-degree/fourth-degree inner product circuit, fourth-degree inner product circuit and third sorting circuit to implement the two-dimensional 4×8 DCT function and, when it is operated in a mode to operate as a two-dimensional 8×8 DCT device, is so configured that control signals, for example, signals with logic “0” are fed from the coefficient switching controller to the first sorting circuit, second sorting circuit, eighth-degree/fourth-degree inner product circuit, fourth-degree inner product circuit and third sorting circuit to implement the two-dimensional 4×8 DCT function. That is, the device is so constructed that, every time input data is sorted in a sorting order determined by the control signals fed from the coefficient switching controller, a predetermined inner product arithmetic operation is performed on the sorted data to obtain DCT coefficients corresponding to control signals output from the coefficient switching controller. As a result, the processing of data sorting is always required not only for the two-dimensional 4×8 DCT but also for the two-dimensional 8×8 DCT. The data sorting and the selection of circuits by control signals fed from the coefficient switching controller to implement functions of the two-dimensional 4×8 DCT or two-dimensional 8×8 DCT, are inseparable from each other. That is, the conventional DCT device is so configured that the selection of circuits adapted to implement the two-dimensional 4×8 DCT or two-dimensional 8×8 DCT is made in a state in which the processing of sorting data is required in an inseparable manner. Therefore, if the data sorting is carried out via a RAM (Random-Access Memory), repeated processing of sorting data is necessary, thus causing much time to be taken sorting the data and a delay in obtaining DCT coefficients and, if the data sorting is carried out via hardware, it causes an increased scale of circuits and a delay in the propagation of signals.

[0025] In view of the above, it is an object of the present invention to provide a method for compressing image signals, a method for decompressing compressed image signals, a compression processing apparatus and decompression processing apparatus which are capable of achieving high-speed transformation by using fixed coefficients in odd-numbered positions as the fixed coefficients used in both 2−2^N−1−2^NDCT and 2−2^N−1−2^NIDCT and which are capable of reducing a circuit scale and of performing high speed transformation by incorporating 2−2^N−1−2^NDCT and 2−1^N−1−2^NIDCT functions into basic 2^N-2^NDCT and 2^N-2^NIDCT devices.

[0026] According to a first aspect of the present invention, there is provided a method for compressing image signals including:

[0027] a selection process of selecting 2^Npieces of picture element data fj (0≦j≦2^N−1) contained in one line or one string constituting a block of 2^N×2^N(N being a natural number) pieces of picture element data forming image data, for each of fixed coefficients Pk given in equation (1) and equation (2) shown below and determined in accordance with a discrete cosine transformation rule;

[0028] a multiplication process of multiplying each piece of the selected picture element data by each of corresponding fixed coefficients Pk to obtain products;

[0029] an addition/subtraction process of performing adding operations and/or subtracting operations between products obtained by the multiplication process and determined in accordance with the discrete cosine transformation rule; and

[0030] an output process of outputting a value obtained by the addition/subtraction process as transformation coefficient data Fj for each line or each string constituting the block of 2^N×2^Npicture element data;

[0031] wherein, in the selection process, first set and second set of picture element data composed of 2^Npieces of picture element data contained in one line or one string constituting the block of 2^N×2^Npieces of picture element data are selected for each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in equation (1) and equation (2) shown below, in a predetermined order, and wherein, in the multiplication process, the first set and second set of picture element data selected for each of the fixed coefficients with k in odd-numbered positions in the equation (1) and equation (2) are multiplied by each of the fixed coefficients with k in odd-numbered positions in the equation (1) and equation (2) to obtain the product; $\begin{array}{cc}{P}_k=\cos \left(\frac{\left(N-1-k\right)\pi }{2^{N+1}}\right)& \left(23\right)\end{array}$

[0032] where

[0033] 0≦k≦2^N−1−2

[0034] however, k=2^N−1−1 is excluded; $\begin{array}{cc}{P}_k=\frac{1}{\sqrt{2}}& \left(24\right)\end{array}$

[0035] where (k=2^N−1−1).

[0036] In the foregoing, a preferable mode is one wherein a sum of 2^Npieces of picture element data having “j” data contained in the picture element data fj, one being lower-numbered data and the other being the next lower-numbered data and thereafter in the same manner, is selected as the first set of picture element data to be selected in the selection process and wherein a difference between 2^Npieces of picture element data having “j” data contained in the picture element data fj, one being lower-numbered data and the other being the next lower-numbered data and thereafter in the same manner, is selected as the second set of picture element data to be selected in the selection process.

[0037] According to a second aspect of the present invention, there is provided a method for decompressing image signals including:

[0038] a selection process of selecting 2^Npieces of transformation coefficient data Fj (0≦j≦2^N−1) contained in one line or one string constituting a block of 2^N×2^N(N being a natural number) pieces of transformation coefficient data block forming transformation coefficient data transmitted after being transformed by a discrete cosine transformation rule, for each of fixed coefficients Pk given in equation (3) and equation (4) shown below and determined in accordance with the discrete cosine transformation rule;

[0039] a multiplication process of multiplying each of the selected transformation coefficient data by each of the corresponding fixed coefficients Pk to obtain products;

[0040] an addition/subtraction process of performing adding operation and/or subtracting operation between products obtained by the multiplication process which is determined in accordance with the discrete cosine transformation rule;

[0041] an output process of outputting a value obtained by the addition/subtraction process as picture element data fj for each line or each string constituting the block of 2^N×2^Npicture element data;

[0042] wherein, in the selection process, first set and second set of transformation coefficient data composed of 2^Npieces of transformation coefficient data contained in one line or one string constituting the block of 2^N×2^Npieces of transformation coefficient data are selected for each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions

[0043] in the equation (3) and equation (4), in a predetermined order, and wherein, in the multiplication process, each of the first set and second set of transformation coefficient data selected for each of the fixed coefficients with k in odd-numbered position in the equation (3) and equation (4) is multiplied by each of the fixed coefficients with k in odd-numbered positions in said equation (3) and equation (4) to obtain the product; $\begin{array}{cc}{P}_k=\cos \left(\frac{\left(N-1-k\right)\pi }{2^{N+1}}\right)& \left(25\right)\end{array}$  0≦k≦2^N−1−2

[0044] where k=2^N−1−1 is excluded; $\begin{array}{cc}{P}_k=\frac{1}{\sqrt{2}}& \left(26\right)\end{array}$

[0045] (k=2^N−1−1).

[0046] In the foregoing, it is preferable that the first set of transformation coefficient data is a sum of transformation coefficient data composed of one data selected from a first transformation data set containing “j” data constituting first half of the 2^Npieces of transformation coefficient data Fj and containing 2^N−1pieces of transformation coefficient data and transformation coefficient data composed of one data selected from a second transformation data set containing “j” data constituting second half of the 2^Npieces of transformation coefficient data Fj and containing 2^N−1pieces of transformation coefficient data and wherein the second set of transformation data is a difference between transformation coefficient data composed of one data selected from the first set of transformation data and transformation coefficient data composed of one data selected from the second set of transformation data.

[0047] According to a third aspect of the present invention, there is provided a method for compressing image signals including:

[0048] a selection process of selecting 2^Npieces of picture element data fj (0≦j≦2^N−1) contained in one line or one string constituting a block of 2^N×2^N(N being a natural number) pieces of picture element data block forming image data, for each of fixed coefficients Pk given in equation (5) and equation (6) shown below and determined in accordance with the discrete cosine transformation rule;

[0049] a multiplication process of multiplying each of the selected picture element data by each of the corresponding fixed coefficients Pk to obtain products;

[0050] an addition/subtraction process of performing adding operation and/or subtracting operation between products obtained by the multiplication process and determined in accordance with the discrete cosine transformation rule; and

[0051] an output process of outputting a value obtained by the addition/subtraction process as transformation coefficient data Fj for each line or each string constituting the block of 2^N×2^Npicture element data;

[0052] wherein, in the selection process, in the case of 2^N-2^Ndiscrete cosine transformation, 2^Npieces of picture element data are selected from 2^Npieces of picture element data contained in one line or one string constituting the block of 2^N×2^Npicture element data for each of the fixed coefficients Pk determined by the 2^N-2^Ndiscrete cosine transformation in a predetermined order and wherein, in the selection process, in the case of 2−2^N−1−2^Ndiscrete cosine transformation, each of first set and second set of picture element data each being composed of 2^Npieces of picture element data to be multiplied by each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (5) and equation (6), is selected from 2^Npieces of picture element data contained in one line or one string constituting the block of 2^N×2^Npieces of picture element data in a predetermined order,

[0053] wherein, in the multiplication process, in the case of the 2^N-2^Ndiscrete cosine transformation, each of the 2^Npieces of picture element data selected in the selection process is multiplied by each of the corresponding fixed coefficients out of the fixed coefficients Pk to obtain products and wherein, in the case of the 2−2^N−1−2^Ndiscrete cosine transformation, each of the first set and second set of picture element data selected based on corresponding fixed coefficient, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (5) and equation (6) is multiplied by the fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (5) and equation (6), and

[0054] wherein, in the addition/subtraction process, in the case of the 2^N-2^Ndiscrete cosine transformation, the adding operations and/or subtracting operations are performed between products obtained by the multiplication process and determined in accordance with the 2^N-2^Ndiscrete cosine transformation and wherein, in the case of the 2−2^N−1−2^Ndiscrete cosine transformation, the adding operations and/or subtracting operations are performed between products obtained by the multiplication process and determined in accordance with the 2−2^N−1−2^Ndiscrete cosine transformation rule; $\begin{array}{cc}{P}_k=\cos \left(\frac{\left(N-1-k\right)\pi }{2^{N+1}}\right)& \left(27\right)\end{array}$  0≦k≦2^N−1−2

[0055] where k=2^N−1−1 is excluded; $\begin{array}{cc}{P}_k=\frac{1}{\sqrt{2}}& \left(28\right)\end{array}$

[0056] (k=2^N−1−1).

[0057] Also, it is preferable that the picture element data to be selected in the block of 2^N-2^Ndiscrete cosine transformation includes the first set of picture element data composed of the 2^Npieces of picture element data contained in one line or one string constituting the block of 2^N×2^Npieces of picture element data to be multiplied by each of the fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (5) and equation (6) and the second set of picture element data composed of the 2^Npieces of picture element data to be multiplied by each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (5) and equation (6).

[0058] Also, it is preferable that the picture element data selected in the selection process is a sum and difference between picture element data constituting a predetermined pair of picture element data.

[0059] Also, it is preferable that the picture element data constituting the predetermined pair of picture element data, in the case of the 2−2^N−1−2^Ndiscrete cosine transformation, are 2^Npieces of picture element data having “j” data contained in the picture element data fj, one being lower-numbered data and the other being the next lower-numbered data.

[0060] Furthermore, it is preferable that the picture element data constituting the predetermined pair of picture element data, in the case of the 2^N-2^Ndiscrete cosine transformation, are picture element data composed of one data selected from the first transformation data set containing “j” data which constitutes first half of the 2^Npieces of picture element data Fj and containing 2^N−1pieces of picture element data and picture element data composed of one data selected from the second picture element data set containing “j” data which constitutes second half of the 2^Npieces of picture element data fj and containing 2^N−1pieces of picture element data.

[0061] According to a fourth aspect of the present invention, there is provided a method for decompressing image signals including:

[0062] a selection process of selecting 2^Npieces of transformation coefficient data Fj (0≦j≦2^N−1) contained in one line or one string constituting a block of 2^N×2^N(N being a natural number) pieces of transformation coefficient data block forming transformation coefficient data transmitted after being transformed by a discrete cosine transformation rule, for each of fixed coefficients Pk given in equation (7) and equation (8) shown below and determined in accordance with the discrete cosine transformation rule;

[0063] a multiplication process of multiplying each of the selected transformation coefficient data by each of the corresponding fixed coefficients Pk to obtain products;

[0064] an addition/subtraction process of performing adding operations and/or subtracting operations between products obtained by the multiplication process and determined in accordance with the discrete cosine transformation rule; and

[0065] an output process of outputting a value obtained by the addition/subtraction process as picture element data fj for each line or each string constituting the block of 2^N×2^Ntransformation coefficient data;

[0066] wherein, in the selection process, in the case of 2^N-2^Ninverse discrete cosine transformation, each of the 2^Npieces of transformation coefficient data is selected from 2^Nm pieces of transformation coefficient data contained in one line or one string constituting the block of 2^N×2^Ntransformation coefficient data for each of the fixed coefficients Pk determined by the 2^N-2^Ninverse discrete cosine transformation method in a predetermined order, and wherein, in the selection process, in the case of 2−2^N−1−2^Ninverse discrete cosine transformation, first set and second set of transformation coefficient data each being composed of 2^Npieces of transformation coefficient data contained in one line or one string constituting the block of 2^N×2^Npieces of transformation coefficient data are selected for each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (7) and equation (8) in a predetermined order,

[0067] wherein, in the multiplication process, in the case of the 2^N-2^Ninverse discrete cosine transformation, each of the 2^Npieces of picture element data selected in the selection process is multiplied by each of corresponding fixed coefficients out of the fixed coefficients Pk to obtain products and wherein, in the case of the 2−2^N−1−2^Ninverse discrete cosine transformation, each of the first set and second set of transformation coefficient data selected based on each of corresponding fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in said equation (7) and equation (8) is multiplied by each of the fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (7) and equation (8) to obtain products,

[0068] wherein, in the addition/subtraction process, in the case of the 2^N-2^Ninverse discrete cosine transformation, the adding operations and/or subtracting operations are performed between products obtained by the multiplication process and determined in accordance with the 2^N-2^Ndiscrete cosine transformation rule and wherein, in the case of the 2−2^N−1−2^Ninverse discrete cosine transformation, the adding operations and/or subtracting operations are performed between products obtained by the multiplication process and determined in accordance with the 2−2^N−1−2^Ndiscrete cosine transformation rule; $\begin{array}{cc}{P}_k=\cos \left(\frac{\left(N-1-k\right)\pi }{2^{N+1}}\right)& \left(29\right)\end{array}$  0≦k≦2^N−1−2

[0069] where k=2^N−1is excluded; $\begin{array}{cc}{P}_k=\frac{1}{\sqrt{2}}& \left(30\right)\end{array}$

[0070] (k=2^N−1).

[0071] In the foregoing, a preferable mode is one wherein the transformation coefficient data to be selected in the 2^N-2^Ninverse discrete cosine transformation includes the first set of transformation coefficient data composed of the 2^Npieces of transformation coefficient data contained in one line or one string constituting the block of 2^N×2^Npieces of transformation coefficient data to be multiplied by each of the fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (7) and equation (8) and the second set of transformation coefficient data composed of the 2^Npieces of transformation coefficient data to be multiplied by each of fixed coefficients, out of the fixed coefficients Pk, with k in even-numbered positions in the equation (7) and equation (8).

[0072] According to a fifth aspect of the present invention, there is provided a device for compressing image signals comprising:

[0073] a selection circuit for selecting 2^Npieces of picture element data fj (0≦j≦2^N−1) contained in one line or one string constituting a block of 2^N×2^N(N being a natural number) pieces of picture element data block forming image data, for each of fixed coefficients Pk given in equation (9) and equation (10) shown below and determined in accordance with a discrete cosine transformation rule;

[0074] a multiplication circuit for multiplying each of the selected picture element data by each of the corresponding fixed coefficients Pk to obtain products;

[0075] an addition/subtraction circuit for performing adding operations and/or subtracting operations between products obtained by the multiplication process and determined in accordance with the discrete cosine transformation rule; and

[0076] an output circuit for outputting a value obtained by the adding operations and/or subtracting operations as trans-formation coefficient data Fj for each line or each string constituting the block of 2^N×2^Npicture element data;

[0077] wherein the selection circuit is that each of the first set and second set of picture element data composed of 2^Npieces of picture element data contained in one line or one string constituting the block of 2^N×2^Npieces of picture element data is selected for each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in equation (9) and equation (10) shown below, in a predetermined order, and wherein, in the multiplication process, each of the first set and second set of picture element data selected for each of the fixed coefficients with kin odd-numbered positions in the equation (9) and equation (10) is multiplied by the fixed coefficient with k in odd-numbered positions in the equation (9) and equation (10) to obtain the product; $\begin{array}{cc}{P}_k=\cos \left(\frac{\left(N-1-k\right)\pi }{2^{N+1}}\right)& \left(31\right)\end{array}$  0≦k≦2^N−1−2

[0078] where k=2^N−1−1 is excluded; $\begin{array}{cc}{P}_k=\frac{1}{\sqrt{2}}& \left(32\right)\end{array}$

[0079] (k=2^N−1−1).

[0080] In the foregoing, a preferable mode is one wherein a sum of 2^Npieces of picture element data having “j” data contained in the picture element data fj, one being lower-numbered data and the other being the next lower-numbered data and thereafter in the same manner, is selected by the selection circuit as the first set of picture element data and wherein a difference between 2^Npieces of picture element data having “j” data contained in the picture element data fj, one being lower-numbered data and the other being the next lower-numbered data, and thereafter in the same manner, is selected by the selection circuit as the second set of picture element.

[0081] According to a sixth aspect of the present invention, there is provided a device for decompressing image signals including:

[0082] a selection circuit for selecting 2^Npieces of transformation coefficient data Fj (0≦j≦2^N−1) contained in one line or one string constituting a block of 2^N×2^N(N being a natural number) pieces of transformation coefficient data block forming transformation coefficient data transmitted after being transformed by a discrete cosine transformation method, for each of fixed coefficients Pk given in equation (11) and equation (12) shown below and determined in accordance with a discrete cosine transformation rule;

[0083] a multiplication circuit for multiplying each of the selected transformation coefficient data by each of the corresponding fixed coefficient Pk to obtain products;

[0084] an addition/subtraction circuit for performing adding operations and/or subtracting operations between products obtained by the multiplication process which is determined in accordance with the discrete cosine transformation rule; and

[0085] an output circuit for outputting a value obtained by the adding operations and/or subtracting operations as picture element data fj for each line or each string constituting the block of 2^N×2^Npicture element data;

[0086] wherein the selection circuit is that each of the first set and second set of transformation coefficient data composed of 2^Npieces of transformation coefficient data contained in one line or one string constituting the block of 2^N×2^Npieces of transformation coefficient datablock is selected for each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (11) and equation 12), in a predetermined order, and wherein the multiplication circuit is that each of the first set and second set of transformation coefficient data selected for each of the fixed coefficients with kin odd-numbered positions in the equation (11) and equation (12) is multiplied by each of the fixed coefficients with k in odd-numbered positions in the equation (11) and equation (12) to obtain the product; $\begin{array}{cc}{P}_k=\cos \left(\frac{\left(N-1-k\right)\pi }{2^{N+1}}\right)& \left(33\right)\end{array}$  0≦k≦2^N−1−2

[0087] where k=2^N−1−1 is excluded; $\begin{array}{cc}{P}_k=\frac{1}{\sqrt{2}}& \left(34\right)\end{array}$

[0088] (k=2^N−1−1).

[0089] In the foregoing, a preferable mode is one wherein the selection circuit, selects, as the first set of transformation coefficient data, a sum of transformation coefficient data composed of one data selected from a first transformation data set containing “j” data which constitutes first half of the 2^Npieces of transformation coefficient data Fj and containing 2^N−1pieces of transformation coefficient data and transformation coefficient data composed of one data selected from a second transformation data set containing “j” data which constitutes second half of the 2^Npieces of transformation coefficient data Fj and containing 2^N−1pieces of transformation coefficient data and wherein the selection circuit, selects, as the second set of transformation data, a difference between transformation coefficient data composed of one data selected from the first transformation data set and transformation coefficient data composed of one data selected from the second transformation data set.

[0090] According to a seventh aspect of the present invention, there is provided a device for compressing image signals including:

[0091] a selection circuit for selecting 2^Npieces of picture element data fj (0≦j≦2^N−1) contained in one line or one string constituting a block of 2^N×2^N(N being a natural number) pieces of picture element data block forming image data, for each of fixed coefficients Pk given in equation (13) and equation (14) shown below and determined in accordance with a discrete cosine transformation rule;

[0092] a multiplication circuit for multiplying each of the selected picture element data by each of the corresponding fixed coefficients Pk to obtain products;

[0093] an addition/subtraction circuit for performing adding operations and/or subtracting operations between products obtained by the multiplication circuit and determined in accordance with the discrete cosine transformation rule; and

[0094] an output circuit for outputting a value obtained by the adding operations and/or subtracting operations as trans-formation coefficient data Fj for each line or each string constituting the block of 2^N×2^Npicture element data;

[0095] wherein the selection circuit, in the case of 2^N-2^Ndiscrete cosine transformation, selects 2^Npieces of picture element data from 2 ^Npieces of picture element data contained in one line or one string constituting the block of 2^N×2^Npicture element data block for each of the fixed coefficients Pk determined by the 2^N-2^Ndiscrete cosine transformation method in a predetermined order and wherein the selection circuit, in the case of 2−2^N−1−2^Ndiscrete cosine transformation, selects each of first set and second set of picture element data each being composed of 2^Npieces of picture element data to be multiplied by each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (13) and equation (14), from 2^Npieces of picture element data contained in one line or one string constituting the block of 2^N×2^Npieces of picture element data block in a predetermined order,

[0096] wherein the multiplication circuit, in the case of the 2^N-2^Ndiscrete cosine transformation, multiplies each of the 2^Npieces of picture element data selected by the selection circuit by each of corresponding fixed coefficients out of the fixed coefficients Pk to obtain products and wherein the multiplication circuit, in the case of the 2−2^N−1−2^Ndiscrete cosine transformation, multiplies each of the first set and second set of picture element data selected based on each of corresponding fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (13) and equation (14) by each of the fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions the equation (13) and equation (14), and

[0097] wherein the addition/subtraction circuit, in the case of the 2^N-2^Ndiscrete cosine transformation, performs the adding operations and/or subtracting operations between products obtained by the multiplication process and determined in accordance with the 2^N-2^Ndiscrete cosine transformation and wherein the addition/subtraction circuit, in the case of the 2−2^N−1−2^Ndiscrete cosine transformation, performs the adding operations and/or subtracting operations between products obtained by the multiplication process and determined in accordance with the 2−2^N−1−2^Ndiscrete cosine transformation rule; $\begin{array}{cc}{P}_k=\cos \left(\frac{\left(N-1-k\right)\pi }{2^{N+1}}\right)& \left(35\right)\end{array}$  0≦k≦2^N−1−2

[0098] where k=2^N−1−1 is excluded; $\begin{array}{cc}{P}_k=\frac{1}{\sqrt{2}}& \left(36\right)\end{array}$

[0099] (k=2^N−1−1).

[0100] In the foregoing, a preferable mode is one wherein the picture element data to be selected in said 2^N-2^Ndiscrete cosine transformation includes the first set of picture element data composed of the 2^Npieces of picture element data contained in one line or one string constituting the block of 2^N×2^Npieces of picture element data block to be multiplied by each of the fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (13) and equation (14) and the second set of picture element data composed of the 2^Npieces of picture element data to be multiplied by each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (13) and equation (14).

[0101] Also, a preferable mode is one wherein the selection circuit, selects, as the picture element data, a sum and difference between picture element data constituting a predetermined pair of picture element.

[0102] Also, a preferable mode is one wherein the selection circuit selects, as the picture element data constituting the predetermined pair of picture element data, in the case of the 2−2^N−1−2^Ndiscrete cosine transformation, 2^Npieces of picture element data having “j” data contained in the picture element data fj, one being lower-numbered data and the other being the next lower-numbered data.

[0103] Also, a preferable mode is one wherein the selection circuit selects, as the picture element data constituting the predetermined pair of picture element data, in the case of the 2^N-2^Ndiscrete cosine transformation, picture element data composed of one data selected from the first transformation data set containing “j” data constituting first half of the 2^Npieces of picture element data Fj and containing 2^N−1pieces of picture element data and picture element data composed of one data selected from the second picture element data set containing “j” data constituting second half of the 2^Npieces of picture element data fj and containing 2^Npieces of picture element data.

[0104] According to an eighth aspect of the present invention, there is provided a device for decompressing image signals including:

[0105] a selection circuit for selecting 2^Npieces of transformation coefficient data Fj (0≦j≦2^N−1) contained in one line or one string constituting a block of 2^N×2^N(N being a natural number) pieces of transformation coefficient data block forming transformation coefficient data transmitted after being transformed by a discrete cosine transformation method, for each of fixed coefficients Pk given in equation (15) and equation (16) shown below and determined in accordance with a discrete cosine transformation rule;

[0106] a multiplication circuit for multiplying each of the selected transformation coefficient data by each of the corresponding fixed coefficients Pk to obtain products;

[0107] an addition/subtraction circuit for performing adding operations and/or subtracting operations between products obtained by the multiplication circuit and determined in accordance with the discrete cosine transformation rule; and

[0108] an output circuit for outputting a value obtained by the adding operations and/or subtracting operations as picture element data fj for each line or each string constituting the block of 2^N×2^Ntransformation coefficient data;

[0109] wherein the selection circuit, in the case of 2^N-2^Ninverse discrete cosine transformation, selects 2^Npieces of transformation coefficient data from 2^Npieces of transformation coefficient data contained in one line or one string constituting the block of 2^N×2^Ntransformation coefficient data for each of the fixed coefficients Pk determined by the 2^N-2^Ninverse discrete cosine transformation rule in a predetermined order, and wherein the selection circuit, in the case of 2−2^N−1−2^Ninverse discrete cosine transformation, selects first set and second set of transformation coefficient data each being composed of 2^Npieces of transformation coefficient data contained in one line or one string constituting the block of 2^N×2^Npieces of transformation coefficient data for each of fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (15) and equation (16) in a predetermined order,

[0110] wherein, the multiplication circuit, in the case of the 2^N-2^Ninverse discrete cosine transformation, multiplies each of 2^Npieces of picture element data selected in the selection process by each of corresponding fixed coefficients, out of the fixed coefficients Pk, to obtain products and wherein the multiplication circuit, in the case of the 2−2^N−1−2^Ninverse discrete cosine transformation, multiplies each of the first set and second set of transformation coefficient data selected based on each of corresponding fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (15) and equation (16) by each of the fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in the equation (15) and equation (16) to obtain products,

[0111] wherein, the addition/subtraction circuit, in the case of the 2^N-2^Ninverse discrete cosine transformation, performs the adding operations and/or subtracting operations between products obtained by the multiplication circuit and determined in accordance with the 2^N-2^Ndiscrete cosine transformation and wherein, the addition/subtraction circuit, in the case of the 2−2^N−1−2^Ninverse discrete cosine transformation, performs the adding operations and/or subtracting operations between products obtained by the multiplication process and determined in accordance with the 2−2^N−1−2^Ndiscrete cosine transformation; $\begin{array}{cc}{P}_k=\cos \left(\frac{\left(N-1-k\right)\pi }{2^{N+1}}\right)& \left(37\right)\end{array}$  0≦k≦2^N−1−2

[0112] where k=2^N−1−1 is excluded; $\begin{array}{cc}{P}_k=\frac{1}{\sqrt{2}}& \left(38\right)\end{array}$

[0113] (k=2^N−1−1).

[0114] Furthermore, a preferable mode is one wherein the selection circuit, in said 2^N-2^Ninverse discrete cosine transformation, as the transformation coefficient data, selects the first set of transformation coefficient data composed of the 2^Npieces of transformation coefficient data contained in one line or one string constituting the block of 2^N×2^Npieces of transformation coefficient data to be multiplied by each of the fixed coefficients, out of the fixed coefficients Pk, with k in odd-numbered positions in said equation (15) and equation (16) and the second set of transformation coefficient data composed of the 2^Npieces of transformation coefficient data to be multiplied by each of fixed coefficients, out of the fixed coefficients Pk, with k in even-numbered in said equation (15) and equation (16).

[0115] The above and other objects, advantages and features of the present invention will be more apparent from the following description taken in conjunction with the accompanying drawings in which:

[0116] FIG. 1 is a schematic block diagram showing electrical configurations of an 8-8/2-4-8 DCT device according to a first embodiment of the present invention;

[0117] FIG. 2 is a schematic block diagram partially showing electrical configurations of the 8-8 /2-4-8 DCT device according to the first embodiment of the present invention;

[0118] FIG. 3 shows apart of an operational timing chart explaining 8-8 DCT in the 8-8/2-4-8 DCT device according to the first embodiment of the present invention;

[0119] FIG. 4 shows a residual part of the operational timing chart explaining the 8-8 DCT in the 8-8/2-4-8 DCT device according to the first embodiment of the present invention;

[0120] FIG. 5 shows apart of an operational timing chart explaining 2-4-8 DCT in the 8-8/2-4-8 DCT device according to the first embodiment of the present invention;

[0121] FIG. 6 shows a residual part of the operational timing chart explaining the 2-4-8 DCT in the 8-8/2-4-8 DCT device according to the first embodiment of the present invention;

[0122] FIG. 7 is a schematic block diagram partially showing configurations of the 8-8/2-4-8 IDCT device according to a second embodiment of the present invention;

[0123] FIG. 8 is also a schematic block diagram partially showing configurations of an 8-8/2-4-8 IDCT device according to the second embodiment of the present invention;

[0124] FIG. 9 is a part of a timing chart explaining operations of an 8-8 IDCT constituting the 8-8 /2-4-8 IDCT device according to the second embodiment of the present invention;

[0125] FIG. 10 is a residual part of the timing chart explaining operations of the 8-8 IDCT constituting the 8-8/2-4-8 IDCT according to the second embodiment of the present invention;

[0126] FIG. 11 is a part of a timing chart explaining operations of an 2-4-8 IDCT constituting the 8-8/2-4-8 IDCT according to the second embodiment of the present invention;

[0127] FIG. 12 is a residual part of the timing chart explaining operations of the 2-4-8 IDCT constituting the 8-8/2-4-8 IDCT according to the second embodiment of the present invention;

[0128] FIG. 13 is a schematic block diagram partially showing electrical configurations of a 16-16/2-8-16 DCT device according to a third embodiment of the present invention;

[0129] FIG. 14 is a schematic block diagram partially showing electrical configurations of the 16-16/2-8-16 DCT device according to the third embodiment of the present invention;

[0130] FIG. 15 is a schematic block diagram partially showing electrical configurations of a 16-16/2-8-16 IDCT device according to a fourth embodiment of the present invention; and

[0131] FIG. 16 is a schematic block diagram partially showing electrical configurations of the 16-16/2-8-16 IDCT device according to the fourth embodiment of the present invention.

[0132] Best modes of carrying out the present invention will be described in further detail using various embodiments with reference to the accompanying drawings.

[0133] FIG. 1 is a schematic block diagram showing electrical configurations of an 8-8 DCT/2-4-8 DCT device according to a first embodiment of the present invention. FIG. 2 is a schematic block diagram partially showing electrical configurations of the 8-8 DCT/2-4-8 DCT device according to the first embodiment. FIG. 3 shows a part of an operational timing chart explaining the 8-8 DCT in the 8-8 DCT/2-4-8 DCT device according to the first embodiment. FIG. 4 shows the residual part of the operational timing chart explaining the 8-8 DCT in the 8-8 DCT/2-4-8 DCT device according to the first embodiment. FIG. 5 shows a part of an operational timing chart explaining the 2-4-8 DCT in the 8-8 DCT/2-4-8 DCT device according to the first embodiment. FIG. 6 shows the residual part of the operational timing chart explaining the 2-4-8 DCT in the 8-8 DCT/2-4-8 DCT device according to the first embodiment. By overlaying a line I—I in FIG. 1 on a line I—I in FIG. 2, an overall configuration of the 8-8 DCT/2-4-8 DCT device can be shown.

[0134] Thus, according to the first embodiment, since the 8-8 DCT/2-4-8 DCT device is so configured that a part of the 8-8 DCT circuit can be used for the 2-4-8 DCT, a high-speed calculating characteristic obtained through a pipeline processing type arithmetic operation in the 8-8 DCT can be maintained in the 2-4-8 DCT and the high-speed calculating characteristics can be still maintained in even miniaturized 8-8 DCT and 2-4-8 DCT devices.

[0135] Prior to description of configurations of the 8-8 DCT/2-4-8 DCT device, operational equations used to perform the 8-8 DCT on every one line or one string constituting 8×8 picture element data are first explained below.

[0136] By expressing the two-dimensional equation (17) described in the above “Description of the Related Art” in the form of one dimensional equation (39) and by decompressing the equation (39) with respect to h and x, the equation (39) is expressed by an equation (40). Where, P₀to P₆are as follows:

[0137] P₀=cos(7π/16)=−cos(9π/16)=0.195090322 . . .

[0138] P₁=cos(6π/16)=−cos(10π/16)=0.382683432 . . .

[0139] P₂=cos(5π/16)=−cos(11π/16)=0.55557023 . . .

[0140] P₃=cos(4π/16)=−cos(12π/16)=0.707106781 . . .

[0141] P₄=cos(3π/16)=−cos(13π/16)=0.831469612 . . .

[0142] P₅=cos(2π/16)=−cos(14π/16)=0.923879532 . . .

[0143] P₆=cos(π/16)=−cos(15π/16)=0.980785280 . . . $\begin{array}{cc}\begin{array}{c}F\left(h\right)=C\left(h\right)\sum _{x=0}^{7}f\left(x,y\right)\cos \left(\frac{\pi h\left(2x+1\right)}{16}\right)\\ =C\left(h\right)\{f\left(0,y\right)\cos \left(\frac{\pi h}{16}\right)+f\left(1,y\right)\cos \left(\frac{3\pi h}{16}\right)+f\left(2,y\right)\cos \left(\frac{5\pi h}{16}\right)+\\ f\left(3,y\right)\cos \left(\frac{7\pi h}{16}\right)+f\left(4,y\right)\cos \left(\frac{9\pi h}{16}\right)+f\left(5,y\right)\cos \left(\frac{11\pi h}{16}\right)+\\ f\left(6,y\right)\cos \left(\frac{13\pi h}{16}\right)+f\left(7,y\right)\cos \left(\frac{15\pi h}{16}\right)\}\end{array}& \left(39\right)\\ \begin{array}{c}\begin{array}{c}\left(i\right)F\left(0\right)=\frac{1}{2\sqrt{2}}\{f\left(0,y\right)·1+f\left(1,y\right)·1+f\left(2,y\right)·1+f\left(3,y\right)·1+\\ f\left(4,y\right)·1+f\left(5,y\right)·1+f\left(6,y\right)·1+f\left(7,y\right)·1\}\\ =\frac{1}{2}\{f\left(0,y\right)·\frac{1}{\sqrt{2}}+f\left(1,y\right)·\frac{1}{\sqrt{2}}+f\left(2,y\right)·\frac{1}{\sqrt{2}}+\\ f\left(3,y\right)·\frac{1}{\sqrt{2}}f\left(4,y\right)·\frac{1}{\sqrt{2}}+f\left(5,y\right)·\frac{1}{\sqrt{2}}+\\ f\left(6,y\right)·\frac{1}{\sqrt{2}}+f\left(7,y\right)·\frac{1}{\sqrt{2}}\}\\ =\frac{1}{2}\{f\left(0,y\right)·P_3+f\left(1,y\right)·P_3+f\left(2,y\right)·P_3+\\ f\left(3,y\right)·P_3+f\left(4,y\right)·P_3+f\left(5,y\right)·P_3+\\ f\left(6,y\right)·P_3+f\left(7,y\right)·P_3\}\end{array}\\ \begin{array}{c}\left(\mathrm{ii}\right)F\left(1\right)=\frac{1}{2}\{f\left(0,y\right)·P_6+f\left(1,y\right)·P_4+f\left(2,y\right)·P_2+\\ f\left(3,y\right)·P_0+f\left(4,y\right)·\left(-P_0\right)+f\left(5,y\right)·\left(-P_2\right)+\\ f\left(6,y\right)·\left(-P_4\right)+f\left(7,y\right)·\left(-P_6\right)\}\end{array}\\ \begin{array}{c}\left(\mathrm{iii}\right)F\left(2\right)=\frac{1}{2}\{f\left(0,y\right)·P_5+f\left(1,y\right)·P_6+f\left(2,y\right)·\left(-P_1\right)+\\ f\left(3,y\right)·\left(-P_5\right)+f\left(4,y\right)·\left(-P_5\right)+f\left(5,y\right)·\left(-P_1\right)+\\ f\left(6,y\right)·P_1+f\left(7,y\right)·P_5\}\end{array}\\ \begin{array}{c}\left(\mathrm{iv}\right)F\left(3\right)=\frac{1}{2}\{f\left(0,y\right)·P_4+f\left(1,y\right)·\left(-P_0\right)+f\left(2,y\right)·\left(-P_6\right)+\\ f\left(3,y\right)·P_2+f\left(4,y\right)·\left(-P_2\right)+f\left(5,y\right)·P_6+\\ f\left(6,y\right)·P_0+f\left(7,y\right)·\left(-P_4\right)\}\end{array}\\ \begin{array}{c}\left(v\right)F\left(4\right)=\frac{1}{2}\{f\left(0,y\right)·P_3+f\left(1,y\right)·\left(-P_3\right)+f\left(2,y\right)·\left(-P_3\right)+\\ f\left(3,y\right)·P_3+f\left(4,y\right)·P_3+f\left(5,y\right)·\left(-P_3\right)+\\ f\left(6,y\right)·\left(-P_3\right)+f\left(7,y\right)·P_3\}\end{array}\\ \begin{array}{c}\left(\mathrm{vi}\right)F\left(5\right)=\frac{1}{2}\{f\left(0,y\right)·P_2+f\left(1,y\right)·\left(-P_6\right)+f\left(2,y\right)·\left(-P_0\right)+\\ f\left(3,y\right)·P_4+f\left(4,y\right)·\left(-P_4\right)+f\left(5,y\right)·P_0+\\ f\left(6,y\right)·P_6+f\left(7,y\right)·\left(-P_2\right)\}\end{array}\\ \begin{array}{c}\left(\mathrm{vii}\right)F\left(6\right)=\frac{1}{2}\{f\left(0,y\right)·P_1+f\left(1,y\right)·\left(-P_5\right)+f\left(2,y\right)·\left(-P_5\right)+\\ f\left(3,y\right)·\left(-P_4\right)+f\left(4,y\right)·\left(-P_1\right)+f\left(5,y\right)·P_5+\\ f\left(6,y\right)·\left(-P_5\right)+f\left(7,y\right)·P_1\}\end{array}\\ \begin{array}{c}\left(\mathrm{viii}\right)F\left(7\right)=\frac{1}{2}\{f\left(0,y\right)·P_0+f\left(1,y\right)·P_2+f\left(2,y\right)·P_4+\\ f\left(3,y\right)·P_6+f\left(4,y\right)·\left(-P_6\right)+f\left(5,y\right)·\left(-P_4\right)+\\ f\left(6,y\right)·\left(-P_2\right)+f\left(7,y\right)·\left(-P_0\right)\}\end{array}\end{array}\}& \left(40\right)\end{array}$

[0144] Then, when the equation (40) is changed by setting so that F(0)=F₀, F(1)=F₁, F(2)=F₂, F(3)=F₃, F(4)=F₄, F(5)=F₅, F(6)=F₆and F(7)=F₇and so that f(0, y)=f₀, f(1, y)=f₁, f(2, y)=f₂, f(3, y)=f₃, f(4, y)=f₄, f(5, y)=f₅, f(6, y)=f₆and f(7, y)=f₇, the equation (40) is expressed by an equation (41). By changing the right side of the equation (41), an equation (42) can be obtained. $\begin{array}{cc}\left[\begin{array}{c}{F}_0\\ F_1\\ F_2\\ F_3\\ F_4\\ F_5\\ F_6\\ F_7\end{array}\right]=\left[\begin{array}{cccccccc}{P}_3& P_3& P_3& P_3& P_3& P_3& P_3& P_3\\ P_6& P_4& P_2& P_0& -P_0& -P_2& -P_4& -P_6\\ P_5& P_1& -P_1& -P_5& -P_5& -P_1& P_1& P_5\\ P_4& -P_0& -P_6& P_2& -P_2& P_6& P_0& -P_4\\ P_3& -P_3& -P_3& P_3& P_3& -P_3& -P_3& P_3\\ P_2& -P_6& -P_0& P_4& -P_4& P_0& P_6& -P_2\\ P_1& -P_5& P_5& -P_4& -P_1& P_5& -P_5& P_1\\ P_0& P_2& P_4& P_6& -P_6& -P_4& -P_2& -P_0\end{array}\right]\left[\begin{array}{c}{f}_0\\ f_1\\ f_2\\ f_3\\ f_4\\ f_5\\ f_6\\ f_7\end{array}\right]& \left(41\right)\\ \left[\begin{array}{c}{F}_0\\ F_1\\ F_2\\ F_3\\ F_4\\ F_5\\ F_6\\ F_7\end{array}\right]=\left[\begin{array}{cccccccc}{P}_3& P_3& P_3& P_3& 0& 0& 0& 0\\ 0& 0& 0& 0& P_6& P_4& P_2& P_0\\ P_5& -P_1& -P_1& -P_5& 0& 0& 0& 0\\ 0& 0& 0& 0& P_4& -P_0& -P_6& P_2\\ P_3& -P_3& -P_3& P_3& 0& 0& 0& 0\\ 0& 0& 0& 0& P_2& -P_6& -P_0& -P_4\\ P_1& -P_5& P_5& -P_1& 0& 0& 0& 0\\ 0& 0& 0& 0& P_0& P_2& P_4& P_6\end{array}\right]\left[\begin{array}{c}{f}_0+f_7\\ f_1+f_6\\ f_2+f_5\\ f_3+f_4\\ f_0-f_7\\ f_1-f_6\\ f_2-f_5\\ f_3-f_4\end{array}\right]& \left(42\right)\end{array}$

[0145] By substituting figures expressed by an equation (43) for f₀+f₇, f₁+f₆, f₂+f₅, f₃+f₄, f₀−f₇, f₁−f₆, f₂−d₅and f₃−f₄contained in the equation (42) and then by changing the right side of the equation (42), an equation (44) can be obtained. By rearranging the equation (44), an equation (45) can be obtained. The 8-8 DCT/2-4-8 DCT device of the first embodiment is adapted to perform the 8-8 DCT in accordance with the equation (45). $\begin{array}{cc}\begin{array}{c}{a}_0=f_0+f_7,a_1=f_1+f_6,a_2=f_2+f_5,a_3=f_3+f_4,\\ a_4=f_0-f_7,a_5=f_1-f_6,a_6=f_2-f_5,a_7=f_3-f_4\end{array}\}& \left(43\right)\\ \left[\begin{array}{c}{F}_0\\ F_2\\ F_4\\ F_6\\ F_1\\ F_3\\ F_5\\ F_7\end{array}\right]=\left[\begin{array}{ccccccc}{a}_0+a_1+a_3+a_2& 0& 0& 0& 0& 0& 0\\ 0& a_0-a_3& a_1-a_2& 0& 0& 0& 0\\ a_0-a_1+a_3-a_2& 0& 0& 0& 0& 0& 0\\ 0& -a_1+a_2& a_0-a_3& 0& 0& 0& 0\\ 0& 0& 0& a_4& a_5& a_6& a_7\\ 0& 0& 0& -a_6& a_4& -a_7& -a_5\\ 0& 0& 0& -a_5& a_7& a_4& a_6\\ 0& 0& 0& -a_7& a_6& -a_5& a_4\end{array}\right]\left[\begin{array}{c}{P}_3\\ P_5\\ P_1\\ P_6\\ P_4\\ P_2\\ P_0\end{array}\right]& \left(44\right)\\ \begin{array}{c}\left[\begin{array}{c}{F}_0\\ F_4\\ F_2\\ F_6\end{array}\right]=\left[\begin{array}{ccc}{a}_0+a_1+a_3+a_2& 0& 0\\ a_0-a_1+a_3-a_2& 0& 0\\ 0& a_0-a_3& a_1-a_2\\ 0& -a_1-a_2& a_0-a_3\end{array}\right]\left[\begin{array}{c}{P}_3\\ P_5\\ P_1\end{array}\right]\\ \left[\begin{array}{c}{F}_1\\ F_3\\ F_5\\ F_7\end{array}\right]=\left[\begin{array}{cccc}{a}_4& a_5& a_6& a_7\\ -a_6& a_4& -a_7& -a_5\\ -a_5& a_7& a_4& a_6\\ -a_7& a_6& -a_5& a_4\end{array}\right]\left[\begin{array}{c}{P}_6\\ P_4\\ P_2\\ P_0\end{array}\right]\end{array}\}& \left(45\right)\end{array}$

[0146] Next, operational equations used to perform the 2-4-8 DCT on every one line or one string of 8×8 picture element data are described below.

[0147] By expressing the two-dimensional equation (21) described in the above “Description of the Related Art” in the form of a one-dimensional equation (46) and by setting F(0), F(1), F(2), F(3), F(4), F(5), F(6), F(7), f(0, z), f(1, z), f(2, z), f(3, z), f(4, z), f(5, z), f(6, z) and f(7, z) obtained by decompressing the equation (46) with respect to h and x so that F(0)=F₀, F(1)=F₁, F(2)=F₂, F(3)=F₃, F(4)=F₄, F(5)=F₅, F(6)=F₆and F(7)=F₇, in the same manner as for the equation (40), and setting so that f(0, z)=f₀, f(1, z)=f₁, f(2, z)=f₂, f(3, z)=f₃, f(4, z)=f₄, f(5, z)=f₅, f(6, z)=f₆, and f(7, z)=f₇, an equation (47) can be obtained. Values of P₀to P6 in the equation (47) are the same as those in the equation (40). $\begin{array}{cc}\begin{array}{c}F\left(h\right)=C\left(h\right)\sum _{x=0}^7\left\{f\left(x,2z\right)+f\left(x,2z+1\right)\right\}\cos \beta \\ F\left(h,v+4\right)=C\left(h\right)\sum _{x=0}^7\left\{f\left(x,2z\right)+f\left(x,2z+1\right)\right\}\cos \beta \end{array}\}\text{where}\beta \equiv \frac{\pi h\left(2x+1\right)}{16}& \left(46\right)\\ \left[\begin{array}{c}{F}_0\\ F_1\\ F_2\\ F_3\\ F_4\\ F_5\\ F_6\\ F_7\end{array}\right]=\left[\begin{array}{cccccccc}{P}_3& P_3& P_3& P_3& 0& 0& 0& 0\\ P_5& P_1& -P_1& -P_5& 0& 0& 0& 0\\ P_3& -P_3& -P_3& P_3& 0& 0& 0& 0\\ P_1& -P_5& P_5& -P_1& 0& 0& 0& 0\\ 0& 0& 0& 0& P_3& P_3& P_3& P_3\\ 0& 0& 0& 0& P_5& P_1& -P_1& -P_5\\ 0& 0& 0& 0& P_3& -P_3& -P_3& P_3\\ 0& 0& 0& 0& P_1& -P_5& P_5& -P_1\end{array}\right]\left[\begin{array}{c}{f}_0+f_1\\ f_2+f_3\\ f_4+f_5\\ f_6+f_7\\ f_0-f_1\\ f_2-f_3\\ f_4-f_5\\ f_6-f_7\end{array}\right]& \left(47\right)\end{array}$

[0148] By changing the right side of the equation (47), an equation (48) can be obtained. By rearranging the equation (48), an equation (49) can be obtained. $\begin{array}{cc}\begin{array}{c}\left[\begin{array}{c}{F}_0\\ F_1\\ F_2\\ F_3\end{array}\right]=\left[\begin{array}{ccc}\left(f_0+f_1\right)+\left(f_2+f_3\right)+\left(f_4+f_5\right)+\left(f_6+f_7\right)& 0& 0\\ 0& \left(f_0+f_1\right)-\left(f_6+f_7\right)& \left(f_2+f_3\right)-\left(f_4+f_5\right)\\ \left(f_0+f_1\right)-\left(f_2+f_3\right)-\left(f_4+f_5\right)+\left(f_6+f_7\right)& 0& 0\\ 0& -\left(f_2+f_3\right)+\left(f_4+f_5\right)& \left(f_0+f_1\right)-\left(f_6+f_7\right)\end{array}\right]\left[\begin{array}{c}{P}_3\\ P_5\\ P_1\end{array}\right]\\ \left[\begin{array}{c}{F}_4\\ F_5\\ F_6\\ F_7\end{array}\right]=\left[\begin{array}{ccc}\left(f_0-f_1\right)+\left(f_2-f_3\right)+\left(f_4-f_5\right)+\left(f_6-f_7\right)& 0& 0\\ 0& \left(f_0-f_1\right)-\left(f_6-f_7\right)& \left(f_2-f_3\right)-\left(f_4-f_5\right)\\ \left(f_0-f_1\right)-\left(f_2-f_3\right)-\left(f_4-f_5\right)+\left(f_6-f_7\right)& 0& 0\\ 0& -\left(f_2-f_3\right)+\left(f_4-f_5\right)& \left(f_0-f_1\right)-\left(f_6-f_7\right)\end{array}\right]\left[\begin{array}{c}{P}_3\\ P_5\\ P_1\end{array}\right]\end{array}\}& \left(48\right)\\ \begin{array}{c}\left[\begin{array}{c}{F}_0\\ F_2\\ F_1\\ F_3\end{array}\right]=\left[\begin{array}{ccc}\left(f_0+f_1\right)+\left(f_2+f_3\right)+\left(f_4+f_5\right)+\left(f_6+f_7\right)& 0& 0\\ \left(f_0+f_1\right)-\left(f_2+f_3\right)-\left(f_4+f_5\right)+\left(f_6+f_7\right)& 0& 0\\ 0& \left(f_0+f_1\right)-\left(f_6+f_7\right)& \left(f_2+f_3\right)-\left(f_4+f_5\right)\\ 0& -\left(f_2+f_3\right)+\left(f_4+f_5\right)& \left(f_0+f_1\right)-\left(f_6+f_7\right)\end{array}\right]\left[\begin{array}{c}{P}_3\\ P_5\\ P_1\end{array}\right]\\ \left[\begin{array}{c}{F}_4\\ F_6\\ F_5\\ F_7\end{array}\right]=\left[\begin{array}{ccc}\left(f_0-f_1\right)+\left(f_2-f_3\right)+\left(f_4-f_5\right)+\left(f_6-f_7\right)& 0& 0\\ \left(f_0-f_1\right)-\left(f_2-f_3\right)-\left(f_4-f_5\right)+\left(f_6-f_7\right)& 0& 0\\ 0& \left(f_0-f_1\right)-\left(f_6-f_7\right)& \left(f_2-f_3\right)-\left(f_4-f_5\right)\\ 0& -\left(f_2-f_3\right)+\left(f_4-f_5\right)& \left(f_0-f_1\right)-\left(f_6-f_7\right)\end{array}\right]\left[\begin{array}{c}{P}_3\\ P_5\\ P_1\end{array}\right]\end{array}\}& \left(49\right)\end{array}$