Apparatus and method for deriving an enhanced decoded reduced-resolution video signal from a coded high-definition video signal

[0001] This is a non-provisional application of provisional application Ser. No. 60/133,429 by M. L. Corner et al, filed 11 May 1999.

[0002] The present invention relates to the decoding of a coded high-definition (HD) video signal to derive an enhanced decoded video signal suitable, for example, for recording or producing a picture-in-picture (PIP) or other reduced-resolution display.

[0003] Known in the art are television receivers that, while displaying a relatively large picture derived from a primary television channel, also simultaneously display a small picture-in-picture (PIP) derived from a secondary television channel. In the case of a high-definition television (HDTV) receiver, the receiver must include a relatively complex and expensive decoder that conforms with the MPEG ISO 13818-2 standard for decoding a received coded HD video signal in real time for high definition display. However, because the PIP is small, there is no need to provide a high definition PIP display because a viewer inherently would not be able to resolve the higher definition components of a high definition PIP. Therefore, to provide the PIP, the HDTV receiver may be supplied with a lower-resolution second simpler and less expensive decoder which still conforms with the ISO 13818-2 standard.

[0004] One approach, known in the art, to providing a lower-resolution second decoder which is somewhat simpler and less expensive than the decoder providing the high definition display, is disclosed in the three U.S. Pat. Nos. 5,614,952, 5,614,957 and 5,635,985, which were, respectively, issued to Boyce et al. on Mar. 25, 1997, Mar. 25, 1997 and Jun. 3, 1997.

[0005] Further, incorporated herein by reference is the teaching of copending U.S. patent application Ser. No. 09/349,865, filed Jul. 8, 1999 and assigned to the same assignee as the present application, which is directed to a lower-resolution second-decoder approach suitable for deriving a PIP display in real time from a received coded HD video signal that is significantly simpler and less expensive to implement than is the second decoder disclosed by Boyce et al, but still conforms with the ISO 13818-2 standard.

[0006] A system involves decoding compressed image data including frequency domain coefficients defining blocks of pixel values representing an image at a first resolution to provide an image at a reduced second resolution. The system includes a motion-compensation-unit (MCU) processor responsive to a selected sub-set of the frequency domain coefficients for deriving the image of the reduced second resolution. The motion-compensation-unit (MCU) processor employs blocks of pixel values representing image data at an intermediate third resolution lower than the first resolution and higher than the reduced second resolution.

[0013] Referring to FIG. 1, there is shown VLD 100, PIP decoding means 102 and HD decoding means 104. In accordance with the known teaching of the MPEG ISO 13818-2 standard, one of the responses of VLD 100 to the input coded HD MPEG data comprising a sequence of MPEG I, P and B frames is to convey coded picture information defined by each of successive 8×8 blocks of quantized discrete cosine transform (DCT) coefficients as an input to HD decoding means 104. Further, in accordance with the known teaching of the MPEG ISO 13818-2 standard, among the functions performed by HD decoding means 104 is to first perform inverse quantization of each successive 8×8 block of DCT coefficients and then perform inverse discrete cosine transformation (IDCT) of the DCT coefficients of each successive 8×8 block. Finally, HD decoding means 104 must perform motion compensation for each P frame and bi-directionally predictive B frame after IDCT has been performed on that P or B frame.

[0014] FIG. 1a shows an 8×8 block of DCT coefficients, wherein (1) the value of coefficient DCT_0,0 (located in the upper left corner of the 8×8 block) represents the average (DC) value (i.e., both the horizontal and vertical frequencies are 0) of the picture defined by the 64 values of a corresponding 8×8 block of pixels prior to having undergone DCT, while (2) the value of coefficient DCT_7,7 (located in the lower right corner of the 8×8 block) represents the highest horizontal frequency and highest vertical frequency components of the picture defined by the 64 values of a corresponding 8×8 block of pixels prior to having undergone DCT. For the case of a HD picture, all, or nearly all, of the 64 DCT coefficients from DCT_0,0 to DCT_7,7 inclusive of FIG. 1a may have non-zero values. This results in a relatively large amount of image-processing computation to accomplish IDCT in real time. Further, motion compensation also involves a large amount of real time image-processing computation. Therefore, HD decoding means 104 requires about 96 Mbits memory to temporarily store MPEG decoded image frames prior to display. HD decoding means 104 requires these frames for motion compensation to reconstruct accurate images for display. Thus, a physical implementation of HD decoding means 104 is relatively expensive.

[0015] Returning to FIG. 1, another of the responses of VLD 100 to the input coded HD MPEG data is to convey only coded picture information defined by a relatively small given number of lower-frequency-defining, quantized DCT coefficients of each successive 8×8 block as an input to PIP decoding means 102. It is to be noted that the PIP processing and images and the term PIP itself is used herein to encompass any form of reduced resolution image and processing and not just television PIP image generation. While the preferred tutorial example of the PIP decoding means described in the aforesaid patent application Ser. No. 09/349,865 employed only the 6 lowest-frequency quantized DCT coefficients, the preferred tutorial example of enhanced-quality PIP decoding means 102, described in detail below, employs 10 DCT coefficients consisting of DCT_0,0, DCT_1,0, DCT_2,0, DCT_3,0, DCT_0,1, DCT_1,1, DCT_2,1, DCT_0,2, DCT_1,2, and DCT_0,3 shown in FIG. 1b for progressive-scan use or, alternatively, consisting of DCT_0,0, DCT_1,0, DCT_2,0, DCT_0,1, DCT_1,1, DCT_0,2, DCT_0,3, DCT_0,4, DCT_0,5, and DCT_0,6 shown in FIG. 1c for interlaced-scan use, thereby providing better high-frequency response for the enhanced PIP display. More specifically, the PIP bit-stream received by VLD 100 has been pre-parsed by a VLD-PIP parser (not shown to simplify FIG. 1) to remove from the bitstream DCT coefficients which are not needed by the PIP decoder.

[0016] The simplified functional block diagram of the embodiment of enhanced-quality PIP decoding means 102 shown in FIG. 2 comprises runlength decoder (RLD) 200, inverse quantizer (IQ) 202, unitary enhanced IDCT, filtering and pixel-decimation processing means 204, base-layer adder 205B, enhancement-layer adder 205E, base and enhancement-layer decimated pixel memory 206, enhancement-layer encoder 207, enhanced motion compensation unit (MCU) processing means 208 and sample-rate converter 210. Although the simplified functional block diagram of FIG. 2 does not show means for controlling the operation of this embodiment of enhanced-quality PIP decoding means 102, it should be understood that suitable control means that conform with the requirements of the ISO 13818-2 standard is included in a physical implementation of this embodiment.

[0017] For illustrative purposes, the following description of elements 200, 202, 204, 205B, 205E, 206, 207, 208 and 210 assumes that each of these elements is being operated in accordance with the above-discussed preferred tutorial example.

[0018] In this example, RLD 200 outputs 10 DCT coefficients for each 8×8 coded block using the 2 scan patterns defined in the ISO 13818-2 standard. The positioning of the 10 DCT coefficients within each 8×8 block is illustrated in FIG. 1b for progressive scan and in FIG. 1c for interlaced scan, as determined by the state of the progressive_—sequence flag. In the FIG. 1b progressive sequence case, if the alternate_—scan flag from the picture coding extension for the current picture is 0, the 10 DCT coefficients correspond to coefficients 0,1,2,3,4,5,6,7,8,9 in 1-dimensional scan order, whereas if alternate_—scan is 1, the 10 DCT coefficients of interest are coefficients 0,1,2,3,4,5,6,7,8,20 in scan order. In the FIG. 1c interlaced sequence case, if the alternate_—scan flag from the picture coding extension for the current picture is 0, the 10 DCT coefficients correspond to coefficients 0,1,2,3,4,5,9,10,20,21 in 1-dimensional scan order, whereas if alternate_—scan is 1, the 10 DCT coefficients of interest are coefficients 0,1,2,3,4,5,6,10,11,12 in scan order. There are two run values that have a meaning in RLD 200 which is different from that described in the ISO 13818-2 standard, depending on the value of the alternate_—scan and progressive_—sequence flags. For progressive sequences, if alternate_—scan is 0, a run value of 10 indicates that the coefficients needed by PIP decoder 102 are all 0 and there is no subsequent non-zero coefficient. Similarly, if alternate_—scan is 1, a run value of 21 indicates that the coefficients needed by decoder 102 are all 0 and there is no subsequent non-zero coefficient. For interlaced sequences, if alternate_—scan is 0, a run value of 22 indicates that the coefficients needed by PIP decoder 102 are all 0 and there is no subsequent non-zero coefficient. Similarly, if alternate_—scan is 1, a run value of 13 indicates that the coefficients needed by decoder 102 are all 0 and there is no subsequent non-zero coefficient. Table 1, summarizes the meaning of run values of 10 and 21 for the two possible values of the alternate_—scan flag for progressive sequence and Table 2, summarizes the meaning of run values of 13 and 22 for the two possible values of the alternate_—scan flag for interlaced sequence. All other alternate_—scan/run value combinations encountered by RLD 200 are interpreted as described in the ISO 13818-2 standard.

[0019]

TABLE 1
Interpretation of run = 10 and run = 21 by RLD 200
for progressive sequences
	Run	Alternate—Scan	Interpretation in PIP RLD
	10	0	All DCT coefficients = 0
	10	1	Same as ISO 13818-2 standard
	21	0	Not allowed
	21	1	All DCT coefficients = 0

[0020]

TABLE 2
Interpretation of run = 13 and run = 22 by RLD 200
for interlaced sequences
	Run	Alternate—Scan	Interpretation in PIP RLD
	13	0	Same as ISO 13818-2 standard
	13	1	All DCT coefficients = 0
	22	0	All DCT coefficients = 0
	22	1	Not allowed

[0021] IQ 202 performs inverse quantization arithmetic and saturation described in the ISO 13818-2 standard on the 10 DCT coefficients shown in FIG. 1b for progressive sequences and shown in FIG. 1c for interlaced sequences. The mismatch control portion of the inverse quantization process is not needed. Conventionally, an extensive computational process requiring three separate steps is needed to convert the coded frequency domain information in an 8×8 block at the output from IQ 202, into spatial domain picture information comprising respective values of a smaller block of decimated pixels of a reduced-resolution PIP display image. The first step is to determine the value of each of the 64 (i.e., full pixel density) pixel values of each 8×8 block of picture information as an IDCT function of the inversely-quantized DCT coefficient values. Thereafter, the second step of lowpass filtering followed by the third step of pixel decimation may be performed on the pixels in each successive 8×8 block to provide the desired smaller block of decimated pixels. For instance, a decimation of alternate horizontal and vertical filtered pixels for the case of a progressive scan would result in a 75% reduction in pixel density. Similarly, a decimation of 3 out of 4 successive filtered pixels in the horizontal direction for the case of an interlaced scan would also result in a 75% reduction in pixel density. Thus, in either case, such a decimation performed for luma pixels and also for chroma pixels would result in a reduction in pixel density from 64 per 8×8 block to only 16 per 8×8 block for each of them. However, the amount of hardware required to implement this conventional three-step computational process is relatively large and, therefore, relatively expensive.

[0022] In accordance with the principles of the invention taught in the aforesaid patent application Ser. No. 09/349,865, the unitary IDCT, filtering and pixel-decimation processing means disclosed therein is able to convert the coded respective values of inversely-quantized DCT coefficients contained in an 8×8 block at the output from IQ 202, into a smaller block of decimated pixels in a single-step computational process. Thus, the amount of hardware required to implement this single-step computational process by means 204 is relatively small and, therefore, relatively inexpensive compared to the aforesaid conventional three-step computational process.

[0023] Specifically, in accordance with the teachings of the aforesaid patent application Ser. No. 09/349,865, the decimated pixel memory thereof (which, because of pixel decimation, requires a storage capacity size of only ¼ the capacity size of a corresponding undecimated pixel memory) comprises a plurality of separate buffers. Each of these buffers is capable of temporarily storing decimated luma and chroma pixels. In conformity with the ISO 13818-2 standard, the decimated pixel memory includes one or more buffers for storing decimated pixels that define reconstructed intracoded (I), predictive-coded (P) and/or bi-directionally predictive-coded (B) frame or field pictures. Further, motion-compensated prediction macroblock output of pixel values from the MCU processing means is added in an adder to each corresponding macroblock output derived in the unitary IDCT, filtering and pixel-decimation processing means. The summed pixel values of the output from the adder are stored into a first buffer of the decimated pixel memory. This first buffer may be a first-in first-out (FIFO) buffer in which the stored decimated pixels may be reordered between (1) being written into the first buffer and (2) being read out from the first buffer and written into another buffer of the decimated pixel memory. In the case of a current P or B frame or field, the decimated pixel memory includes a buffer for storing a macroblock for input to the MCU processing means to provide motion compensation.

[0024] In accordance with the principles of the present invention, two layers of decimated pixels are advantageously stored respectively in base and enhancement-layer decimated pixel memory 206 to achieve high-quality motion compensation and improve PIP image quality. The first of these two layers is a base-layer of decimated pixels and the second of these two layers is an enhancement-layer of vector-quantized values of luma macroblock decimated pixels that are employed in enhanced MCU processing means 208 during decoding of P pictures The enhanced layer is used to provide a reduced resolution image of greater resolution than is obtainable by using just the decimated pixels of the base-layer. Both the base-layer and this enhancement-layer are employed by enhanced MCU processing means 208, in a manner described in detail below.

[0025] Preferred embodiments of IDCT, filtering and pixel-decimation processing means 204, enhancement-layer encoder 207 and enhanced MCU processing means 208 for implementing the present invention will now be described in detail.

[0026] The unitary enhanced IDCT, filtering and pixel-decimation processing means 204 provides the following sets of 16 decimated pixel values for each of the base and enhancement-layers used for each of progressive scan and interlaced scan (each set being a function of 10 DCT coefficient values).
Progressive-Scan, Base-Layer Set of Decimated Pixel Values $\begin{array}{c}{g}_1\left(0,0\right)=[8{\mathrm{DCT}}_{0,0}+10{\mathrm{DCT}}_{1,0}+7{\mathrm{DCT}}_{2,0}+4{\mathrm{DCT}}_{3,0}+10{\mathrm{DCT}}_{0,1}+\\ 13{\mathrm{DCT}}_{1,1}+9{\mathrm{DCT}}_{2,1}+7{\mathrm{DCT}}_{0,2}+9{\mathrm{DCT}}_{1,2}+4{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(1,0\right)=[8{\mathrm{DCT}}_{0,0}+4{\mathrm{DCT}}_{1,0}-7{\mathrm{DCT}}_{2,0}-9{\mathrm{DCT}}_{3,0}+10{\mathrm{DCT}}_{0,1}+\\ 5{\mathrm{DCT}}_{1,1}-9{\mathrm{DCT}}_{2,1}+7{\mathrm{DCT}}_{0,2}+4{\mathrm{DCT}}_{1,2}+4{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(2,0\right)=[8{\mathrm{DCT}}_{0,0}-4{\mathrm{DCT}}_{1,0}-7{\mathrm{DCT}}_{2,0}+9{\mathrm{DCT}}_{3,0}+10{\mathrm{DCT}}_{0,1}-\\ 5{\mathrm{DCT}}_{1,1}-9{\mathrm{DCT}}_{2,1}+7{\mathrm{DCT}}_{0,2}-4{\mathrm{DCT}}_{1,2}+4{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(3,0\right)=[8{\mathrm{DCT}}_{0,0}-10{\mathrm{DCT}}_{1,0}+7{\mathrm{DCT}}_{2,0}-4{\mathrm{DCT}}_{3,0}+10{\mathrm{DCT}}_{0,1}-\\ 13{\mathrm{DCT}}_{1,1}+9{\mathrm{DCT}}_{2,1}+7{\mathrm{DCT}}_{0,2}-9{\mathrm{DCT}}_{1,2}+4{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(0,1\right)=[8{\mathrm{DCT}}_{0,0}+10{\mathrm{DCT}}_{1,0}+7{\mathrm{DCT}}_{2,0}+4{\mathrm{DCT}}_{3,0}+4{\mathrm{DCT}}_{0,1}+\\ 5{\mathrm{DCT}}_{1,1}+4{\mathrm{DCT}}_{2,1}-7{\mathrm{DCT}}_{0,2}-9{\mathrm{DCT}}_{1,2}-9{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(1,1\right)=[8{\mathrm{DCT}}_{0,0}+4{\mathrm{DCT}}_{1,0}-7{\mathrm{DCT}}_{2,0}-9{\mathrm{DCT}}_{3,0}+4{\mathrm{DCT}}_{0,1}+\\ 2{\mathrm{DCT}}_{1,1}-4{\mathrm{DCT}}_{2,1}-7{\mathrm{DCT}}_{0,2}-4{\mathrm{DCT}}_{1,2}-9{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(2,1\right)=[8{\mathrm{DCT}}_{0,0}-4{\mathrm{DCT}}_{1,0}-7{\mathrm{DCT}}_{2,0}+9{\mathrm{DCT}}_{3,0}+4{\mathrm{DCT}}_{0,1}-\\ 2{\mathrm{DCT}}_{1,1}-4{\mathrm{DCT}}_{2,1}-7{\mathrm{DCT}}_{0,2}+4{\mathrm{DCT}}_{1,2}-9{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(3,1\right)=[8{\mathrm{DCT}}_{0,0}-10{\mathrm{DCT}}_{1,0}+7{\mathrm{DCT}}_{2,0}-4{\mathrm{DCT}}_{3,0}+4{\mathrm{DCT}}_{0,1}-\\ 5{\mathrm{DCT}}_{1,1}+4{\mathrm{DCT}}_{2,1}-7{\mathrm{DCT}}_{0,2}+9{\mathrm{DCT}}_{1,2}-9{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(0,2\right)=[8{\mathrm{DCT}}_{0,0}+10{\mathrm{DCT}}_{1,0}+7{\mathrm{DCT}}_{2,0}+4{\mathrm{DCT}}_{3,0}-4{\mathrm{DCT}}_{0,1}-\\ 5{\mathrm{DCT}}_{1,1}-4{\mathrm{DCT}}_{2,1}-7{\mathrm{DCT}}_{0,2}-9{\mathrm{DCT}}_{1,2}+9{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(1,2\right)=[8{\mathrm{DCT}}_{0,0}+4{\mathrm{DCT}}_{1,0}-7{\mathrm{DCT}}_{2,0}-9{\mathrm{DCT}}_{3,0}-4{\mathrm{DCT}}_{0,1}-\\ 2{\mathrm{DCT}}_{1,1}+4{\mathrm{DCT}}_{2,1}-7{\mathrm{DCT}}_{0,2}-4{\mathrm{DCT}}_{1,2}+9{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(2,2\right)=[8{\mathrm{DCT}}_{0,0}-4{\mathrm{DCT}}_{1,0}-7{\mathrm{DCT}}_{2,0}+9{\mathrm{DCT}}_{3,0}-4{\mathrm{DCT}}_{0,1}+\\ 2{\mathrm{DCT}}_{1,1}+4{\mathrm{DCT}}_{2,1}-7{\mathrm{DCT}}_{0,2}+4{\mathrm{DCT}}_{1,2}+9{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(3,2\right)=[8{\mathrm{DCT}}_{0,0}-10{\mathrm{DCT}}_{1,0}+7{\mathrm{DCT}}_{2,0}-4{\mathrm{DCT}}_{3,0}-4{\mathrm{DCT}}_{0,1}+\\ 5{\mathrm{DCT}}_{1,1}-4{\mathrm{DCT}}_{2,1}-7{\mathrm{DCT}}_{0,2}+9{\mathrm{DCT}}_{1,2}+9{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(0,3\right)=[8{\mathrm{DCT}}_{0,0}+10{\mathrm{DCT}}_{1,0}+7{\mathrm{DCT}}_{2,0}+4{\mathrm{DCT}}_{3,0}-10{\mathrm{DCT}}_{0,1}-\\ 13{\mathrm{DCT}}_{1,1}-9{\mathrm{DCT}}_{2,1}+7{\mathrm{DCT}}_{0,2}+9{\mathrm{DCT}}_{1,2}-4{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(1,3\right)=[8{\mathrm{DCT}}_{0,0}+4{\mathrm{DCT}}_{1,0}-7{\mathrm{DCT}}_{2,0}-9{\mathrm{DCT}}_{3,0}-10{\mathrm{DCT}}_{0,1}-\\ 5{\mathrm{DCT}}_{1,1}+9{\mathrm{DCT}}_{2,1}+7{\mathrm{DCT}}_{0,2}+4{\mathrm{DCT}}_{1,2}-4{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(2,3\right)=[8{\mathrm{DCT}}_{0,0}-4{\mathrm{DCT}}_{1,0}-7{\mathrm{DCT}}_{2,0}+9{\mathrm{DCT}}_{3,0}-10{\mathrm{DCT}}_{0,1}+\\ 5{\mathrm{DCT}}_{1,1}+9{\mathrm{DCT}}_{2,1}+7{\mathrm{DCT}}_{0,2}-4{\mathrm{DCT}}_{1,2}-4{\mathrm{DCT}}_{0,3}]/64\\ g_1\left(3,3\right)=[8{\mathrm{DCT}}_{0,0}-10{\mathrm{DCT}}_{1,0}+7{\mathrm{DCT}}_{2,0}-4{\mathrm{DCT}}_{3,0}-10{\mathrm{DCT}}_{0,1}+\\ 13{\mathrm{DCT}}_{1,1}-9{\mathrm{DCT}}_{2,1}+7{\mathrm{DCT}}_{0,2}-9{\mathrm{DCT}}_{1,2}-4{\mathrm{DCT}}_{0,3}]/64\end{array}$
Progressive-Scan, Enhancement-Layer Set of Decimated PIXEL Values $\begin{array}{c}{g}_0\left(0,0\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+1{\mathrm{DCT}}_{0,1}+\\ 1{\mathrm{DCT}}_{1,1}+1{\mathrm{DCT}}_{2,1}+3{\mathrm{DCT}}_{0,2}+4{\mathrm{DCT}}_{1,2}+6{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(1,0\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+1{\mathrm{DCT}}_{0,1}+\\ 0{\mathrm{DCT}}_{1,1}-1{\mathrm{DCT}}_{2,1}+3{\mathrm{DCT}}_{0,2}+2{\mathrm{DCT}}_{1,2}+6{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(2,0\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+1{\mathrm{DCT}}_{0,1}+\\ 0{\mathrm{DCT}}_{1,1}-1{\mathrm{DCT}}_{2,1}+3{\mathrm{DCT}}_{0,2}-2{\mathrm{DCT}}_{1,2}+6{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(3,0\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+1{\mathrm{DCT}}_{0,1-}\\ 1{\mathrm{DCT}}_{1,1}+1{\mathrm{DCT}}_{2,1}+3{\mathrm{DCT}}_{0,2}-4{\mathrm{DCT}}_{1,2}+6{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(0,2\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+2{\mathrm{DCT}}_{0,1}+\\ 3{\mathrm{DCT}}_{1,1}+2{\mathrm{DCT}}_{2,1}+3{\mathrm{DCT}}_{0,2}+4{\mathrm{DCT}}_{1,2}-2{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(1,2\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+2{\mathrm{DCT}}_{0,1}+\\ 1{\mathrm{DCT}}_{1,1}-2{\mathrm{DCT}}_{2,1}+3{\mathrm{DCT}}_{0,2}+2{\mathrm{DCT}}_{1,2}-2{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(2,2\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+2{\mathrm{DCT}}_{0,1}-\\ 1{\mathrm{DCT}}_{1,1}-2{\mathrm{DCT}}_{2,1}+3{\mathrm{DCT}}_{0,2}-2{\mathrm{DCT}}_{1,2}-2{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(3,2\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+2{\mathrm{DCT}}_{0,1}-\\ 3{\mathrm{DCT}}_{1,1}+2{\mathrm{DCT}}_{2,1}+3{\mathrm{DCT}}_{0,2}-4{\mathrm{DCT}}_{1,2}-2{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(0,4\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+2{\mathrm{DCT}}_{0,1}+\\ 3{\mathrm{DCT}}_{1,1}+2{\mathrm{DCT}}_{2,1}-3{\mathrm{DCT}}_{0,2}-4{\mathrm{DCT}}_{1,2}-2{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(1,4\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+2{\mathrm{DCT}}_{0,1}+\\ 1{\mathrm{DCT}}_{1,1}-2{\mathrm{DCT}}_{2,1}-3{\mathrm{DCT}}_{0,2}-2{\mathrm{DCT}}_{1,2}-2{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(2,4\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+2{\mathrm{DCT}}_{0,1}-\\ 1{\mathrm{DCT}}_{1,1}-2{\mathrm{DCT}}_{2,1}-3{\mathrm{DCT}}_{0,2}+2{\mathrm{DCT}}_{1,2}-2{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(3,4\right)=0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+2{\mathrm{DCT}}_{0,1}-\\ 3{\mathrm{DCT}}_{1,1}+2{\mathrm{DCT}}_{2,1}-3{\mathrm{DCT}}_{0,2}+4{\mathrm{DCT}}_{1,2}-2{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(0,6\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+1{\mathrm{DCT}}_{0,1}+\\ 1{\mathrm{DCT}}_{1,1}+1{\mathrm{DCT}}_{2,1}-3{\mathrm{DCT}}_{0,2}+49{\mathrm{DCT}}_{1,2}+6{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(1,6\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+1{\mathrm{DCT}}_{0,1}+\\ 0{\mathrm{DCT}}_{1,1}-1{\mathrm{DCT}}_{2,1}-3{\mathrm{DCT}}_{0,2}-2{\mathrm{DCT}}_{1,2}+6{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(2,6\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+1{\mathrm{DCT}}_{0,1}+\\ 0{\mathrm{DCT}}_{1,1}-1{\mathrm{DCT}}_{2,1}-3{\mathrm{DCT}}_{0,2}+2{\mathrm{DCT}}_{1,2}+6{\mathrm{DCT}}_{0,3}]/64\\ g_0\left(3,6\right)=[0{\mathrm{DCT}}_{0,0}+0{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{3,0}+1{\mathrm{DCT}}_{0,1}-\\ 1{\mathrm{DCT}}_{1,1}+1{\mathrm{DCT}}_{2,1}-3{\mathrm{DCT}}_{0,2}+4{\mathrm{DCT}}_{1,2}+6{\mathrm{DCT}}_{0,3}]/64\end{array}$
Interlaced-Scan, Base-Layer Set of Decimated Pixel Values $\begin{array}{c}{g}_1\left(0,0\right)=[8{\mathrm{DCT}}_{0,0}+7{\mathrm{DCT}}_{1,0}+11{\mathrm{DCT}}_{0,1}+10{\mathrm{DCT}}_{1,1}+10{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}+9{\mathrm{DCT}}_{0,3}+8{\mathrm{DCT}}_{0,4}+6{\mathrm{DCT}}_{0,5}+4{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(1,0\right)=[8{\mathrm{DCT}}_{0,0}-7{\mathrm{DCT}}_{1,0}+11{\mathrm{DCT}}_{0,1}-10{\mathrm{DCT}}_{1,1}+10{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}+9{\mathrm{DCT}}_{0,3}+8{\mathrm{DCT}}_{0,4}+6{\mathrm{DCT}}_{0,5}+4{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(0,1\right)=[8{\mathrm{DCT}}_{0,0}+7{\mathrm{DCT}}_{1,0}+9{\mathrm{DCT}}_{0,1}+9{\mathrm{DCT}}_{1,1}+4{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}-2{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}-11{\mathrm{DCT}}_{0,5}-10{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(1,1\right)=[8{\mathrm{DCT}}_{0,0}-7{\mathrm{DCT}}_{1,0}+9{\mathrm{DCT}}_{0,1}-9{\mathrm{DCT}}_{1,1}+4{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}-2{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}-11{\mathrm{DCT}}_{0,5}-10{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(0,2\right)=[8{\mathrm{DCT}}_{0,0}+7{\mathrm{DCT}}_{1,0}+6{\mathrm{DCT}}_{0,1}+6{\mathrm{DCT}}_{1,1}-4{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}-11{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}+2{\mathrm{DCT}}_{0,5}+10{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(1,2\right)=[8{\mathrm{DCT}}_{0,0}-7{\mathrm{DCT}}_{1,0}+6{\mathrm{DCT}}_{0,1}-6{\mathrm{DCT}}_{1,1}-4{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}-11{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}+2{\mathrm{DCT}}_{0,5}+10{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(0,3\right)=[8{\mathrm{DCT}}_{0,0}+7{\mathrm{DCT}}_{1,0}+2{\mathrm{DCT}}_{0,1}+2{\mathrm{DCT}}_{1,1}-10{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}-6{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}+9{\mathrm{DCT}}_{0,5}-4{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(1,3\right)=[8{\mathrm{DCT}}_{0,0}-7{\mathrm{DCT}}_{1,0}+2{\mathrm{DCT}}_{0,1}+2{\mathrm{DCT}}_{1,1}-10{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}-6{\mathrm{DCT}}_{0,3}+8{\mathrm{DCT}}_{0,4}+9{\mathrm{DCT}}_{0,5}-4{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(0,4\right)=[8{\mathrm{DCT}}_{0,0}+7{\mathrm{DCT}}_{1,0}-2{\mathrm{DCT}}_{0,1}-2{\mathrm{DCT}}_{1,1}-10{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}+6{\mathrm{DCT}}_{0,3}+8{\mathrm{DCT}}_{0,4}-9{\mathrm{DCT}}_{0,5}-4{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(1,4\right)=[8{\mathrm{DCT}}_{0,0}-7{\mathrm{DCT}}_{1,0}-2{\mathrm{DCT}}_{0,1}+2{\mathrm{DCT}}_{1,1}-10{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}+6{\mathrm{DCT}}_{0,3}+8{\mathrm{DCT}}_{0,4}-9{\mathrm{DCT}}_{0,5}-4{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(0,5\right)=[8{\mathrm{DCT}}_{0,0}+7{\mathrm{DCT}}_{1,0}-6{\mathrm{DCT}}_{0,1}-6{\mathrm{DCT}}_{1,1}-4{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}+11{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}-2{\mathrm{DCT}}_{0,5}+10{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(1,5\right)=[8{\mathrm{DCT}}_{0,0}-7{\mathrm{DCT}}_{1,0}-6{\mathrm{DCT}}_{0,1}+6{\mathrm{DCT}}_{1,1}-4{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}+11{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}-2{\mathrm{DCT}}_{0,5}+10{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(0,6\right)=[8{\mathrm{DCT}}_{0,0}+7{\mathrm{DCT}}_{1,0}-9{\mathrm{DCT}}_{0,1}-9{\mathrm{DCT}}_{1,1}+4{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}+2{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}+11{\mathrm{DCT}}_{0,5}-10{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(1,6\right)=[8{\mathrm{DCT}}_{0,0}-7{\mathrm{DCT}}_{1,0}-9{\mathrm{DCT}}_{0,1}+9{\mathrm{DCT}}_{1,1}+4{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}+2{\mathrm{DCT}}_{0,3}-8{\mathrm{DCT}}_{0,4}+11{\mathrm{DCT}}_{0,5}-10{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(0,7\right)=[8{\mathrm{DCT}}_{0,0}+7{\mathrm{DCT}}_{1,0}-11{\mathrm{DCT}}_{0,1}-10{\mathrm{DCT}}_{1,1}+10{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}-9{\mathrm{DCT}}_{0,3}+8{\mathrm{DCT}}_{0,4}-6{\mathrm{DCT}}_{0,5}+4{\mathrm{DCT}}_{0,6}]/64\\ g_1\left(1,7\right)=[8{\mathrm{DCT}}_{0,0}-7{\mathrm{DCT}}_{1,0}-11{\mathrm{DCT}}_{0,1}+10{\mathrm{DCT}}_{1,1}+10{\mathrm{DCT}}_{0,2}+\\ 0{\mathrm{DCT}}_{2,0}-9{\mathrm{DCT}}_{0,3}+8{\mathrm{DCT}}_{0,4}-6{\mathrm{DCT}}_{0,5}+4{\mathrm{DCT}}_{0,6}]/64\end{array}$
Interlaced-Scan, Enhancement-Layer Set of Decimated Pixel Values $\begin{array}{c}{g}_0\left(0,0\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}+4{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}+\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(2,0\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}+4{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}-\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(0,1\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}+4{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}+\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(2,1\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}+4{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}-\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(0,2\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}+2{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}+\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(2,2\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}+2{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}-\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(0,3\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}+1{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}+\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(2,3\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}+1{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}-\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(0,4\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}-1{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}+\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(2,4\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}-1{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}-\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(0,5\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}-2{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}+\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(2,5\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}-2{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}-\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(0,6\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}-4{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}+\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(2,6\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}-4{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}-\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(0,7\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}-4{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}+\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\\ g_0\left(2,7\right)=[0{\mathrm{DCT}}_{0,0}+3{\mathrm{DCT}}_{1,0}+0{\mathrm{DCT}}_{0,1}-4{\mathrm{DCT}}_{1,1}+0{\mathrm{DCT}}_{0,2}-\\ 7{\mathrm{DCT}}_{2,0}+0{\mathrm{DCT}}_{0,3}+0{\mathrm{DCT}}_{0,4}+0{\mathrm{DCT}}_{0,5}+0{\mathrm{DCT}}_{0,6}]/64\end{array}$