BlockGraphTransforms
Edgeadaptive transforms for efficient depth map coding
In this work a new set of edgeadaptive transforms (EATs) is presented as an alternative to the standard DCTs used in image and video coding applications. These transforms avoid ?ltering across edges in each image block, thus, they avoid creating large high frequency coef?cients. These transforms are then combined with the DCT in H.264/AVC and a transform mode selection algorithm is used to choose between DCT and EAT in an RDoptimized manner. These transforms are applied to coding depth maps used for view synthesis in a multiview video coding system, and provides up to 29% bit rate reduction for a ?xed quality in the synthesized views.
Edge adaptive transform (EAT) design
The EAT design process consists of three steps, i.e., (i) edge detection is applied on the residual block to find edge locations, (ii) a graph is constructed based on the edge map, then (iii) an EAT is constructed and EAT coefficients are computed. The EAT coefficients are then quantized using a uniform scalar quantizer and the same runlength coding used for DCT coefficients is applied. The 2 × 2 sample block in Fig. 1 is used to illustrate the main ideas. We describe the encoder operation when applied to blocks of prediction residuals, though the same ideas can be easily applied to original pixel values.
[[1] Figure 1]

First edge detection is applied on the residual block to ?nd
edge locations and a binary edge map is generated that indicates the locations of edges. If no edges are found in a residual block, then a DCT is used and no further EAT processing
is performed. Otherwise, the encoder computes an EAT. A
graph is generated from this edge map, where each pixel in the
residual block is connected to each of its immediate neighbors
(e.g., 4connected neighbors) only if there is no edge between
them. From this graph, we derive the adjacency matrix, diagonal degree matrix, and the Laplacian matrix. Projecting the signal of a graph
G onto the eigenvectors of the Laplacian matrix yields a spectral decomposition of the signal, i.e., it provides a “frequency
domain” interpretation of the signal on the graph. EAT coef?cients are computed as the projection of the onedimensionalized imagesignal onto the eigenvectors
of the Laplacian matrix. The coefficients are quantized with a uniform scalar quantizer. This vector of coef?cients is then reformed
into an N × N block by placing the coefficients in the
standard zigzag fashion used for the DCT.

The EATs minimize the number of nonzero coef?cients
that must be encoded for a piecewise constant(PWC) image, thus, they provide
a highly ef?cient representation for depth maps (since depth
maps are nearly PWC). Since residual depth maps (resulting
from intra or inter prediction) are also PWC, EATs also provide an ef?cient representation for
residual depth maps. However, since edge information must be encoded and sent to the
decoder, these EATs are not necessarily RDoptimal, i.e., if
the edge map bit rate is too high, the RD cost for these EATs
may actually be greater than the RD cost for the DCT. Therefore,
it would be better to choose between EAT and DCT in an
RDoptimal fashion.
In other words, an
EAT should only be used in place of the DCT when it yields
lower RD cost than the DCT. Otherwise, the DCT should be
used. This leads to an RDoptimized transform mode selection algorithm as shown in Fig. 2. The edge maps and transform mode information (e.g., EAT or DCT) are encoded using
contextadaptive binary arithmetic coding (CABAC).
[[2] Figure 2]
 G. Shen, W.S. Kim, S. K. Narang, A. Ortega, J. Lee, and H. Wey, "Edgeadaptive transforms for efficient depthmap coding," in Proc. of 28th Picture Coding Symposium (PCS ‘10), Nagoya, Japan, Dec. 2010. PDF format
 W.S. Kim, S. K. Narang, and A. Ortega, "Graph based transforms for depth video coding," Proc. of IEEE Intl. Conf. Acoustics, Speech, and Signal Proc. (ICASSP 2012), Kyoto, Japan, Mar. 2012.
 G. Cheung, W.S. Kim, A. Ortega, J. Ishida, and A. Kubota, "Depth map coding using graph based transform and transform domain sparsification," Proc. of IEEE Intl. Workshop on Multimedia Signal Proc., Hangzhou, China, Oct. 2011. (Top 10% Paper Award)