Under the DSP G framework, the gradient of a signal at a node is a scalar value. However, let us define a vector gradient ? i f at a node i such that the j th element of the gradient vector is ( ? i f)( j) = ? wij ( f ( i ) ? f ( j)). Therefore, the length of the gradient vector at node i is the number of incoming edges at the node.

(a) An image signal can also be considered as a graph signal lying on the graph shown in

Figure 10.11, where the edges are pointing rightward and downward. For an arbitrary

node, what is the vector gradient at the node?

(b) Describe under what conditions on a graph signal

f the relation ||?if||1= |(Linf)(i)| will

hold? Here Lin is the directed Laplacian (in-degree) of the graph.

(c) Do you think that defining a vector gradient might be a better choice for quantifying

the local variations in a graph signal and subsequently for quantifying global variation as

well? Why?

Consider the node labeling as shown in Figure 10.12. The gradient vector at an arbitary


node i will be zero except at four entries which are