Biomedical Engineering Reference
In-Depth Information
Definition 2 (NN) The one-dimensional nearest neighbor interpolation of
a discrete image I at position x is defined as
NN(I;x) := I([x]) ;
(7.10)
where [x] is the rounded value of x.
Definition 3 (Linear) The one-dimensional linear interpolation of a dis-
crete image I at position x is defined as
Linear(I;x) := (1
x
)I(bxc) +
x
I(dxe) ;
(7.11)
where
x
= xbxc. b
c and d
e are the floor and ceiling functions.
·
·
Definition 4 (Spline) The one-dimensional spline interpolation of a dis-
crete image I is defined as the function minimizing the bending energy
Z
(Spline(I;x)
00
)
2
dx
E(I) =
(7.12)
subject to
Spline(I;i) = I(i); 8i 2 \N:
(7.13)
The NN and linear interpolation as defined above can also be interpreted as
a convolution of the discrete signal with an accordant interpolation function.
The appropriate filter function for the sinc, NN, and linear interpolation are
plotted in Figure 7.5 for x 2 [3; 3]. These plots illustrate that the linear
interpolation function is a better approximation of the ideal sinc function
compared to NN.
A comparison of the described interpolation schemes is illustrated for a
1D example in Figure 7.6. The given data is indicated by the five big dots.
The solid lines show the nearest neighbor interpolation with unwanted jump
discontinuities. Linear interpolation is shown with a dashed line. The dotted
line represents the smooth spline interpolation. As a reference, the ideal sinc
interpolation is plotted with a dash-dot line.
1
1
1
0.5
0.5
0.5
−3 −2 −1
1 2 3
−3 −2 −1
1 2 3
−3 −2 −1
1 2 3
(a) Sinc
(b) NN
(c) Linear
FIGURE 7.5: Interpolation functions for x 2 [3; 3]. (a) Sinc interpolation
function. (b) NN interpolation function. (c) Linear interpolation function.
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