Image Processing Reference

In-Depth Information

of these measures is used to construct a decision map which in turn forms an actual

weight of the fusion.

In [201], authors have explained the use of non-subsampled contourlet transform

for fusion of multi-focus images. In this technique, low frequency coefficients have

been combined with selection and averaging rules, while the high frequency coef-

ficients have been selected on the basis of standard deviation at the corresponding

level. In [197] the use of contourlet transform has been proposed for medical image

fusion where the fusion rules include weighted average and local energy which are

applied in the transform domain. The CT and MRI images have been fused using

contourlet transform in [26] where the fusion weights are inversely proportional to

the distance of the pixel from the current value of the fused pixel.

Curvelets have been another popular choice to represent edges [23, 52]. Curvelets

are a multi-scale transform that more efficiently represent the edges and other

singularities along curves than the wavelets. The efficiency refers to the ability to

represent the data in a fewer number of coefficients for a given accuracy of recon-

struction [52]. Choi et al. have experimented with curvelet transform for fusion of

satellite images [40]. A finite ridgelet transform is obtained by taking a DWT of the

coefficient vector of the finite Radon transform. Fusion of remote sensing images

using the ridgelet transform has been proposed in [36].

A few researchers have applied the estimation theory to the problem of image

fusion. The problem of fusion has been formulated as the following: The fused

image is considered to be the underlying
true scene
. The input images obtained from

multiple imaging sensors are assumed to depict the partial scene. The images contain

an incomplete scene contaminated with some noise. The fusion problem thus gets

transformed into the problem of estimation of the underlying true sceneâ€”which

is the fused image itself. Once the relationship between the input images and the

fused image is modeled, one can employ suitable techniques from the estimation

theory which has a highly rich literature. Sharma [163], and Sharma et al. [164] have

modeled the input images as noisy, locally affine functions of the true scene to be

estimated. A Bayesian framework has been employed to obtain either the maximum

likelihood (ML), or the maximum
a posteriori
(MAP) estimates of the true scene,

i.e., the fused image. The parameters of this model have been estimated from the

local covariance of the images. This model also offers flexibility to add a prior about

the scene, if known; however, it assumes the noise component to follow a Gaussian

distribution. Blum [15], and Yang and Blum [196] have improved the probabilistic

fusion by allowing non-Gaussian noise distribution. An expectation maximization

(EM)-based solution to detect concealed weapon through image fusion has been

presented in [196]. For robustness against noise, a total variation (TV)-based prior

has been incorporated into the probabilistic model by Kumar [94], and Kumar and

Dass [95]. Xu et al. have proposed a Markov random field (MRF)-based prior for the

fused image, and a Gaussian assumption of the parameters of the image formation

model for hyperspectral image fusion [192, 193].

When the number of images to be fused is higher than three, the fusion problem

can be viewed as a dimensionality reduction of the input data. Principal component

analysis (PCA) is a powerful tool used for dimensionality reduction of multispectral