Curvelet Transform: Medical Image Fusion Techniques Report

This report explores the application of Curvelet Transform techniques for medical image fusion, focusing on enhancing image quality and integrating data from different imaging modalities like CT and MRI. The discussion covers the limitations of DWT and the advantages of curvelet and ridgelet transforms in handling image singularities. The process involves histogram matching, wavelet-based fusion, subband filtering, and ridgelet transform application, followed by image interpolation to improve resolution. The report details the steps involved in curvelet transform algorithms, including image registration, curvelet transformation, fusion rules, and inverse curvelet transformation, with an emphasis on enhancing image edges and overall visual quality. Desklib provides this assignment as a resource for students studying image processing and data science.

2.0 The proposed curvelet fusion of the resonance of the magnetic and
the computation of the images that are tomography.
DWT is a famous tool that is used to analyse images and signals as it
has a property advantage that make it to localize the singularity in a
point in a given signal or an image. Just like any other tool DWT has
one advantage where when we are processing the image it sometimes
gives very big number coefficients in all the scales that are
corresponding to the image edges. This disadvantage has led the DWT
to be very inefficient in handling the long linear and singularities of an
image that is curvilinear. Approaches that have been made of late such
as the ridgelet and the curvelet transform have become very efficient
to handle the long linear singularities and the curvilinear singularities
in a given image. Where we have the curvelet transform we always
prefer to use the AWT rather than using the DWT as AWT decomposes
an image in to many different subbands referred to as the
approximation and the detailed planes respectively where each and
every subband in the detail plane is divided in to other tiles which are
more smaller. Lastly after understanding the above we can say that
the specific ridgelet transform is now applied on each tile. In this
concept we say that the edges of the image are now represented
efficiently by the ridgelet transform as now the edges of the images
will be seen like straight lines which are very small. Thus this has led
the curvelet transform to be very effective when we extend it with RT
in detecting the curved edges. The curvelet transform algorithms can
be summarized using the following guideline below
1. We apply the histogram that matches with the MRI images.
2. We split the images in to four sub-bands namely the ∆1, ∆2, ∆3, and
the P3, where in this case we use AWT.
3. We have to perform some tiling on the three subbands namely the
∆1, ∆2, and ∆3 respectively.
1

4. We perform a discrete ridgelet scale on all the tiles of the subbands
∆1, ∆2, and ∆3.
We have used figure 3 in explaining the schematic diagram of the
curvelet transform and the steps they depict. See the below detailed
steps where we consider the approach on the proposed curvelet fusion
of the CT and the MR images
1. The first step is registering the MR and the CT images
2. We perform a curvelet transform on the MR and the CT images
3. We will use the maximum fusion rule when fusing the ridgelet
transform of the specific tiles in the respective subbands ∆1, ∆2,
and ∆3 in the MR and the CT images.
4. We will perform an inverse curvelet transform on p3 in the MR
image and subbands which are fused denoted as ∆1f, ∆2f, and ∆3f.
5. We may use a post processing of high-pass for filtering which w be
applied for sharpening the images which are fused in cases where
we have some blurred images that may be resulted to the digital
estimation of such ridgelet transform.
2.1 Enhancing the image on the basis of the Histogram Match
This approach will entirely depend on the match with the histogram
which will be a tool that will be used to enhance the images of IR.it is
good to note that the images which are visible will have a better
histogram shuch is distributed as in comparison with the other IR
images which are known to have a band and limited and histograms. In
solving this we may think to modify the the histogram representing the
IR image which is spread over a given range of the images which are
visible, thus this has made the IR image to be enhanced visually. The
above process is known as the histogram matching and is carried out
2

where we modify the mean and the given variance of all images of IR
which are on the basis of the counterparts in the specific visual
images. We can summarize the ablove using the below equations and
caluclations.
1. This the computation of the mean of the image of IR in its
original form g(m,n)
In this case the letter m and n represents the IR images
dimensions.
2. Computation of the visual image mean reference f(k, l)
In this case k and l are the reference image dimensions.
3. estimation of the standard deviation of the Image of IR r1.
4. estimation of the STDEV of the given reference image that is
th r2.
5. Estimation of factor of correction where we divide the StdEv of the
reference image with ths STDeV of the given IR image.
3

6. Computation of the mean factor that hsd been modified fc.
7. Estimation of the histogram image that matches FH
3.0 Wavelet Based Fusion
Wavelet is a multiple resolution decomposition of an image tool.in the
Wavelet based transform the features of the image are classified in to
two that is the high freq components and the low freq components as
they filter the processes using multiple scales. DWT is used to
decompose the signals in to the mutually-orthogonal sets of the
wavelet scales where the main difference is from the CWT.
In the decomposition of the wavelets as shown in the below equation
we can say the image sources are and which ate
decomposed to give the approximates and the detailed coefficients
that is required at the lower levels where we will have to use the the
two-dimensional DWT which helps in conversion of the image from its
spatial domain to the required freq domain. Both images coefficients
are subsequently combined where we use the rule of fusion as
explained below. The image that has been fused can then be
obtained using the inverse DWT like shown below.
4

The image is partitioned in to vertical1 and horizontal1 lines
respectively representing the first order of the DWT and then the
image is separated using the four parts namely LL1, LH1, HL1, HH1
[11].
The following is the systematic process of the wavelet transform of
image that has been fused. We can explain the following process
where we have use the steps below in explaining the general process
in details.
1. The first step is implementing the DWT on the sources of the
images in building the wavelet lower that has been decomposed.
2. The second step is fusing each and every decomposition where
we have to average the approximates and the detailed
coefficients in each sub-band using the largest greatest
magnitude.
3.1 Curvelet transform
The curvelet transform (CT) is that method that is applied on the basis
of the segmentation where we have to divide the image inputted in
small overlapping tiles and the ridgelet transforms of which they must
be applied to each and every tile for good performance in detecting
the edges. CT has resulted in giving a better performance than the
wavelet transform where we consider the ratio value of the signals to
the surrounding noise.
4.0 Subbands Filtering
5

The main reason for doing this step is decomposing the additive
components of the image in each and every set of the subband on the
image. The step have been known in isolating the freq components of
images in to different planes without necessary down sampling the
planes as it is in the traditional WT. the trous steps given below are
used for the above purpose. Where are given an image p then it will
be possible in constructing the approximations sequence.
(2)
In this case n is the integer which is preferred to be 3. In constructing
the above sequence we will have successive convolutions with a
certain low pass-kernel must be performed. The functions such as f 1, f
2, f 3, and f n meaning the kernel convolutions which is stated and
given by the [J43]
The computations of the wavelet planes is very different between the
two consecutive approximations such as the
Pl−1 and Pl, i.e., as shown below.
This will help in giving the reconstruction formula to be
In this case we will the detailed planes I containing the high freq
details and the Pn which is low freq component of the approximation.
6

We can decompose and reconstruct as shown below in the two figures
and equations below respectively.
The other step is to finding the transformation that is capable of
representing the edges that have slopes which are very different and
orientations. There is a possible solution where the ridgelet transform
may be interpreted as in cases of the one-dimensional (1D) WT in the
Radon domain.
Figure 1Decomposition in the AWT.
.Figure 2Reconstruction in the AWT.
Tiling is a method that is used for splitting the image so that it can
overlap the other tiles where we apply the same transformation in the
curved lines that must be present in the subbands of straight lines
helping to handle the edges which are curved. We can employ as scale
selection algorithm in the approach that has been proposed in
modifying the default curvelets choice on the number of scales. The
7

algorithm above will ensure the high freq that has been centered in the
region they are smooth. In this part of the inner curvelet scale will not
be smoothened by the divisions present in the angular. The equations
of the scale selections is given by,
In this case d represent the square length that covers the high freq of
the magnitude values, where the N1 and N2 represents the horizontal
numbers and the vertical image pixels respectively. The decomposition
of the curvelet levels in the default is always given by
J=log2 (min (N1, N2))-3)
6.0 Ridgelet Transform
The ridgelet transform (RT) rose because of the need in finding a
sparse that will represent all the functions that have many
discontinuities along the lines. RT belong to the family of the DT which
employs the basic functions. In facilitating the mathematical
representation we can view this wavelet analysis in the radon domain
where we can obtain and define the 2D continuous RT through the
intro to the basic functions. The following equation will give us the
transform method of RT where the coefficients of the image are
represented as f ((x1, x2),) where f acts as the function can be
mathematically be presented as follow.
8

In this case Ѱ is the basic RT function where the a must be greater
than o and ϵ [0, 2π] is always constant. The main challenge in the RT is
that it will only work on the line singularities. In overcoming the above
challenge we must apply the projection slice theorem of the RT by
combining it with the ridgelet transform in transforming the line
singularities to point singularities. So, the radon transform for any
image may it be f will be a collection of the integrals of all the indexed
which are defines as follows. The combination of the equations 8 and 9
can be able to give us the following
7.0 Image Interpolation
This is the process through which the number of pixels per the unit
area in any specific image can be increased immensely to help us
obtain the HR image from the LR image. It can be applied in so many
ways in the image processing systems. It has allowed the users in
varying the sizes of the images they interact with in concentrating on
some of the details.
Interpolation of the images that have been fused with the MR and the
CT scans are very significant to the specialists using them. We will
9

need a sophisticated technique for interpolating that will help in
preserving the edges which are sharp in the process of interpolation.
Polynomial based image type of an interpolation technique is one of
the traditional techniques used for the interpolation of the images. The
traditional techniques works on the basis that the sysnthesis of the
unknown pixels values can be known with the use of the
neighbourhood which are known. The following relation can help in
estimation of the interpolated function from the from the equally
spaced 1D sampled data sequence .
In this case - interpolation basis function
x and xk will represent the discrete and the continuous spatial
distances respectively
Values of the of c (xk ) are known as the interpolation coefficients.
8.0 Performance metrics
In performance metrics there exists various number of objectives that
varies depending on the degree of the complexity and are known for
hosting a number of different approaches. In the above research we
have considered the three major metrics in evaluating the
performance of the approach that has been proposed.
1. The Root Mean Square Errror (RMSE)- which is metric measure
that is used to show the differences of the references in the input
with those of the fused output images and it is carried as shown
in the equation below.
10

In this case m and n will represent the dimensions of R(M,N)
respectively reperesenting the general reference of the input images.
F(m, n) – will always be used to represent the reference of the fused
output image.
2. Peak Signal to Noise Ratio (PSNR) which a metric measure that is
used in measuring the maximum power of the corrupting noise
affecting the relaiblity of the given represenatation. It is shown
using the following equation mathematically.
11