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British Journal of Healthcare and Medical Research - Vol. 12, No. 02
Publication Date: April 25, 2025
DOI:10.14738/bjhmr.1202.18387.
Martínez-Hernandez, C. A., Rodríguez-Lelis, J. M., Pérez, O. D., Rodríguez-Ramírez, J. A., Licona, I. L., & Joaquin, P. O. (2025).
Application of Continuous Wavelet Transform to Raw Magnetic Resonance Signals to Differentiate Tissue Features. British Journal
of Healthcare and Medical Research, Vol - 12(02). 13-26.
Services for Science and Education – United Kingdom
Application of Continuous Wavelet Transform to Raw Magnetic
Resonance Signals to Differentiate Tissue Features
C. A. Martínez-Hernandez
Departamento de Ingeniería Mecánica,
TECNM/Centro Nacional de Investigación y Desarrollo Tecnológico,
Cuernavaca, Morelos, México
J. M. Rodríguez-Lelis
Departamento de Ingeniería Mecánica,
TECNM/Centro Nacional de Investigación y Desarrollo Tecnológico,
Cuernavaca, Morelos, México
Oscar Domínguez Pérez
Departamento de Ingeniería Mecánica,
TECNM/Centro Nacional de Investigación y Desarrollo Tecnológico,
Cuernavaca, Morelos, México
J. A. Rodríguez-Ramírez
4Centro de Investigación en Ingenieía y Ciencias Aplicadas,
Universidad Autónoma de Morelos, Cuernavaca, Morelos, México
Irving Lecona Licona
Departamento de Ingeniería Mecánica,
TECNM/Centro Nacional de Investigación y Desarrollo Tecnológico,
Cuernavaca, Morelos, México
Joaquin, P. O.
ABSTRACT
It is quite useful to diagnose different health conditions such as cancer, osteolysis,
amongst others, by the use of images for clinical practice or research. The study and
to interpret, anatomical areas of interest has always been a core subject of imaging
systems. Technological development in this area has produce had focuses on
enhance temporal and spatial resolutions. Although a lot of research has been done
to improve images characteristics, the final diagnostic relies on the judgment and
experience of the medical specialist since there is not a numerical relationship to
the mechanical properties of the human tissues. Based on the former, this work is
aimed to develop a procedure to identify the characteristics of tissue and relate
them to mechanical properties as an aid to medical diagnosis. Here, the continuous
wavelet transform is applied as a passband filter tool to raw magnetic resonance
signals. A relation between frequency content and tissues present in the image was
identified. From here,it was found that tissue regions exhibiting higher stiffness and
toughness emit signals with low frequency, while tissues with lower stiffness and
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toughness emit signals with high frequency. The method proposed allows to extract
information that can help to generate parameters for classification, detection and
mechanical properties of human tissue, for the tissue disease diagnosis.
Keywords: Raw, resonance magnetic signal, filter, mechanical properties, passband filter,
tissue.
INTRODUCTION
The magnetic resonance image (MRI) is a noninvasive technique from where the body can be
represented through images, that are generated from a radio wave energy, a strong magnetic
field, and the atomic-nucleus quantum-mechanical properties of the Hydrogen [1]. The images
help to differentiate soft tissue, bone, and some synovial liquids with great clarity. The
differentiation is possible by the properties contained on the different tissues present in the
body, which depends on their compositions: quantities of lipids, proteins, and water [2-6].
Although clear images are obtained, the interpretation of the images is a function of the medical
experiences of the medical specialist.
Nyman et. al. [7] related the mechanical properties of bone with measurement of the free and
bound water by nuclear magnetic resonance NMR. Bound water was directly related to bone
strength and toughness while free water was inversely related to the modulus of elasticity.
Horch et al [8] found that signals of the spectrum 1HNMR are betterthan X-ray to predict yield
stress, peak stress, and pre-yield toughness. The 1HNMR signals can be extracted from clinical
magnetic resonance MRI, thus offering the potential for improved clinical assessment of
fracture risk [8].
A challenge to relate the mechanical properties with the information form the MRI signals is
that the acquired RMI signals are saved according to specific data formats, which depends on
the vendor. Nowadays, a data sets of raw magnetic resonance signals are available in a standard
format, the International Society for Magnetic Resonance in Medicine (ISMRM), proposed a Raw
Data Format in 2013. These signals, called ISMRMRD [9]. The MRI signals contain indirect
evidence of anatomic, cellular, and atomic features. However, the information into these signals
is seldom used for diagnosis and analysis tasks [10].
Many techniques of signal processing have been designed for feature extraction, and time- frequency and multiresolution analysis. The analysis based on the Fourier transform (FT)is the
most common, from where the frequency content of the signal is found. Nevertheless,
knowledge of the frequency content is not enough for bio-signals due to its non-stationary
feature [11]. For analysis of a bio-signals, it is necessary a time-based information for the
frequency content. Th e short time Fourier frequency (STFT) is an alternative to achieving the
former. However, the fact that the resolution restricted to a preselected window length the
analysis becomes limited. Another technique widely used in the signal analysis is the Wavelet
Transform (WT). This technique does not require preselected window length and does not have
fixed time- frequency resolution over the time-frequency space.
Two types of WT can be recognized the literature, these are discrete wavelet transform DWT
and continuous wavelet transform CWT. The DWT is used in denoising, compression, and filter
of signals [12–16]. The Continuous Wavelet Transform (CWT) provides a method for displaying
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Martínez-Hernandez, C. A., Rodríguez-Lelis, J. M., Pérez, O. D., Rodríguez-Ramírez, J. A., Licona, I. L., & Joaquin, P. O. (2025). Application of
Continuous Wavelet Transform to Raw Magnetic Resonance Signals to Differentiate Tissue Features. British Journal of Healthcare and Medical
Research, Vol - 12(02). 13-26.
URL: http://dx.doi.org/10.14738/bjhmr.1202.18387.
and analyzing characteristics of signals that are dependent on time and scale [12]. The CWT is
used as a tool for diagnosis of faults, analysis of responses of a linear mechanical system, feature
extraction, detecting and identifying signals with exotic spectral features, transient information
content, or other nonstationary properties [17–19].
A key feature of the CWT is that it acts as a passband filter. When the CWT is applied to a signal,
the frequency content that occurs at a particular time is extracted. The center frequency and
shape of the passband filter is based around the mother wavelet function [20].
Because of the difference in the composition portions of water, proteins and lipids that exist in
tissue, we seek determi- nate and compute the relation between the frequencies and its
mechanical properties. We used the CWT as a passband filter with the aim to differentiate the
frequencies of the diverse tissues present in the resonance magnetic image study.
THEORETICAL FRAMES
Raw Magnetic Resonance Signals
The process of obtain MR signals is shown in Fig.1. Here, the strong magnetic field created by
the MRI scanner causes the atoms in a body to align in the same direction of the magnetic field.
Radio waves are then sent from the MRI machine and move these atoms out of the original
position. As the radio waves are turned off, the atoms return to their original position and send
back radio signals. The radio signals are then captured by a receiver transducer, and then stored
into a k-space [21].
Fig. 1: MR signal schema. The hydrogen spins of the sample are affected by a static magnetic
field and a radiofrequency pulse, at Larmor frequency, this allows the energy absorption, later
the sample starts to relax and emits energy that is recorded as the resonance magnetic signal,
in its frequential domain.
The k-space contains each generated signal by the tissue region of interest. After the excitation
pulse has been applied there is a variation of the phase gradient, this information can be saved
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into the k-space using the trajectories: Cartesian, Spiral, Radial, and Zig-Zag. The stored signals
represent the relationship between the domain frequency and the spin- density signal in time
[1, 22, 23]. An example of MR signals acquisition into the k-space using a Cartesian trajectory is
shown in the Fig.2.
The International Society for Magnetic Resonance in Medicine (ISMRM) through a
subcommittee at 2013, proposed a Raw Data Format, aimed to capture the details of the MRI in
a such a way that could allow image reconstruction [24].
The ISMRM Raw Data (ISMRMRD) is structured in two sections: an XML header and the Raw
data. The XML section contains all information about acquisition protocol and parameters for
image reconstruction. The Raw data is organized as a sequence of data items consisting of fixed- size data headers and the corresponding k-space data for each set of samples or data chunk
[24].
Fig. 2: Example of a raw data matrix, k space, where each transformed signal in the frequency
domain is stored into a row of the k space through a Cartesian trajectory.
Tissue Properties
The magnetic resonance imaging uses the variation of the physical properties of tissues, mainly
the biochemical composition, to get the images with widely contrasts range. This is because the
different tissue properties depend on their composition: lipid portions, proteins, and water [5,
6].
Tissues are formed by similar cells and an extracellular matrix, whose origin is embryonic. They
work in combination to develop specialized activities. Many tissues can be grouped by its basic
functions or similar morphology, which can be correlated [5]. The mechanical properties are
then determined by its composition level. In the elemental level can be found the collagen
molecule, which through parallel linking, form the fibrils. These are linked by proteoglycans
matrix, which contribute to mechanical behavior too [25].
The contribution of each signal depends besides of the type of tissue to the particular pulse
sequence used. As already stated the major contributions sources are water, lipids, small
organic molecules, and macromolecules, however the macromolecules cannot be represented
by a conventional image from MRI due to their very short T2 value [26]. The behavior of mobile
fatty acids is slightly different compared to oxygen-bounded hydrogens in water molecules.
Therefore, the interaction between the RF pulse and Larmor frequency in tissues are a function
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Martínez-Hernandez, C. A., Rodríguez-Lelis, J. M., Pérez, O. D., Rodríguez-Ramírez, J. A., Licona, I. L., & Joaquin, P. O. (2025). Application of
Continuous Wavelet Transform to Raw Magnetic Resonance Signals to Differentiate Tissue Features. British Journal of Healthcare and Medical
Research, Vol - 12(02). 13-26.
URL: http://dx.doi.org/10.14738/bjhmr.1202.18387.
of the portion of water and fatty content. This phenom is called the chemical shift. As the
frequency information is used for spatial encoding, the chemical shift is present over the pixels
of the MRI [27].
Continuous Wavelet Transform
The wavelet transform, proposed by Grossman and Morlet [29], is widely used in signal and
image processing. The continuous wavelet transform (CWT) is used as a tool for analysis and
feature determination. The CWT applied to a signal can be used as filter and to extract the
frequency content at a particular time. The transform of signal s(t) at a scale a and at time b, is
defined by equation 1. The CWT equation can be solved at a range of scales, a can alter the
center frequency and shape of the mother wavelet, to find out different time and frequency
information [13].
CWT(a, b) =
1
√|a|
∫ s(t)ψ
∗
(
(t−b)
a
) dt ∞
−∞
(1)
where
• s(t) is the signal in time domain.
• ψ is the wavelet mother.
• a is the scale factor.
• b is the translation factor.
• asterisk ∗ represents the complex conjugate.
the equation 1 is equivalent to convolution of the signal s(t) with an impulse response [13, 20].
The convolution between the signal s(b) and ψ∗(b) implies the presence of a passband filter.
Convolution is important because relates three signal of interest: the input signal, the output
signal and the impulse response. Convolution describes the procedure to use the impulse
decomposition. If the system is considered a filter, the impulse response is called the filter
kernel, the convolution kernel or simply, the kernel.
The filtering consists of attenuate o block some frequencies which not are required. The filters
with frequency response are classified as (1) High pass filters (2) low pass filters (3) pass- band
filters and (4) bandstop filters [28]. In the convolution, is just necessary to know the system’s
impulse response to calculate what the output will be for any possible input signal [28].
MATERIAL AND METHODS
A Left Tibia (LT) study from Stanford 2D Fast Spin Echo (FSE) project [9] is used in this work.
The TL study consists of 28 slices in a sagittal tomographic cut and it was acquired using 16
receiver coils. An example of the slice of LT study is shown in Fig.3.
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Fig. 3: Left Tibia study, sagittal tomographic cut [9].
Table 1: TL study attributes based on the parameters described in the XML header.
System Field Strength 3.0 T
Receiver Bandwidth (rBW) 41.6 KHz
Number of Channels 16
Matrix Size (Np, Ns, Nsl) 352 x 202 x 28
Field of View 270.0 mm x 270.0 mm x 4.5 mm
Number of Slices 28
Number of Phases 1
Number of Contrasts 1
Trajectory Cartesian
Repetition Time 888 ms
Echo Time 9.3 ms
Flip Angle 111◦
Sequence Type SE
A summary of TL study attributes is described in table 1. The completed description can be
found in XML header of the study. From the attributes shown in table 1, where the signal is in
the frequency domain, each of then is transformed in the time domain such that a CWT cold be
applied. An Inverse Fast Fourier Transform (IFFT) is used to transform between the Frequency
to the Time domain of the signal.
In Fig.4 shows the relationship between the signal domain.
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Martínez-Hernandez, C. A., Rodríguez-Lelis, J. M., Pérez, O. D., Rodríguez-Ramírez, J. A., Licona, I. L., & Joaquin, P. O. (2025). Application of
Continuous Wavelet Transform to Raw Magnetic Resonance Signals to Differentiate Tissue Features. British Journal of Healthcare and Medical
Research, Vol - 12(02). 13-26.
URL: http://dx.doi.org/10.14738/bjhmr.1202.18387.
Fig. 4: Example of the relationship between the Frequency and the Time domain, the signal can
be transformed using an FFT or IFFT. The signal in the Frequency and the Time domain of TL
study [9] and its components, real and imaginary, are shown.
The maximum frequency described by equation 2 and Bandwidth per pixel by equation 3, were
computed using the receiver bandwidth (rBW), and signal size (Number of samples, Ns), such
that:
Fmax =
rBW
2
(2)
BWPix =
RBW
Ns
(3)
The CWT implementation is then:
1) In accordance to the application requirements, the mother wavelet is chosen.
2) Determine the scales at which the analysis must be carried out.
In the wavelet analysis, the way in which can be related the scale to frequency is to determine
the center frequency of the wavelet, using the following relationship:
Fa =
Fc
a
Where: a is a scale; Fc is the center frequency of the wavelet in Hz.; Fa is the pseudo-frequency
corresponding to the scale a, in Hz. From the above relationship, it can be seen, that the scale is
inversely proportional to pseudo-frequency, then if one knows the maximum frequency one
can compute the minimum scale (amin) as:
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amin =
Fc
Fmax
(5)
The CWT and analysis of frequencies are applied to each signal of the k space. After that, one
can reconstruct an image using the whole output signals with a particular scale. The analysis is
focused on the tissues that are present in the reconstructed image and the frequencies of the
signals after using the CWT as a passband filter. Afterwards, the frequency content is analyzed
through the Fourier Transform. As was mentioned the advantage of using the CWT as a
passband filter is that it helps to identify the moment im which the frequencies are present.
This can be used to know the content frequency in each pixel with the aim to relate the energy
and the mechanical properties. To understand the influence of frequencies on the pixel the
analysis was based on the commutative property of IFFT on 2D. First, we applied the IFFT on
1D over the columns to know the relation between its phases. After, it is applied the step by
step the IFFT on 1D over each row, to know what frequency has the maximum value in the
spectrum. Then one can compare the maximum value of (Mv) and the final value of (Fv), by the
use of the relationship Mv /Fv. If the value of relationships is 1, then the frequency has a
maximum influence.
RESULTS
The intervals to be implemented for the CWT analysis are shown in table 2. These intervals
were computed according to equations 2 and 3 and table 1. In this work the raw signals of the
11th slice from the Left tibia study [9] are used. The CWT analysis can be supported by
scalograms, which are images that allow knowing correlation levels between the wavelet and
the signal. In figure 5 the black regions represent the maximum correlation. it can be noted that
the correlation occurs near to the maximum frequency, independent of the mother wavelet.
Table 2: Maximum and Minimum scales used according to the mother wavelet used.
The difference in the scales is due to each wavelet has a particular center frequency.
Name Center freq. (Hz) minimum maximum
Coiflet 0.7059 1.4118 285.1765
Daubechies 0.6923 1.3846 279.6923
Morlet 0.8125 1.6250 328.2500
Mexican Hat 0.2500 0.5000 101.0000