HARP
Synthesize tag lines
Construct displacement fields for small motions
Motion tracking – phase based optical flow (CINE-HARP)
Calculate 2D Lagrangian strain (using CINE-HARP tracking)
Calculate 2D Eulerian strain (one shot HARP)
MR tagging (MRT) uses a special pulse sequence to spatially modulate the longitudinal magnetization of the
subject prior to acquiring image data.
It is based
on the fact that SPAMM-tagged MR images [5,6] have
a collection of distinct spectral peaks in the Fourier domain,
and that each spectral peak contains information
about the motion in a certain direction. The inverse
Fourier transform of just one of these peaks, extracted
using a bandpass filter, is a complex image whose phase
is linearly related to a directional component of the
true motion. We define an angle image to be exactly
this phase image, except that it is constrained to lie in
the range [-T, T ) by the action of the standard inverse
tangent operator. Despite this angle-wrapping artifact,
an angle image can be used to estimate synthetic tag
lines, and pairs of angle images can be used to measure
small displacement fields, optical flow in image
sequences, and two-dimensional strain.
The spectral peaks in Fig. lb arise because the
sinusoids in the SPAMM tag pattern amplitude modulate
the underlying image, acting as as carriers that
shift the underlying spectrum into various positions.
The number and distribution of the spectral peaks vary
according to the specific tagging pulse sequence
It is intuitive that local motion would cause the
phase of sinusoidal patterns to be locally changed. In
fact, the motion produces an angle modulation of the
tag pattern
These complex images can be approximately determined
by filtering the spectral peaks of tagged MR images using
bandpass filters. Alternatively, they can be directly
imaged using selective k-space imaging.
Eulerian: Coordinates used in fluid dynamics which are fixed in space
Lagrangian: Coordinates used in fluid dynamics in which the coordinates are fixed to a
given parcel of fluid, but move in space
Regenerating MR Tagged Images Using
Harmonic
Phase (HARP) Methods
Because tag patterns are typically periodic, they can be expanded
in a Fourier series. Multiplication of such a tag pattern with the
object creates an amplitude modulation that causes space to
have periodically distributed “lumps” of energy called harmonic
peaks. The spacing between the peaks is inversely proportional
to tag spacing, while the number of the harmonic peaks and
their energy ratios affect the width and profile of the resulting
tag lines. In general, a smaller number of peaks produce images
with wider tag lines than those with a larger number of peaks.
For example, 1-1 SPAMMuses only two RF pulses, resulting in
three harmonic peaks (including the “dc” peak at the origin of
space) and the tag profile is a simple sinusoid.
To obtain crisp thin tag lines, more RF pulses are used, and the
resulting images have a larger number of peaks, in general.
The basic concept behind HARP is that each (non-dc) harmonic
peak has complete information about a particular one-dimensional
(1-D) component of the tag’s deformation, which relates
to tissue motion. Because of this, HARP uses two harmonic
peaks, either directly acquired or extracted using bandpass filters,
to estimate the two-dimensional (2-D) motion in a plane.
Cardiac Motion Tracking Using CINE Harmonic Phase
(HARP) Magnetic Resonance Imaging
The locations of the spectral
peaks in Fourier space are integer multiples of the fundamental
tag frequency determined by the SPAMM tag pulse
sequence.
In (31), we described what
might be referred to as single-shot HARP image analysis
techniques: reconstructing synthetic tag lines, calculating
small displacement fields, and calculating Eulerian strain
images. These methods require data from only a single
phase (time-frame) within the cardiac cycle, but are limited
because they cannot calculate material properties of the
motion. In this study, we extend these methods to image
sequences—CINE tagged MR images—describing both a
material point tracking technique and a method to use
these tracked points to calculate Lagrangian strain, including
circumferential and radial strain.
application of a tag pattern to a given image
slice is an amplitude modulation of the tag pattern by the
anatomy D(x, t). From Eq. (2), we see that the displacement
u(x, t) causes a phase modulation of the underlying
tag pattern, where the phase modulation index is −ωT0
. The
Eulerian strain is the spatial derivative of displacement; therefore,
Eulerian strain can be interpreted as the “signal” that
frequency modulates the carrier. Accordingly, the harmonic
image I(x, t) is an AM-FM signal whose instantaneous amplitude
isD(x, t) and whose instantaneous frequency (IF) is
Eulerian strain.
Given this interpretation, HARP analysis can be viewed
as a phase or frequency demodulation technique, and the
estimation of Eulerian strain is essentially the estimation of
the IF of an AM-FM image. In HARP the IF is estimated by
taking derivative of the phase of the harmonic image, which
is similar to the analytic signal used in the AM-FM signal
analysis
HARP WEB PAGE
In fact, the phase of a given point does not change due to motion — the phase is a
material property. The slope of the phase change, however, in direct correspondence to
the change in frequency of the sinusoid, which in turn reflects the underlying strain.
HARP analysis methods exploit the following two properties: 1) for a given point,
harmonic phase is constant with time and 2) the slope of the harmonic phase is linearly
related to the underlying mechanical strain.
a k (y, t ) I k (y, t )
ImI k
I k (y , t ) tan 1
Re I k
Position of material point at time t is given by the reference map
ak (y, t ) W [w k y ak (y, t )]
W angle unwrapping function w tag frequencies
a (y, t )
u 2 (y, t ) (W T H ) 1 k
al (y, t )
Limited to finding small motions.
H image orientation, W tag frequencies
q( y , t ) y u 2 ( y , t )
OR
y q( y , t) (W T H ) 1 y ai (y , t )
(y, t ) y q(y, t )1 1
Problems
Unwrapping sensitive to noise
Decay of tagging
Interference from harmonics
Dynamic range limitations