HARP (algorithm)

Harmonic phase (HARP) algorithm (ref TMI paper) is a medical image analysis technique capable of extracting and processing motion information from tagged magnetic resonance image (MRI) sequences. Developed by N. F. Osman and J. L. Prince et al., the method uses spectral peaks in Fourier domain of tagged MRI, calculates the phase images of their inverse Fourier transforms which are called harmonic phase (HARP) images, and tracks the motion of material points through time under the assumption that the HARP value of a fixed material point is invariant. The method is fast and accurate, therefore has been accepted as one of the most popular tagged MRI analysis methods in medical image processing field.

Background In cardiac magnetic resonance imaging, tagging techniques (1-6) make it possible to capture and store the motion information of the myocardium in vivo (7-11). MR tagging uses a special pulse sequence to create temporary features – tags in the myocardium. Tags deform together with the myocardium as the heart beats and are captured by MR imaging. Analysis of the motion of the tag features in many images taken from different orientations and at different times can be used to track material points on the myocardium (11-12). Tagged MRI is widely used to develop and refine models of normal and abnormal myocardial motion (7,8,12-14) to better understand the correlation of coronary artery disease with myocardial motion abnormalities (15) and the effects of treatment after myocardial infarction (17). However, suffered from long imaging and post-processing times (19), tagged MRI was slow in entering into routine clinical use until HARP algorithm was developed and published in 1999 (ref paper).

Description HARP processing A tagged MRI showing motion of a canine heart is shown in Fig. 1a. The effect of tagging can be described as a multiplication of the underlying image by a sinusoid tag pattern having a certain fundamental frequency, causing an amplitude modulation of the underlying image and replicating its Fourier transform into the pattern shown in Fig. 1b. a: An MR image with vertical SPAMM tags. b: Shows the magnitude of its Fourier transform. By extracting the spectral peak inside the circle in b, a complex image is produced with a magnitude (c) and a phase (d). HARP processing uses a bandpass filter to isolate one of the spectral peaks. For example, the circle drawn in Fig. 1b is the -3 dB isocontour of the bandpass filter used to process this data. Selection of the filters for optimal performance is discussed in (31). The inverse Fourier transform of the filtered image yields a complex harmonic image I(y, t) at image coordinates y = [y1 y2]T and time t: Eq 1. where Dk is called the harmonic magnitude image and ϕk is called the harmonic phase image. The harmonic magnitude image in Fig. 1c extracted from Fig. 1a using the filter in Fig. 1b shows the geometry of the heart. And the harmonic phase image in Fig. 1d contains the motion of the myocardium in horizontal direction. In practice, tagged images from two directions (both horizontal and vertical) are processed to provide a 2D motion map in the image plane. Notice that the harmonic phase images is computed by taking the inverse tangent of the imaginary part divided by the real part of I(y, t), such that the range of this computation is only in [-π, +π). In other words, Fig. 1d is only the wrapped value of the actual phase. We denote this principle value by ak(y, t); it is mathematically related to the true phase by: Eq.2and3 Either ak or ϕk might be called a harmonic phase (HARP) image, but only ak can be directly calculated and visualized. It is the basis for HARP tracking.

HARP tracking For a fixed material point with a HARP value, only one of the points sharing the same HARP value in a later time frame is the correct match. If the apparent motion is small from one image to the next, it is likely that the nearest of these points is the correct point. The tracking result is very accurate in this case.

Consider a material point located at ym at time tm. If ym+1 is the apparent position of this point at time tm+1, we have: Eq The Newton-Raphson interative method is used to find a solution, which is: Eq In practice, since ϕ is not available, a is used in its place. This equation can be formally rewritten as: EQ The result of HARP tracking of one frame of cardiac MRI is shown in figure 2. It is obtained by calculating both motions from horizontal direction and vertical direction, resulting in a 2D vector field showing the motion of every material point on the myocardium at this time frame. The entire HARP algorithm takes only a few minutes to perform on a normal computer and the motion tracking result is accurate (with a typical error range of +-1 pixel). As a result, it is now widely adopted by the medical image analysis community a standard processing technique for tagged MRI. References External Links