News Column

Patent Application Titled "Non-Linear Solution for 2D Phase Shifting" Published Online

July 18, 2014



By a News Reporter-Staff News Editor at Health & Medicine Week -- According to news reporting originating from Washington, D.C., by NewsRx journalists, a patent application by the inventors BONE, DONALD JAMES (Wentworth Falls, AU); WONG, SAMPSON SZE CHUNG (Dundas, AU); LARKIN, KIERAN GERARD (Putney, AU); PAN, RUIMIN (Macquarie Park, AU), filed on December 11, 2013, was made available online on July 3, 2014 (see also Canon Kabushiki Kaisha).

The assignee for this patent application is Canon Kabushiki Kaisha.

Reporters obtained the following quote from the background information supplied by the inventors: "In the last century or so a large number of techniques have been developed to make (normally invisible) phase variations visible in imaging devices. These techniques include: Zernike phase contrast, Nomarski differential interference contrast, generalized phase contrast, Foucault knife edge, schlieren, shadowgraph, dark-field and wire-test. More recently there has been progress in extending some of these techniques to X-ray imaging. A number of new phase-contrast techniques have been developed for the particularly difficult nature of X-rays, arising primarily from the difficulty of focusing and imaging X-ray beams. These techniques include TIE (Transport of Intensity Equation) phase contrast imaging and ptychography. Another such technique is known as 'X-ray Talbot moire interferometry' (XTMI), that yields intermediate images which encode one or more differential phase images. The XTMI method when implemented using simple linear gratings gives one differential phase image. XTMI implemented with two dimensional (crossed) gratings yields two (crossed or orthogonal) differential phase images.

"Phase shifting methods are well known and have been used extensively for the analysis of interferometric fringe patterns. Phase shift interferometry (PSI) was first introduced in 1974. PSI involves moving the reference mirror in small linear increments and storing the interferogram image at each step. The value of each point in the interferogram image and the amount of artificially introduced phase shift are used to calculate corresponding optical path length variation of that point. Phase shifting techniques with multiple sinusoidal components have also been proposed.

"Phase shifting methods have also been applied to x-ray Talbot moire interferometry. With linear gratings, which are only sensitive to one gradient direction, it is known that if higher harmonics can be ignored, then a minimum of three images are required to recover the low frequency background intensity, the fringe modulation amplitude and the phase parameters, at a point in the image. However this system is only sensitive to the differential behaviour orthogonal to the grating. Rotating the grating and taking a further two exposures allows the differential phase and modulation parameters in the orthogonal direction to be obtained with a total of only five exposures. Mechanically, this is complex and required a rotation of the grating by 90.degree. during the measurement.

"Crossed or two-dimension (2D) gratings allow the system to be sensitive to the differential phase and the modulation in two orthogonal directions at the same time. Mathematical models of the imaging process for X-ray Talbot systems with crossed gratings using phase shifting assume that the captured intensity, Z.sub.n, in the n.sup.th image in the sequence at a pixel can be represented in the form

"Z n = a + m 1 , 0 cos ( .xi. 1 , 0 + .phi. 1 , 0 , n ) + m 0 , 1 cos ( .xi. 0 , 1 + .phi. 0 , 1 , n ) + m 1 , - 1 cos ( .xi. 1 , - 1 + .phi. 1 , - 1 , n ) + m - 1 , 1 cos ( .xi. - 1 , 1 + .phi. - 1 , 1 , n ) ( 1 ) ##EQU00001##

"where a is a low frequency background intensity, the second term, m.sub.1,0 cos(.xi..sub.1,0+.phi..sub.1,0,n), is a modulated sinusoid with modulation strength m.sub.1,0 and a phase induced by the object, .xi..sub.1,0 and an imposed phase step .phi..sub.1,0,n. This modulated sinusoid arises from one element of the pair of gratings that form the crossed grating. The third term has the same general form but arises from the second element of the crossed grating. The fourth and fifth terms appear similar in form but arise from interactions of the second and third terms. As such the phase steps, .phi..sub.1,-1,n and .phi..sub.-1,1,n are not independently imposed but arise from the first two terms as

".phi..sub.1,1,n=.phi..sub.1,0,n+.phi..sub.0,1,n

".phi..sub.1,-1,n=.phi..sub.1,0,n-.phi..sub.0,1,n (2)

"This formulation of the intensity has nine unknown parameters, so at least nine phase-stepped images, producing nine intensity values (Z.sub.1 to Z.sub.9) associated with nine equations are required to solve for the nine unknowns. A need exists for methods that can reduce the number of phase steps below the currently limit of 9 steps.

"Recently, other researchers have proposed a model with fewer parameters for a windowed Fourier Transform (WFT) based analysis of x-ray Talbot moire interferograms. This could be adapted to a phase shifting method and analysis. In the context of a phase shifting method, this simplified model would take the form

"Z.sub.n=a+b(1-cos(.xi..sub.1,0+.phi..sub.1,0,n))(1-cos(.xi..sub.0,1+.phi- ..sub.0,1,n)) (3)

"However, though this model has fewer parameters, it forces the modulation amplitude b of the two crossed cosine terms to be the same. This is not true in practice. There is a need therefore for a simplified model that better fits the behaviour of real x-ray Talbot moire interferograms.

"Another problem with phase shifting methods for crossed grating x-ray Talbot moire interferometry is that the phase steps must have the precise values imposed by the model used in order to fit the observations. Any variation in the phase step from the nominal phase step required by the analysis will induce errors in the resulting parameters recovered by the analysis since the analysis methods explicitly assume the nominal phase steps. However, because the phase steps are produced by very small mechanical movements in the system, significant errors typically occur in the resulting phase steps. Although there are known methods for correcting for these phase step errors for interferometric phase shifting with one-dimensional (1D) fringe systems, there are no known methods for correcting for phase step errors for crossed grating x-ray Talbot moire interferometry. A need exists for a method of analysis for these systems that can correct for these phase errors."

In addition to obtaining background information on this patent application, NewsRx editors also obtained the inventors' summary information for this patent application: "Disclosed is a method of reconstructing a representative detailed phase image from a set of fringe pattern interferogram images of an object. The images are captured by an x-ray interferometer having a crossed diffraction grating. A set of captured fringe pattern interferogram images from the x-ray interferometer are provided, the set comprises no more than eight captured fringe pattern interferogram images. The method determines an estimate of an absorption parameter (a), two-dimensional amplitude modulation parameters (m.sub.x, m.sub.y), and two-dimensional phase modulation parameters (.xi..sub.x and .xi..sub.y) from an appropriate closed-form solution using the received set of captured fringe pattern interferogram images, and reconstructs the representative detailed phase image using the parameter estimates.

"Desirably the determining step revises the parameter estimates. Preferably the parameter estimates for each pixel are revised by generating a simulated set of fringe intensities for that pixel and comparing the simulated set of fringe intensities with the set of captured fringe intensities at that pixel. Alternatively the parameter estimates for each pixel are revised by generating a simulated set of fringe intensities for that pixel and minimising an error in corresponding pixels in the simulate set of fringe intensities and the set of captured fringe intensities.

"In a particular implementation, the determining step (comprises: obtaining an initial estimate of the parameters; using a phase estimation method to correct phase step errors in the parameters; and iteratively determining an optimal solution for the parameters using the corrected phase steps.

"In specific implementations, the set comprises: 5 captured fringe pattern interferogram images and the phase steps of the crossed diffraction grating are .phi..sub.x: [0, 2.pi./5, 4.pi./5, 6.pi./5, 8.pi./5] and .phi..sub.y: [0, 6.pi./5, 2.pi./5, 8.pi./5, 4.pi./5]; 6 captured fringe pattern interferogram images and the phase steps of the crossed diffraction grating are .phi..sub.x: [0, .pi./2, .pi./2, .pi., .pi., 3.pi./2] and .phi..sub.y: [.pi./2, 0, .pi., .pi./2, 3.pi./2, .pi.]; 7 captured fringe pattern interferogram images and the phase steps of the crossed diffraction grating are .phi..sub.x: [0, .pi./2, .pi., 3.pi./2, 0, .pi./2, .pi.] and .phi..sub.y: [0, 3.pi./2, .pi., .pi./2, .pi., .pi./2, 0]; or 8 images captured fringe pattern interferogram images and the phase steps of the crossed diffraction grating are .phi..sub.x: [0, .pi./2, .pi., 3.pi./2, 0, .pi./2, .pi., 3.pi./2] and .phi..sub.y: [0, 3.pi./2, .pi., .pi./2, .pi., .pi./2, 0, 3.pi./2].

"In another aspect, disclosed is a method of reconstructing a representative detailed phase image from a set of fringe pattern interferogram images of an object captured by an x-ray interferometer having a crossed diffraction grating. This method comprises providing a set of captured images; providing a reconstruction technique capable of reconstructing the image from no more than eight images by relating cross components associated with the fringe pattern to the principle components associated with the fringe pattern; and reconstructing the representative detailed phase image from the provided set of captured images using the provided reconstruction technique.

"Other aspects are disclosed.

BRIEF DESCRIPTION OF THE DRAWINGS

"At least one embodiment of the present invention will now be described with reference to the following drawings, in which:

"FIG. 1 is a schematic flow diagram illustrating a method of phase demodulation for fringe pattern interferogram images according to one implementation;

"FIG. 2 is a schematic flow diagram illustrating a method of phase demodulation for fringe pattern interferogram images according to a further implementation;

"FIG. 3 is a schematic flow diagram depicting the details of the iterative phase demodulation method from FIG. 2;

"FIG. 4 schematically depicts an experimental set-up for an X-ray Talbot interferometer relevant to the arrangements described;

"FIGS. 5A-5C illustrate an examples of the gratings and the self-image described in FIG. 4;

"FIG. 6 is a schematic flow diagram exemplifying detail of the image capture process described in FIG. 1 and FIG. 2;

"FIG. 7 illustrated the phase shift pattern of a 9-step image capture process;

"FIG. 8 illustrated the phase shift pattern of an 8-step image capture process;

"FIG. 9 illustrated the phase shift pattern of another 8-step image capture process;

"FIG. 10 illustrated the phase shift pattern of a 7-step image capture process;

"FIG. 11 illustrated the phase shift pattern of a 6-step image capture process;

"FIG. 12 illustrated the phase shift pattern of a 5-step image capture process;

"FIGS. 13A and 13B form a schematic block diagram of a general purpose computer system upon which arrangements described can be practiced;"

For more information, see this patent application: BONE, DONALD JAMES; WONG, SAMPSON SZE CHUNG; LARKIN, KIERAN GERARD; PAN, RUIMIN. Non-Linear Solution for 2D Phase Shifting. Filed December 11, 2013 and posted July 3, 2014. Patent URL: http://appft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.html&r=4247&p=85&f=G&l=50&d=PG01&S1=20140626.PD.&OS=PD/20140626&RS=PD/20140626

Keywords for this news article include: Bone Research, Canon Kabushiki Kaisha.

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Source: Health & Medicine Week


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