Using the Local Phase of the Magnitude of the Local Structure Tensor for Image

Registration

Anders Eklund, Daniel Forsberg, Mats Andersson, Hans Knutsson

Linköping University

Agenda

• Phase based image registration• The local structure tensor• Extending the phase idea -

combining local phase and tensor magnitude• Results on synthetic data• Results on real data

Phase based image registration

• Image registration is normally done by using the image intensity, for example by maximizing the correlation or the mutual information between the images

• Easy to create test images for which intensity based algorithms fail…

Phase based image registration

• A better approach is to use the local phase, instead of the image intensity

• The local phase is better suited for the assumptions made in the optical flow algorithm(no intensity difference between images, the image is locally smooth)

• The local phase can for example be estimated by using quadrature filters

Quadrature filters

Frequency domain Spatial domain

Real part

Imaginary part

Phase based image registration

• The complex valued filter response q is a bandpass filtered version of the analytical signal

• Lines are even functions, related to cosine• Edges are odd functions, related to sine

• The quotient of the real and the imaginary filter responses gives information about the neighbourhood (edge or line, bright line, dark line, etc)

Local phase

Bright lineDark line

Dark to bright edge

Bright to dark edge

Phase based registration

• Phase based registration works wellfor images with different intensity

• There are, however, images for which even the phase based approach fails

• Example, the shapes of the objects are consistent between the images, but a dark to bright edge in one image is a bright do dark edge in the other image

Two nice test images

The image intensity as well as the local phase has changed

What is constant between these images? The shape of the objects!

Reference image Altered image

Real images?

• Are there images in the real world for which the shapes are constant, but the intensity and the local phase changes?

Quantitative MRI

Same subject, same slice, different MR

settingsSlice 1 Slice 2

Slice 3 Slice 4

Extending the phase idea

• We want a similarity measure that is invariant both to image intensity and to the local phase, i.e. that only considers the shape of the objects

• The local phase is invariant to the image intensity, but is different for dark to bright edges and bright to dark edges

Local phase

Reference image Altered image

The local structure tensor

• The local structure tensor represents the local structure as a 2 x 2 matrix (for 2D) in each pixel

• The local structure tensor can, for example, be estimated by using quadrature filters

Tensor magnitude

• The magnitude of the local structure tensor is invariant to the local phase (the orientation of a line and an edge in the same direction is the same)

• The tensor magnitude is not invariant to the image intensity

• The tensor magnitude is high where there is a well defined orientation

Tensor magnitude

Reference image Altered image

Local phase of tensor magnitude

• Combine the local phase and the tensor magnitude to achieve invariance both to image intensity and to local phase

• Apply quadrature filters to the original image and calculate the tensor magnitude in each pixel

• Apply the quadrature filters again, now to the tensor magnitude

Local phase of tensor magnitude

Reference image Altered image

Evaluation of similarity measure

• How good is our new similarity measure?• Mutual information of the image intensity (blue)• Mutual information of the local phase (green) • Mutual information of local phase of tensor

magnitude (pink)• One image was rotated between -30 and 30

degrees• Normalized values to have a maximum of 1

Reference and altered test image

MRI slices 1 and 3

Evaluation of registration

performance

• Tested the new similarity measure both with rigid and non-rigid registration

• No big increase in processing time,possible to use existing algorithms

• Calculate the tensor magnitude of the reference image and the source image

• Send the tensor magnitude images to the registration algorithm

• Apply the found displacement field to the original images

Rigid registration

• For rigid registration of the MRI slices the local phase of the image intensity works as well as the local phase of the tensor magnitude

• If the fat border is removed (constant for all four slices), only the local phase of the tensor magnitude works

Non-rigid registration

• Non-rigid registration is harder since it is not possible to use global information

• For the non-rigid registration the Morphon was used, which is a phase based non-rigid registration algorithm

Shifted test imagesintensity difference

Registration results

Local phase of image intensityLocal phase of tensor magnitude

MRI slices 1 and 3absolute intensity difference

Original Shifted

Registration results

Local phase of image intensityLocal phase of tensor magnitude

Summary

• A new similarity measure for image registration has been presented

• The idea of phase based registration was extended by using the local phase of the tensor magnitude, instead of the local phase of the image intensty

• Promising results for synthetic and real data have been presented

Thank you for your attention!