Deconvolution questions
For the ICTM method: none! However, this is only true when the a-priory knowledge that the object is non-negative is absolutely true, as is the case for fluorescence emission.
Yes. Huygens treats the image as the only known plane of a 3D stack and proceeds as usual. Before deconvolving, go to the Microscopic Parameters and set the z-sampling distance to the Nyquist rate as explained in 'sampling densities' (Huygens User Guide). For optimal results we do advise to image in 3D when possible, especially when performing quantitative analysis after deconvolution.
The deconvolution process enhances the fine structure of the cells, but unfortunately if the image contains artefacts (because of unstable laser power etc.) they will also become more apparent in the deconvolved image. Scanner instability is mainly a not-reproducible phenomenon, thus deconvolution is not the solution. In some cases the instability IS reproducible like slow thermal drift. If this instability is also present in the PSF measurement the deconvolution can reconstruct the image.
The absolute value of the final Quality factor much depends on the data, the microscope type, and the background. It is a global value computed over the entire image, so the contribution of a local resolution increase can be small.
For example, suppose you have a large featureless image with one tiny object. While the tiny object may be restored very well, the change in the featureless part is negligible. The quality factor will therefore hardly change, though the restoration is successful.
Usually, widefield images show a much higher quality increase than for example confocal images.
For example, suppose you have a large featureless image with one tiny object. While the tiny object may be restored very well, the change in the featureless part is negligible. The quality factor will therefore hardly change, though the restoration is successful.
Usually, widefield images show a much higher quality increase than for example confocal images.
TIRF (Total Internal Reflection Fluorescence) deals mostly with 2D images. Huygens can handle 2D images by internally treating them as part of a 3D stack from which most planes happen to be missing.
Strictly speaking Huygens does not generate TIRF Theoretical PSFs, but customers report good results with high NA confocal PSFs. Proceed by setting the Microscope type of your image to confocal and NA ~ 1.4.
Strictly speaking Huygens does not generate TIRF Theoretical PSFs, but customers report good results with high NA confocal PSFs. Proceed by setting the Microscope type of your image to confocal and NA ~ 1.4.
Yes, Huygens will certainly improve the image.
The first test should be done using a Wide Field Microscope Theoretical Psf with the actual parameters used for the acquisition, maybe with a lower Numerical Aperture to enlarge the PSF a little bit.
Still, remember that deconvolution assumes that the Image Formation is linear, and transmission is not, due to possible interference effects. These effects are lesser for thicker samples, but they can create restoration artifacts if they are noticeable. These can be balanced by properly tuning the Signal To Noise Ratio and the Max Num Of Iterations.
The ideal particle to distill an Experimental Psf with the Psf Distiller is probably a sub-resolution gold particle. Note that all the images should be inverted (negative) in Huygens before distilling a PSF and deconvolution: high intensities (gray values closer to white) should describe high object density, but raw transmission images provide the opposite. In Huygens Essential, this can be done with Tools > Invert image.
The Ideal Sampling constraints are about half as stringent as for the confocal imaging mode. For an objective with high N.A. (1.4) the voxel sizes should be in the range of 100-150 nm laterally and 350-500 nm axially.
The first test should be done using a Wide Field Microscope Theoretical Psf with the actual parameters used for the acquisition, maybe with a lower Numerical Aperture to enlarge the PSF a little bit.
Still, remember that deconvolution assumes that the Image Formation is linear, and transmission is not, due to possible interference effects. These effects are lesser for thicker samples, but they can create restoration artifacts if they are noticeable. These can be balanced by properly tuning the Signal To Noise Ratio and the Max Num Of Iterations.
The ideal particle to distill an Experimental Psf with the Psf Distiller is probably a sub-resolution gold particle. Note that all the images should be inverted (negative) in Huygens before distilling a PSF and deconvolution: high intensities (gray values closer to white) should describe high object density, but raw transmission images provide the opposite. In Huygens Essential, this can be done with Tools > Invert image.
The Ideal Sampling constraints are about half as stringent as for the confocal imaging mode. For an objective with high N.A. (1.4) the voxel sizes should be in the range of 100-150 nm laterally and 350-500 nm axially.
This strongly depends on the refractive indexes (lens immersion medium and specimen embedding medium). If there is no refractive index mismatch then restoration up to around 200 microns can be expected.
The restoration improves the resolution in all directions, but more so in the z-direction. In typical confocal images it is easy to increase the z-resolution by a factor 2. With a measured PSF a factor 4 is attainable.
However, the z-resolution in the measured image is often 4x worse than the lateral resolution. So at best you can compensate for that, but due to the lateral resolution gain the result will still be non-spherical. The gain in lateral resolution can be spoiled by applying a Gaussian filter (Operation window -> Restoration -> Gaussian filter/Quick Gaussian filter) to decrease lateral resolution again in a controlled manner. Because, in our experience, practically no one wishes to reduce resolution the restoration tools do not do that automatically.
Still, if the sphere is physically small enough (of the order of a voxel) the restoration can reduce it to close to a single voxel. Provided that the z-sampling is small enough. This situation occurs easily in the deconvolution of widefield images with 100nm x 100nm x 100nm voxel size.
However, the z-resolution in the measured image is often 4x worse than the lateral resolution. So at best you can compensate for that, but due to the lateral resolution gain the result will still be non-spherical. The gain in lateral resolution can be spoiled by applying a Gaussian filter (Operation window -> Restoration -> Gaussian filter/Quick Gaussian filter) to decrease lateral resolution again in a controlled manner. Because, in our experience, practically no one wishes to reduce resolution the restoration tools do not do that automatically.
Still, if the sphere is physically small enough (of the order of a voxel) the restoration can reduce it to close to a single voxel. Provided that the z-sampling is small enough. This situation occurs easily in the deconvolution of widefield images with 100nm x 100nm x 100nm voxel size.
For fluorescence and from a theoretical viewpoint MLE restoration is the only proper choice. However, for high SNR's as in widefield images this becomes rather academic. Since the ICTM method in Huygens Pro is computationally more efficient than the MLE implementation we suggest using ICTM for WF images. As to 'quality', an examination based on subjective criteria such as visual inspection might lead to ICTM winning in low noise cases; an examination based on mathematically sound criteria has shown MLE to be the best choice. For noisy images (most confocal ones) the MLE algorithm is not only scientifically more correct, it also produces visually more pleasing images: far less background noise artifacts than ICTM.
Huygens Deconvolution is optimized for fast performance without compromising quality. The exact computation time depends on several factors, including:
For more information on this, visit the GPU Benchmarks page.
- Microscope type: images from widefield microscopes tend to require more iterations than those from confocal or 2-photon microscopes.
- Object type: sparse objects can be restored more effectively than dense objects. The more resolution gain is possible, the more iterations are needed, even if the iterations themselves also become more effective ('bigger steps').
- Noise: low noise makes a large resolution gain possible so then more iterations are needed.
- Algorithm: our Good's Rougnness MLE (Maximum Likelihood Estimation) needs less time than our Classic MLE; see Deconvolution Algorithms.
- Hardware: the number and type of CPU processors and GPU cards influence performance. Additionally, if the memory is insufficient, the processing speed depends on the type and specifications of virtual memory, as well as I/O performance.
For more information on this, visit the GPU Benchmarks page.
The deconvolution itself is done frame by frame, but the pre and post processing need the whole time sequence. The preprocessing consists of bleaching correction and background estimation. In Huygens Pro there is also the possibility to apply a time or full 4D prefilter. In the postprocessing the Z-drift is corrected over time (see Zdrift Correction). Most of these operations work on multiple adjacent frames, so working on a file by files bases would limit the current and future processing possibilities.
This does not imply that the 4D stack image needs to be loaded in RAM memory. Provided the swap space is large enough most of it can be swapped out. In fact, paging out means that the processed data is written in raw form back to disk. This will slow down the preprocessing operations, but the deconvolution speed would hardly be impeded, paging being only necessary between frames.
A second possibility is to use a script to process the frames individually. Because the CMLE and QMLE deconvolution methods allow you to specify a background level as a percentage of the estimated background that would handle varying backgrounds also nicely.
This does not imply that the 4D stack image needs to be loaded in RAM memory. Provided the swap space is large enough most of it can be swapped out. In fact, paging out means that the processed data is written in raw form back to disk. This will slow down the preprocessing operations, but the deconvolution speed would hardly be impeded, paging being only necessary between frames.
A second possibility is to use a script to process the frames individually. Because the CMLE and QMLE deconvolution methods allow you to specify a background level as a percentage of the estimated background that would handle varying backgrounds also nicely.
There can be two major reasons for this:
- There is a problem with the measured PSF:
- bead images were saturated or undersampled.
- beads have formed aggregates.
- beads were moving while being imaged.
- insufficient signal from beads causing inaccuracies in the averaging procedure.
- strongly varying background.
- The conditions under which the PSF was measured do not match the imaging conditions.
- The most important parameter is the medium refractive index. To exclude magnification calibration problems, best record the bead images at the same sampling density as the specimen. To check whether there is a matching problem between the measured PSF and the specimen data, deconvolve the specimen with a theoretical PSF. If the result is better then there can be a matching problem.
When you deconvolve a 2D-time image as a 3D image, you force the software to assume that the relationships between the 2D slices, if any, are due to the axial imaging properties of the microscope, but this is obviously not the case! The results will be therefore incorrect.
Instead, when you correctly process the 2D-Time Series as such with the Time option the software will, after correcting for bleaching and variable background, properly deconvolve each 2D image as a 3D data set with one plane recorded and the rest missing, as explained here Is deconvolution on 2D or 2D-time images possible?.
Some File Formats having indexes in the file names are interpreted as 3D stacks by default: you may need to convert the dataset from XYZ to XYT once opened.
As an alternative you could write a Tcl script for Huygens Scripting or Huygens Professional to deconvolve the 2D-time series frame by frame. Still, your images would not be corrected for bleaching and varying background, a task automatically performed if you have the Time option.
In case of a 3D-time series, with the Time option your data will be also corrected for axial drift.
Instead, when you correctly process the 2D-Time Series as such with the Time option the software will, after correcting for bleaching and variable background, properly deconvolve each 2D image as a 3D data set with one plane recorded and the rest missing, as explained here Is deconvolution on 2D or 2D-time images possible?.
Some File Formats having indexes in the file names are interpreted as 3D stacks by default: you may need to convert the dataset from XYZ to XYT once opened.
As an alternative you could write a Tcl script for Huygens Scripting or Huygens Professional to deconvolve the 2D-time series frame by frame. Still, your images would not be corrected for bleaching and varying background, a task automatically performed if you have the Time option.
In case of a 3D-time series, with the Time option your data will be also corrected for axial drift.
Yes. The results are especially remarkable for Widefield Microscopes. A 2D image recorded with a microscope can be considered as a slice from a 3D image. In this case the Huygens Software treats the data as a (severely) truncated 3D image, but 3D nonetheless. When deconvolving it Huygens attempts to reconstruct blur sources outside the slice and remove blur from it, regardless of the time frames.
If you have a 2D time series, make sure Huygens interprets the series as such. Some File Formats having indexes in the file name are interpreted as 3D stacks by default: you may need to convert the dataset once opened from XYZ to XYT. See Convert The Data Set. The restoration might not be optimal otherwise, as explained in Can a 2D-time series be deconvolved as 3D stack?.
Proper parameters
When doing 2D deconvolution of widefield images, set the Z Sample Size to the ideal Nyquist value. You can calculate this by using the Nyquist Calculator.If you have a 2D time series, make sure Huygens interprets the series as such. Some File Formats having indexes in the file name are interpreted as 3D stacks by default: you may need to convert the dataset once opened from XYZ to XYT. See Convert The Data Set. The restoration might not be optimal otherwise, as explained in Can a 2D-time series be deconvolved as 3D stack?.
Yes, this is certainly possible in Huygens.
For versions 21.10 and higher, this can best be done in the Deconvolution Wizard in either Huygens Essential or Huygens Professional. If the microscopic parameters specify the image as "brightfield" the image will first be inverted automatically by the Deconvolution Wizard. This is done because dense objects should have high intensity vallues in our deconvolution algorithms. Automatically the Tikhonov Miller algorithm will be selected. This algorithm is strongly advised for all brightfield or equivalent images as this algorithm does not amplify the background noise.
For the other deconvolution tools, the PSF distiller and the renderers the image should first be inverted manually. In Huygens Essential this can be done by choosing the option 'invert image' from the 'Tools' menu. In Huygens Professional select the image and go to 'Deconvolution', 'Operations window', 'Arithmetic', 'One image', and then 'Invert'.
Brightfield imaging is not a 'linear imaging' process. In a linear imaging process the image formation can be described as the linear convolution of the object distribution and the point spread function (PSF), hence the name deconvolution for the reverse process. So in principle one cannot apply deconvolution based on linear imaging to non linear imaging modes like brightfield and reflection. Fortunately, in the brightfield case the detected light is to a significant degree incoherent. Because in that case there are few phase relations the image formation process is largely governed by the addition of intensities, especially if one is dealing with a high contrast image, 'linearizing' the problem. In short, a Bright Field Microscope is not exactly a linear imaging device, but can be made to behave almost like one.
Dr. Marcel Oberlaender et al. from the MPI of Neurobiology in Martinsried proved the validity of the Huygens deconvolution for brightfield data with the linear Tikhonov Miller algorithm.
In practice one goes about deconvolving brightfield images by inverting them and processing them further as incoherent fluorescence widefield images. Still, one should watch out for interference patterns (periodic rings and fringes around objects) in the measured image. These could become pronounced in low contrast images.
For versions 21.10 and higher, this can best be done in the Deconvolution Wizard in either Huygens Essential or Huygens Professional. If the microscopic parameters specify the image as "brightfield" the image will first be inverted automatically by the Deconvolution Wizard. This is done because dense objects should have high intensity vallues in our deconvolution algorithms. Automatically the Tikhonov Miller algorithm will be selected. This algorithm is strongly advised for all brightfield or equivalent images as this algorithm does not amplify the background noise.
For the other deconvolution tools, the PSF distiller and the renderers the image should first be inverted manually. In Huygens Essential this can be done by choosing the option 'invert image' from the 'Tools' menu. In Huygens Professional select the image and go to 'Deconvolution', 'Operations window', 'Arithmetic', 'One image', and then 'Invert'.
Brightfield imaging is not a 'linear imaging' process. In a linear imaging process the image formation can be described as the linear convolution of the object distribution and the point spread function (PSF), hence the name deconvolution for the reverse process. So in principle one cannot apply deconvolution based on linear imaging to non linear imaging modes like brightfield and reflection. Fortunately, in the brightfield case the detected light is to a significant degree incoherent. Because in that case there are few phase relations the image formation process is largely governed by the addition of intensities, especially if one is dealing with a high contrast image, 'linearizing' the problem. In short, a Bright Field Microscope is not exactly a linear imaging device, but can be made to behave almost like one.
Dr. Marcel Oberlaender et al. from the MPI of Neurobiology in Martinsried proved the validity of the Huygens deconvolution for brightfield data with the linear Tikhonov Miller algorithm.
In practice one goes about deconvolving brightfield images by inverting them and processing them further as incoherent fluorescence widefield images. Still, one should watch out for interference patterns (periodic rings and fringes around objects) in the measured image. These could become pronounced in low contrast images.
Using the Operations Window in Huygens Professional all channels are processed in one run. To work with only one channel proceed to split the multichannel image (Select Edit->Split or ALT-S) into single channel images. The single channel images can be deconvolved individually with specific parameter values. For example, the SNR can be edited in the "Signal/Noise per channel" input line of the Operations Window. The Join operation can be used after the deconvolution of single channel images to combine them into a multichannel image.
If using the Devonvolution Wizard of Huygens Professional or Essential, you can choose which channel you want to process and skip those you are not interested in.
If using the Devonvolution Wizard of Huygens Professional or Essential, you can choose which channel you want to process and skip those you are not interested in.
Besides the Huygens deconvolution FAQ there is plenty of information on image acquisition and restoration, deconvolution, and the way the Huygens Software works available at the SVI site. Below, a number of links to help you get started:
It can be rigorously proven that when dealing with non-negative objects (fluorescing objects) the I-divergence criterion (MLE) is the only consistent choice, whereas for objects which can be both positive and negative (e.g. sound) least squares (ICTM) is the best choice. Another viewpoint is that I-divergence incorporates the Poisson nature of the emitted fluorescence light whereas least squares incorporates Gaussian noise. With this in mind one could expect that for low noise levels where the differences in the Poisson and Gaussian distributions are small there is no preference between the methods other than their computational efficiency. One reason why we still have a preference for the MLE algorithm is that it handles noisy backgrounds much better than the ICTM. A disadvantage of MLE is that it easily overemphasizes small structures, but we constrain this.
See also: MLE versus ICTM - Will one method be faster or more rigorous or give a better quality result?.
See also: MLE versus ICTM - Will one method be faster or more rigorous or give a better quality result?.
As a rule of thumb one gets easily 2x in Z and a bit less in XY. This applies to even noisy images. To gain more resolution a measured PSF is necessary. Some published figures (Bioimaging 4 1996 pp. 187-197): HIW (nm) z x y
- raw bead image 790 270 265
- restored 221 116 93
- bead object function 83 83 83 (i.e. the `true' bead)
- difference 138 33 10
The quality factor (QF) can only be used in a relative way, to compare between iterations, and is used for the stop criterion. The comparison of these values makes no sense if applied to deconvolutions of different images. The Quality Factor of the MLE algorithm is directly derived from the I-divergence; in the ICTM algorithm it is derived from the Tikhonov-Miller functional as described in the literature.
Sometimes one channel of the dataset contains Reflected light signal, i.e., the light which passes straight through various beam splitters and on into the Reflection Detector.
- Is there any validity in deconvolving this Reflected light component?
- If so what emission / excitation wavelength's would be applicable?
Yes. Single plane widefield deconvolution works because the data is extrapolated into a region above and below the plane spanning typically between 10-20 planes of 100-300 nm in Z. The software generates an appropriate PSF.
The QMLE iterations are approximately 5 times more efficient than the CMLE iterations, whereas they also take slightly less time per iteration. So 10 QMLE iterations are equivalent to 50 iterations in CMLE. CMLE is superior in handling low Signal to Noise (SNR) data, like low light level confocal images. In principle the CMLE algorithm with an SNR setting > 60 converges to the same result as the QMLE algorithm, but after many more iterations. Also, for good quality widefield images QMLE is the best choice.
See Restoration Methods.
See Restoration Methods.
The deconvolution process increases the Dynamic Range of the dataset, i.e, the intensity range increases. A computer screen cannot display high intensities arbitrarily, there is a maximum for the brightness of a pixel in the screen. Therefore the maximum intensity in an image is normally mapped to the maximum intensity a screen can display, and all the other pixels in the image are scaled accordingly. Read more here.
People have done simulations with synthetic objects:
- van der Voort, H.T.M and K.C. Strasters, "Restoration of confocal images for quantitative image analysis" JoMi 178, 1995, pp 165-181.
- van Kempen, G.M.P. et al, "Comparing Maximum Likelihood Estimation and Constrained Tikhonov-Miller Restoration". IEEE Eng. in Med. And Biology 15 No 1. pp 76-83, 1996.
The extra microscopic parameters for STED images are needed to define the effect of the depletion laser and will improve the deconvolution. There are five microscopic parameters per channel which are only neccessary for STED images:
- Excitation Fill Factor
- Saturation factor
- Wavelength
- Immunity fraction
- Shape coefficients
Indeed go to the Batch-Processor ( Under the header Deconvolution) you can do much:
You are interested in the latter:
- You can deconvolve massive amounts of images automatically
- You can use the Batch-Processor as a file-converter for massive amounts of data (use 'skip' in the algorithm choice in the Deconvolution template)
- You can tune the optimal setting for a particular data-set very quickly
You are interested in the latter:
- Press 'Add task' and a second connected window will open up.
- Choose 'Select Files' and select the dataset you want to apply the deconvolution algorithm with different parameters settings to.
- Continue and either make a NEW MICROSCOPY TEMPLATE or select an EXISTING MICROSCOPY TEMPLATE (If that contains the correct information about the acquisition of the dataset already).
- If you are sure that the meta data are 100% correct already you can choose ANY MICROSCOPY TEMPLATE and choose in the toggle window 'Keep all meta data from file'.
- Continue and either make a NEW DECONVOLUTION TEMPLATE or select an EXISTING DECONVOLUTION TEMPLATE (If that contains the correct information about the dataset already).
- Press DONE and you will be back in the first BP window.
- Click on the little window under 'Task', go t Edit and choose duplicate one or more times depending on the number of copies you want
- You see now four identical jobs.
- As you want to change the DECONVOLUTION TEMPLATES setting simply click on it .
- The second connected window will open again and you can change the setting in the DECONVOLUTION TEMPLATE via the tabs. (e.g. different algorithms, different SNR, adding a Chromatic Aberration template etc), save it before closing.
- You will see that after the name of the DECONVOLUTION TEMPLATE ':0' is added. If you do it again for another in the list you will see ':1' added.
The voxel sizes are not changed by the restoration! The slicer takes the exact aspect ratio into account. No rounding off to integers.
Usually, when this happens you have run out of swap space. If you work with large images it is better to switch off the undo system. As a rule of thumb, bear in mind that 3 or 4 times the size of the float image can be reserved by Huygens out of the available RAM memory. The rest can be swapped out. When working with large images in Huygens then the disk I/O speed is an additional bottleneck. As a result, the time needed to swap pages in and out during an iteration determines computing speed.
Consider a 300MB dataset (~25Mvoxel image): the system has to write AND read *at least* 300MB per iteration. To speed up swapping it is advantageous to count on fast disks. Additionally, the current ICTM algorithm cannot process large datasets brick by brick. The CMLE and QMLE methods both can process datasets brick by brick. We recommend to use CMLE in general and QMLE in particular for low noise widefield data.
Consider a 300MB dataset (~25Mvoxel image): the system has to write AND read *at least* 300MB per iteration. To speed up swapping it is advantageous to count on fast disks. Additionally, the current ICTM algorithm cannot process large datasets brick by brick. The CMLE and QMLE methods both can process datasets brick by brick. We recommend to use CMLE in general and QMLE in particular for low noise widefield data.
We have a set of batch scripts (TclTk) available on our webserver: See Batch Script. Notes:
- The script is able to launch multiple parallel jobs to make full usage of a multiple cpu system.
- These scripts use Huygens Scripting (license needed).
The multiphoton feature can be set up in the 'Excitation photon count' field in the Microscopic Parameter Editor. For a 2-photon system set this field to '2'. See Multi Photon Microscope.
These steps will help to lower the amount of memory needed for deconvolution:
- Switch off the undo system (Options menu).
- Reduce the size of the image (Crop tool). In particular, widefield images often contain many slices with blur.
- A theoretical PSF will be generated on-the-fly by the MLE and QMLE restoration methods. There's no need to create a theoretical PSF by hand.
When you convolve two images the resulting intensity is spread over the corners of the image. This phenomenon is caused by the following two properties of the FFT:
`Real' Fourier transforms contain only the positive frequencies in the u-direction (with u, v, w , spatial frequencies corresponding with the x, y, z axis). However, they do contain negative frequencies in the other dimensions. For visualization purposes real-FFTs are therefore less suited than complex-FFTs.
As a consequence of defining the frequency origin (frequency space) at the (0,0,0) voxel, convolutions with funtions which are not centered around (0,0,0) will cause a shift in the resulting image. For example, when you convolve an image with a sphere located at the center ( xc , yc , zc ) of the second image, the result will be shifted over a vector ( xc , yc , zc ). Because images are interpreted as periodic all octants in the resulting image will appear `swapped'. You can prevent this from happening by using the following methods:
- Images are interpreted as periodic, i.e. the image has infinite size but repeats itself in each dimension with periods that are equal to the size of the original image in the corresponding dimension.
- The frequency origin (frequency space) is located at the zero voxel (bottom-left-plane) of the 3D image. The positive frequencies are located in the first octant of the image. Depending on the type of transform you selected, `complex' or `real', the negative frequencies are present in the other octants.
`Real' Fourier transforms contain only the positive frequencies in the u-direction (with u, v, w , spatial frequencies corresponding with the x, y, z axis). However, they do contain negative frequencies in the other dimensions. For visualization purposes real-FFTs are therefore less suited than complex-FFTs.
As a consequence of defining the frequency origin (frequency space) at the (0,0,0) voxel, convolutions with funtions which are not centered around (0,0,0) will cause a shift in the resulting image. For example, when you convolve an image with a sphere located at the center ( xc , yc , zc ) of the second image, the result will be shifted over a vector ( xc , yc , zc ). Because images are interpreted as periodic all octants in the resulting image will appear `swapped'. You can prevent this from happening by using the following methods:
- Center the image with which you are going to convolve around (0,0,0). When this image, for instance a sphere generated with Generate sphere , is centered use To/from optic rep. to shift it to the origin. In other cases use the following method:
- (Alternative centering method) Determine the Center of Mass (CM) of the second image with Image statistics. Then use Shift image to move the CM to the origin. Since Shift image also interprets the image as periodic, this will produce the desired result. An advantage of this method is that you can shift the CM over a non-integer distance.
- When you have convolved with a function located at the center of the image, you can undo the shifting effect by applying To/from optic rep. to the convolution result.
Yes, key publications are:
- Csiszar, I., 1991; Why least squares and maximum entropy? An axiomatic approach to interference for linear inverse problems. Ann. Stat., 19, No. 4, pp. 2033-2066 (PDF 8 MB).
- Kempen, G.M.P., van der Voort, H.T.M., 1006; Comparing Maximum Likelihood Estimation and Contrained Tikhonov-Miller Restoration. IEEE Engineering in Medicine and Biology vol 15, No 1, pp. 76-83