Linear Dimensionality Reduction Of Motion Information

From ILabWiki

Introduction

Like many other types of image information, motion information can be analyzed at multiple scales and in multiple directions. For instance, it is easy to imagine motion information in multiple directions. An object may move left or an object may move right. Thus, to catch motion, many filter schemes and biological systems detect motion in certain directions (Rall 1970; Prodöhl, Würtz and von der Malsberg 2003). Motion may also be detected in the absence of direction but suffers from the problem that all motion is treated equally. Thus, detecting motion without direction is no different than detecting flicker. Since we need directional information to be informative about the nature of a motion being observed we must introduce the first few dimensions of motion information, which is the direction of motion.

In addition to direction, the way in which we detect or process motion creates a problem of scale. The scale dimension comes about in many ways, but in particular when images are analyzed using a kernel-based method. For instance, when detecting image features using steerable gabor wavelets each image feature is in essence taken over the product of a gabor wavelet with a specific wavelength and phase. That is, a gabor patch, has a specific width, for instance 4 pixels wide in wavelength. If an image feature is much larger than that, or much smaller than that then the response of the gabor filter will be limited. As such, when using a kernel-based method, it is common to use multiple scales of gabors. This allows one to pick up image features at many different scales (Freeman and Adelson, 1990). However, this introduces new information in the form of new dimensions. This has the drawback that the information over motion increases for scales s and dimensions das s*d. Thus, data size gain may be large as dimensions and scales are added. This creates a difficulty if the motion information is desired for further processing. For instance, in order the classify the motion of any object in space using a scheme with 6 scales and 4 directions, one has to processes 24 dimensions. If one adds in other features for classification such as color, texture and luminance, the dimensions can grow very quickly. We can see that as more features are added the number of dimensions can quickly become a problem. As such if we wish to add any features to a classifier, including motion, we probably want to keep such dimensions scaled down since extraneous dimensions lead to Bellman’s Curse of Dimensionality.

Figure 1. The left image is the original from frame 300 of a 600 frame movie. The right image shows the motion channel response to motion at 6 different scales starting from the smallest. The direction response is right, top, left, bottom, horizontal and vertical. For each response seen, the output is the average over that type.

An immediate question one may have is, are all these dimensions really necessary for the understanding of motion information. That is, we introduced directionally tuned filters to differentiate motion direction, but we introduced multiple scales of processing not so much to increase the amount of information we have, but so much as to increase the likelihood of catching information. As such one might have an intuition that we can simply combine the information of motion filters, for instance, by summing all the filters in one direction and dividing by the number of filters. That is, we might reduce the dimensionality by finding the average left motion filter. An example of this approach is seen in figure 1. However, there is a problem with this approach. The information from each filter is not necessarily independent (Perona, Date). That is, a medium scale leftward motion filter may pick up a small motion object that a small scale leftward filter would also pick up. The medium scale filter would not provide a strong response, but its response would covary with a smaller filter. If the responses are not independent then the mean response is biased by the selection of filters. The motion filter rather than showing a strong response for the size and velocity of motion will favor objects whose features the filters bias the most. That is, the mean response for all leftward motion filters might favor an object whose shape fits best over several kernels. Thus, ideally in order to get a true idea of motion with dimensions reduced, we need to account for possible dependencies across scales.

Additionally another drawback to averaging is that by averaging over all the data across scales, we in essence throw away any information about the scales. Thus, this method of reduction suffers from the problem of not being able to deduce much about the scale information from the original data. As such, if we decide that the information contained in the scales is useful, we might want some way to preserve some of that information.

What we then want is a way to reduce the dimension of motion information that attempts to account for the interaction between scales. Here we discuss three basic methods. The first is independent component analysis/principle component analysis (ICA/PCA) (Hyvärinen and Oja, 2000; Hyvärinen, 1999; Bell and Sejnowski 1995). The other methods discussed are variations on basic orthonormal basis reduction. The first approach reduces the number of dimensions while trying to reduce mutual information between outputs. The other methods try to reduce the dimensions while providing all the information in a compact representation. All methods use linear transformation and are fairly similar. However, while ICA/PCA assumes that the information output is the full representation of the information the basic orthonormal basis reduction makes the opposite assumption that the information input is the full representation of the information. Both methods have their pros and cons which will be discussed later.

Before continuing we wish to mention how our use of ICA differs from other vision uses of ICA. For instance, a common use in vision of ICA is to find a basis set of vision filters (Karklin and Lewicki 2003; Hyvärinen and Oja, 2000; Hateren and Schaaf, 1998; Olshauson and Field, 1998, 1996; Bell and Sejnowski 1997). Thus, ICA is used to find a first level filter representation for vision. Here, we are using ICA to find a second level representation following filtering by kernel process filter. As such, what we will derive is a second level abstract set of post process filters.

ICA/PCA

ICA/PCA is a mixed method that combines the dimensionality reduction of PCA with the independence factoring of ICA. To describe how this works we will start out with a description of ICA and PCA on their own and then mention how they are merged together. For ICA the idea is that the information you have has been mixed by some unknown linear process and you wish to find a linear unmixing process. In general we wish to find some matrix W that unmixes some mixed input x to give us s:

were we have observed x as a mixing of a mixing matrix A and our data s

So for instance, you might have two microphones in a room with two people talking. Each microphone input signal has both voices on them. In essence, the space in the room has acted as a linear mixer conducting both voices onto both microphones. The question then is how do you get these signals apart?

ICA works by maximizing joint entropy between the two signals in this case. That is, it applies a linear transformation such that each output from channel 1 is as different as possible given the linear transformation from channel 2. For instance if the output from the linear transform is 1 for channel 1 then it would like the output to be 0 for channel 2 and visa versa. Thus, the first ICA implementations used a neural network with an anti-hebbian like rule. What this then provides is a solution to the cocktail party problem where many voices that are mixed can be unmixed. For ICA we can find the unmixing matrix using a neural net with the following rules using the method of Bell and Sejnowski (1995):

Here x is the input and y the output with W being the resulting unmixing matrix. j is the jacobian of the determinates of the partial derivative of y over x.

We can then apply this processes to images. What this gives us is an output where we have linearly reduced the dependence as much as possible assuming one information source for each input. Additionally, information redundancies have been reduced which gives us a more crisp view of our information.

However, prior to ICA we might apply PCA. Here PCA works by finding a linear projection of one data dimension onto another (Jollife 1986). This is sort of opposite of what ICA does in that PCA mixes signals or dimensions together that it assumes are redundant while ICA unmixes dimensions which it assumes to overlap. PCA accomplishes this mixing by finding planes to project data on where covariance is strong. The issue in combining ICA and PCA comes from the fact that ICA eliminates any covariance, thus PCA cannot work when ICA has been performed on data. Thus, PCA is performed first.

Figure 2. From the FastICA web page, we can see that ICA is achieved by whitening and then rotating the data and using projection persuit to find the independent components.

For combined ICA/PCA we use FastICA (Hyvärinen and Oja, 2000; Hyvärinen, 1999). This gives us the best of both worlds. We can assume that we have both mixed information as well as redundant information. For instance, if you have 20 dimensions, ICA will assume that you have 20 signals to separate. PCA allows you to reduce that number to any dimension. Thus we apply PCA and reduce 20 channels to 10 then we apply ICA to unmix those 10 channels. We thus are allowed to make the assumption that some information is redundant and covaries while some information is not redundant and covaries. So what we are doing is combining what we think is overlapping information and then sepperating what is left over. For instance, if you have four microphones and two voices, you would combine the four channels into two using PCA and then separate out the two voices from the two combined channels using ICA. An example of the working of FastICA is given in figure 2.

Basic Orthonormal Basis Reduction

ICA/PCA makes the assumption that more information is in the original data then one would like. Thus, it assumes that if you reduce 20 channels to 10, then there are really only 10 interesting pieces of information. The disadvantage to this approach is that we do not guarantee any preservation of the original 20 channels. However, we can reduce the information using a specially constructed orthonormal basis set that contains clues about the original data. The upside to this type of reduction is that general information from individual filter responses is recoverable. That is, each input channel contributes uniquely to the output such that it can be determined which inputs gave a positive response. The downside to this type of reduction is that we lose the ability to determine dependence between dimensions. However, since we have preserved some of the original information, we are permitted some ability to backward analyze the dimensionality reduced data.

Dimensionality Reduction on Motion Channels

The motion we are interested in processing is a part of the iLab saliency program (Itti and Koch, 2001). The saliency code has 24 motion channels in all. These breakdown into 6 different scale channels in the up, down, left and right directions. When a video is run through the saliency code each frame of video will produce output for all 24 channels. ICA/PCA can be computed by taking a large number of samples from motion video from a large sample of clips. In this case, ICA/PCA is computed with 65,000 samples limited by the 16 bit addressing of arrays in Matlab. These samples are taken from 20 video clips that are of length 10 seconds each. Each video is taken from a natural scene, but an effort is made to get a good general representation of a variety of scenes. In order to limit the sample size to 65,000 samples, only approximately 10 samples are taken per video frame. The 10 sub-samples per frame are taken from a randomized sampling of salient locations in each frame. The random sampling is done using a Monte Carlo method which treats the saliency map as a probability map (Mundhenk et al, 2004). Thus, the image is treated as a retinotopic region of neurons which have a probability of firing related to the saliency at their location. Samples are taken based on whether a neuron fires or not. The sub-sample is taken by decimating over the samples (Taking every nth sample). Since the first set of samples is random, the decimated sub-samples should be as well.

Figure 3. The 24 input channels are multiplied by a corresponding number in a filter. Each response is then summed. The filters can be interpreted by noticing that each chart is in the shape of a plus sign. The top arm corresponds to motion in the up direction, left to left etc. Larger scale components are at the tips of the arms.

The ICA unmixing matrix is determined using the FastICA package in Matlab. Processing is very fast in takes < 30 seconds on a Pentium IV 2.6 GHz based machine. If the dimensionality reduction is < 4 then > 90% of variance is accounted for. That is, if we assume that the data is generally linear, and well formed, then from the original 24 channels, quantitatively there is really not much more than 5 or 6 channels worth of data. Thus, an interesting observation is that approximately 75% of the data in the 24 motion channels is redundant and possibly unnecessary. This mirrors similar results with other saliency channels where for instance Red-Green, Blue-Yellow and Luminance channels are all reducible to 2 channels from 6 each while accounting for > 90% variance. Thus, similarly, about 66% of the data may be redundant (Mundhenk at al 2004).

A 24 to 6 reduction was chosen and the ICA/PCA filters were qualitatively examined. Figure 3 shows how the reduction is handled. In general with a 24 to 6 reduction we describe as:

Here x is the input from the 24 motion features. We can visualize W from directions starting with each motion direction over its six scales n as:

where d is a matrix of one direction a for instance leftward. In this case, d is a 6x6 matrix. We then have four of these for each motion direction and as such W is:

We then describe 6 filters that are part of a bank were each filter is a column in W and W is a bank of filters. Thus, the filters that are described in the following text contain one column from each da. Appendix A contains the raw data of several filters and can give one an idea of how these actually look. It should be noted that W is transposed in the way you view it. As such equations 1.1 and 1.2 still hold.

Figure 4. These are the 6 basic filter components derived by ICA/PCA. These charts can be interpreted using figure 3 as a reference.

As one can see from figure 4 the filters are a little odd, but there is an underlying symmetry that was intriguing. For instance, on many of the filters there is a anti-symmetry between up and down or right and left. That is, a filter might give a strong negative response to left motion (Negative as in less than zero, not as in Null) while giving a strong positive response to right motion. That is, a right and left motion response could cancel out. Additionally, there seems to be some specificity to scales. For instance small left moving features will only cancel out with other small right moving features but not with large right moving features. Figure 5 shows an output from this filter over a frame of motion from a video clip.

Figure 5. These are the outputs from frame 300 for all the three filter types mentioned. In general notice that there is a similarity among all three filter banks. For the orthonormal and semi-orthonormal notice that the orthonormal 1,2 matches closely semi-orthonormal 3,4 and orthonormal 3,4 matches very closely semi-orthonormal 1,2.

Orthonormal Basis Filter

The ICA/PCA analysis provided an intriguing idea; it might be that dimensionality reduction in the motion channel lies less with rather abstract feature relationships, but more with very concrete representations of directionality and scale cancellation. For instance, if you have a small leftward motion and a small rightward motion, you can let them cancel out as noise. Thus we developed a hand made dimensionality reduction filter to do just this. The filter was designed to be orthonormal which means that each of the 6 filters in the bank are orthogonal to each other in terms of information. We can define a set as orthonormal as following. For any matrix set:

as well as:

But the matrix must be normal as well. That is, the sum of squares for all columns must equal 1:

For all elements a in column m the sum of squares is 1. This means that the length of all the vectors in the matrix A are equal to one. Thus, it is normal. In Addition to orthonormality, the sum output for each filter is 0.

As such, each filter bank can cancel itself out if the input is high in entropy. In general terms, this also means that the filter bank is unbiased. It can also be observed that the sum of the absolute values for each of the 24 inputs across the filter bank is 6 in all cases. Thus, each of the 24 inputs carries equal weight across all the filters. We can express this for all rows n as:

Which when we take into account the number of columns N along row n we see that:

for all filter values. The orthonormal basis filter bank we derived can be seen in figure 6.

Another interesting feature of this filter bank is that each of the 24 input channels will provide a unique output signature. This can be seen in figure 7. As such, there is information that can provide a backward construction of the 24 input channels from the reduced set. Finally, it should be noticed that the filter bank has cancellation for left v. right and up v. down with specificity for scales. For instance, rightward motion tends to cancel out leftward motion, but that the cancellation is sometimes dependant on the motion being at the same scale.

Figure 6. These are the orthonormal filters derived. There exact normalized value can be seen in appendix A.

Another feature of this filter bank is that the six filters can be classified into three different sub-types. The first sub-type (1A and 1B) has a strong sensitivity to scale. Thus, motion in the same direction at different scales can cancel out. What this means is that this filter sub-type is particularly selective for the scale of the motion. The second sub-type (2A and 2B) has limited selectivity for scale, but strong selectivity for direction. Thus, for the first two filter types, we preserve some information specifically about the scale and direction of motion. The third sub-type (C and D) complete the orthonormality of the filter bank. Filter C is more sensitive to lateral v. horizontal movements while filter D is sensitive to large v. small scales without directional specificity.

Figure 7. Placed side by side, it can be seen that for each of the 24 channels, the filter bank gives a unique combination of positive and negative responses. Additionally, it is easy to see how the response values are normalized for each channel as the absolute value of each response sums to six. These exact values can be seen in their normalized form in appendix A.

Information recoverability

Information from the orthonormal basis set is partially recoverable depending on the type of input. If only one channel is activated then it can be determined easily which channel was activated since each channel has a unique combination of positive and negative activation combinations for each filter. However, if two or more input channels are activated then the information becomes more ambiguous. As such, for two or more channel activations, there are three types of combinations

1. Independant – The channels activate filter responses that do not conflict or create ambiguous information. For instance a combination of channel 12 and 22.
2. Dominant – One channel which activates with other channels has a dominant signature and as such would selected if inputs are equal. For instance, a combination of channel 19 and 20.
3. Anti – Two channels cancel each other out if they are symmetric to each other. For instance channels 8 and 22.

In general, this creates a situation where the information recovery is blurry, but reasonable. That is, dominant channels tend to dominate other channels that are similar. As such, the wrong answer is still close to reality. Thinking channel 20 activated with 20 and 19 activated is a minor error in general. What this means is that the output information about motion reflects the original input data, but does lose some information. We can then say that if more than one channel is activated at a time, the reduction is losey. In a way, this makes the dimension reduction of motion using such an orthonormal basis almost JPEGesque, and in general, this method does bier a resemblance to Fourier transformation methods.

Semi-orthonormal basis filters

Creating basis filters has some limitations. The first is that it is often difficult to stipulate exactly what you would like the basis filters to be like. For instance, using Gram-Schmidt orthonormalization (Arfken 1985) one can create a basis set from a set of linearly independent functions. However, the problem is that you are not guaranteed that the basis will resemble the exact filters you want. Indeed, it can be seen that many basis filter sets (for instance Olshausen and Field 1995, 1997) have an odd appearance, and while they account well for the information in an image, they lack the elegance of basic steerable filters. Additionally, the odd shape of each filter makes interpretation of the information difficult. That is, steerable filters for instance have an easy to understand response, but a gabor patch sticking half off to one side is much harder to interpret. Thus, a basis set constructed in this manner yields data that is harder to understand.

Figure 8. These are the filters derived from the semi-orthonormal process. There exact value can be seen in appendix A.

In addition, a true basis set may be less desirable since it may be more sensitive to noise. A non-basis set might average out spikes experienced in one filter since more than one filter can respond to the same feature. Thus, the redundancy of a non-basis filter set allows for smoothing of noise in an image. A possible way to address these two concerns is with the construction of a semi-othonormal basis. That is, we create a bank of filters which is constrained in its form to meet the simplicity of form requirement from the first requirement and by allowing it to have redundancies, it should have some smoothing properties.

To create a semi-orthonormal basis set, we start with a rigid set of parameters about the symmetry and form of the filters. We then define an orthogonality constraint which we also want to meet, for instance eq. 1.10. Using gradient decent and Powell’s method (Powell 1977) we adjust the free parameters until we have a filter bank that is as close to eq. 1.10 as possible, yet still does not violate the rigid parameters set down. An interesting result was obtained by using the well known symmetric Laguerre polynomials as a basis for the form of each directional section of each filter. Thus for l being a Laguerre polynomial, we derived 6 different sets of coefficients, one for each filter, with symmetry and anti-symmetry defined between up, right, down and left.

This created an interesting semi-basis set that was sensitive to motion by direction and scale, or only by scale. The form is essentially the same as orthonormal basis filters and exhibits information recoverability in a similar manner. However, we allow some redundant information and as such, it may be more resilient to noise. Indeed, as one can see from figure 5 the responses are very similar. However, in this case the redundancies seem to work against us since motion noise seems to be more prevalent following semi-orthonormal filtering when compared with the orthonormal basis set. This is most likely due to the fact that most of the noise is at the smallest scale. Redundantly repeating information from the smallest scale replicates this noise. However, while it does have this noise, it seems to lack some of the shadow output seen in ICA and the orthonormal basis set. Thus, we have removed one source of noise but perhaps introduced another.

Results and Discussion

Redundancy

An interesting result from these experiments is that the motion channels of the saliency model are extremely redundant, perhaps far more than could be beneficial. The evidence for this can be seen in two ways. First, ICA/PCA reduction of the 24 motion channels to 6 accounts for 90% of the variance. As mentioned this suggests that as much as 75% of the information in the motion channels is redundant. In general this may be quite a bit considering that most compression utilities get around 50% compression on most input data. Thus, in general, one might suspect that the average computer data is only 50% redundant.

The redundancy can also be seen by comparing the output from the orthonormal basis filter bank with the semi-orthonormal filter bank. The outputs are extremely similar despite the fact that the orthonormal basis filter tends to favor mid-scales more while the semi-orthonormal filter strongly favors extreme scales. Thus, the mid-scales and extreme scales may be redundant with each other. Redundancy is eliminated by both methods during their working. ICA/PCA eliminates redundancy during the PCA step. So long as the data is generally linear and well behaved, this tends to work well. However, PCA can fail for instance in the presence of multi-modal data. The orthonormal basis filter bank also removes some redundancy by allowing some entropy factors to cancel out. That is, a response across several scales will cause a cancellation effect. As such, very broad activations of channels that one would see with redundant data tends to be suppressed. However, the orthonormal basis is not designed expressly to eliminate redundancy. As such, the redundancy elimination claim is not as strong.

Noise

General noise reduction seems to be the greatest with the orthonormal basis filter. However, shadow noise seems to be more reduced by the semi-orthonormal filter. ICA/PCA seems to introduce more noise than the other two. In general, ICA should filter out noise while treating it as another channel. However, if the relationship between noise and non-noise is non-linear then ICA will perform poorly. This may be the case since the noise may be related to artifacts introduced by the camera. For instance, cameras perform adaptations during operation. As such, as objects move through a scene, the camera may introduce artifacts at different rates as it adapts to the changing scene. As such, the components we desire from the objects in the scene may have a non-linear dependence with artifacts introduced by the camera. In such a case, ICA may be inclined to treat such noise artifacts as bound to the object rather than as a separate entity. If such is the case, then ICA will partially fail. The solution is then to either use a camera more free of artifacts or develop a non-linear ICA, which is a thesis project on its own. Thus, it may be the case that ICA is undesirable for motion video.

It should be noted in addition that it is not surprising that the orthonormal and semi-orthonormal sets seemed less noisy than the original or the ICA/PCA result. This is because the form used acts in a band pass capacity. That is, each filter has limited sensitivities to certain scales. Additionally, information that contradicts cancels. This should allow it to filter out nonsense data, some of which may be noise. So for instance, if a location in an image has both small scale rightward and leftward responses, one might expect this to be a noise artifact since continuous objects moving in a scene should not exhibit such behavior. Thus, the anti-symmetry may contribute to noise cancellation.

Information Independence

ICA/PCA by its very design creates independent components such that the information output contains the least possible amount of mutual information. An orthonormal basis set functions differently making sure that no information is biased. Thus, the data does not pick up a new bias from this filter. It should be noted that an orthonormal basis set does not necessarily unmix data. Thus, an ICA matrix W is not the same thing as an orthonormal basis W. As such while dimension reduction by the orthonormal set meets the criteria of dimension reduction and does so in an unbiased manner, it does not guarantee that the outputs will be independent. As such, if we value information independence the ICA/PCA process is more likely meet our needs.

An interesting observation one can make from the different methods of reduction presented here is that the proper one to choose depends on what kind of data is desired. However, the othornormal method is interesting in that it is more a competitor with PCA than with ICA. As such, following the orthonormal operation, the data can have ICA computed on it. As such we can add independence to the data.

Biological relevance

In terms of biological relevance there are a few issues. The first is the question about plausibility. This question is simple to answer since some neural network operations can be reduced to a matrix multiplication. As such, since the reductions in their final form are merely matrix multiplications, they can be easily mapped to the functions of neurons. Additionally, the derivations can be plausible since ICA can be derived using a neural network. As to whether this is what is going on in the brain, in part yes and part no. The yes part is linked to the idea that the brain is constantly reducing features to simpler components. As such, the motion dimension reduction in this paper is functionally similar to gabor wavelet convolution in that it reduces and extracts more meaningful information from information already in place. However, it cannot be said with any authority at this point that this is a method used by the brain to reduce motion information.

Conclusion

To summarize, the different dimension reduction techniques have their pros and cons. The ICA/PCA method is still the method of choice to insure independence of the data output. However, the orthonormal set is better for data interpretation. Additionally, it seemed better at handling the noise than the ICA/PCA method, which was somewhat surprising. The other advantage of the Orthonormal matrix reduction is that it allows some ability to recover the original data. In general both approaches remove data redundancy since the orthonormal basis tends to pull apart data into an interpretation while ICA/PCA accomplishes this during the PCA step.

Appendix A: Filters

A.1 Semi-Orthonormal Laguerre Filter Bank

2.74E-02	2.74E-02	6.52E-01	6.52E-01	4.58E-02	-6.16E-01
8.17E-03	8.17E-03	1.95E-01	1.95E-01	1.37E-02	-1.83E-01
3.12E-02	3.12E-02	-1.33E-03	-1.33E-03	3.83E-02	2.84E-03
-1.29E-03	-1.29E-03	-3.12E-02	-3.12E-02	-2.08E-03	2.79E-02
-1.87E-01	-1.87E-01	9.69E-03	9.69E-03	-2.29E-01	-2.09E-02
-6.24E-01	-6.24E-01	2.61E-02	2.61E-02	-7.65E-01	-5.69E-02
-2.74E-02	2.74E-02	-6.52E-01	6.52E-01	4.58E-02	-6.16E-01
-8.17E-03	8.17E-03	-1.95E-01	1.95E-01	1.37E-02	-1.83E-01
-3.12E-02	3.12E-02	1.33E-03	-1.33E-03	3.83E-02	2.84E-03
1.29E-03	-1.29E-03	3.12E-02	-3.12E-02	-2.08E-03	2.79E-02
1.87E-01	-1.87E-01	-9.69E-03	9.69E-03	-2.29E-01	-2.09E-02
6.24E-01	-6.24E-01	-2.61E-02	2.61E-02	-7.65E-01	-5.69E-02
-2.74E-02	-2.74E-02	-6.52E-01	-6.52E-01	4.58E-02	-6.16E-01
-8.17E-03	-8.17E-03	-1.95E-01	-1.95E-01	1.37E-02	-1.83E-01
-3.12E-02	-3.12E-02	1.33E-03	1.33E-03	3.83E-02	2.84E-03
1.29E-03	1.29E-03	3.12E-02	3.12E-02	-2.08E-03	2.79E-02
1.87E-01	1.87E-01	-9.69E-03	-9.69E-03	-2.29E-01	-2.09E-02
6.24E-01	6.24E-01	-2.61E-02	-2.61E-02	-7.65E-01	-5.69E-02
2.74E-02	-2.74E-02	6.52E-01	-6.52E-01	4.58E-02	-6.16E-01
8.17E-03	-8.17E-03	1.95E-01	-1.95E-01	1.37E-02	-1.83E-01
3.12E-02	-3.12E-02	-1.33E-03	1.33E-03	3.83E-02	2.84E-03
-1.29E-03	1.29E-03	-3.12E-02	3.12E-02	-2.08E-03	2.79E-02
-1.87E-01	1.87E-01	9.69E-03	-9.69E-03	-2.29E-01	-2.09E-02
-6.24E-01	6.24E-01	2.61E-02	-2.61E-02	-7.65E-01	-5.69E-02

A.2 Orthonormal Basis Filter Bank (normalized)

3.86E-01	-3.86E-01	0.00E+00	0.00E+00	0.00E+00	0.00E+00
1.29E-01	-1.29E-01	2.58E-01	-2.58E-01	0.00E+00	0.00E+00
0.00E+00	0.00E+00	3.29E-02	-3.29E-02	-3.54E-01	-3.54E-01
-3.29E-02	3.29E-02	0.00E+00	0.00E+00	-3.54E-01	3.54E-01
-2.58E-01	2.58E-01	1.29E-01	-1.29E-01	0.00E+00	0.00E+00
0.00E+00	0.00E+00	3.86E-01	-3.86E-01	0.00E+00	0.00E+00
-3.86E-01	-3.86E-01	0.00E+00	0.00E+00	0.00E+00	0.00E+00
-1.29E-01	-1.29E-01	2.58E-01	2.58E-01	0.00E+00	0.00E+00
0.00E+00	0.00E+00	3.29E-02	3.29E-02	3.54E-01	-3.54E-01
3.29E-02	3.29E-02	0.00E+00	0.00E+00	3.54E-01	3.54E-01
2.58E-01	2.58E-01	1.29E-01	1.29E-01	0.00E+00	0.00E+00
0.00E+00	0.00E+00	3.86E-01	3.86E-01	0.00E+00	0.00E+00
-3.86E-01	3.86E-01	0.00E+00	0.00E+00	0.00E+00	0.00E+00
-1.29E-01	1.29E-01	-2.58E-01	2.58E-01	0.00E+00	0.00E+00
0.00E+00	0.00E+00	-3.29E-02	3.29E-02	-3.54E-01	-3.54E-01
3.29E-02	-3.29E-02	0.00E+00	0.00E+00	-3.54E-01	3.54E-01
2.58E-01	-2.58E-01	-1.29E-01	1.29E-01	0.00E+00	0.00E+00
0.00E+00	0.00E+00	-3.86E-01	3.86E-01	0.00E+00	0.00E+00
3.86E-01	3.86E-01	0.00E+00	0.00E+00	0.00E+00	0.00E+00
1.29E-01	1.29E-01	-2.58E-01	-2.58E-01	0.00E+00	0.00E+00
0.00E+00	0.00E+00	-3.29E-02	-3.29E-02	3.54E-01	-3.54E-01
-3.29E-02	-3.29E-02	0.00E+00	0.00E+00	3.54E-01	3.54E-01
-2.58E-01	-2.58E-01	-1.29E-01	-1.29E-01	0.00E+00	0.00E+00
0.00E+00	0.00E+00	-3.86E-01	-3.86E-01	0.00E+00	0.00E+00

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External Links

This paper can also be found in PDF format at: http://www.mundhenk.com/publications/WEBPUBLISHED-2004-Linear-Motion-Reduction.pdf