Alignment and Matching

Alignment and Matching
In this part of the course we will focus on the challenges of matching 3D shapes once a particular shape descriptor has been chosen. Thomas Funkhouser and Michael Kazhdan Princeton University

Challenge The shape of model does not change when the model is translated, scaled, or rotated Translation Rotation = One of the challenges of matching 3D models stems from the fact that we consider the shape of a model to be unchanged under the action of similarity transformations. So if you are given two models to match, you don’t want to simply match them in whatever alignment they are given to you, instead, you want to find the optimal pairwise alignment between the two models and then compare them at that alignment. In general, there are three different approaches to resolving this problem. Scale

Outline Matching Conclusion Alignment Exhaustive Search Invariance
Normalization Part vs. Whole Conclusion One of the challenges of matching 3D models stems from the fact that we consider the shape of a model to be unchanged under the action of similarity transformations. So if you are given two models to match, you don’t want to simply match them in whatever alignment they are given to you, instead, you want to find the optimal pairwise alignment between the two models and then compare them at that alignment. In general, there are three different approaches to resolving this problem.

Exhaustive search for optimal rotation
Search for the best aligning transformation: Compare at all alignments Match at the alignment for which models are closest The first approach is to exhaustively search over the space of all transformations for the transformation minimizing the distance between the models. We then define the distance between the models as the distance at the minimizing transformation. Consider, for example, the case in which we would like to find the best rotation for aligning the cow to the dog. To do this we would compare the dog to the cow at all possible rotations, computing the shape descriptor of the rotated cow and comparing it to the shape descriptor of the dog. This would give us the distance between the two models as a function of rotation. Exhaustive search for optimal rotation

Exhaustive Search Search for the best aligning transformation:
Compare at all alignments Match at the alignment for which models are closest By searching for the rotation that minimizes the distance between the dog and the cow we can find the optimal alignment of the models, and get a measure of the model similarity at that alignment.

Exhaustive Search Search for the best aligning transformation:
Use signal processing for efficient correlation Represent model at many different transformations Properties: Gives the correct answer Is hard to do efficiently In general, the search for the optimal alignment is performed in one of two ways. In the first approach, a signal processing framework is used to efficiently compute the distance between one model and all transformations of the other one. This approach is well suited in the case when we would like to match over any possible transformation as the signal processing technique gives the measure over a uniformly sampled set of transformations. In some cases however, we can limit the space of possible aligning transformations, and in this case it can be more efficient to explicitly represent a model in the different valid transformations and directly compare within this subset of transformations. While either approach can be used to find the correct measure of similarity – explicitly solving for the optimal alignment of two models, these approaches require a high computation overhead and tend to inhibit the efficiency of retrieval. In the second approach, instead of storing only one shape descriptor for a model, a collection of shape descriptors are stored, corresponding to the shape descriptor of the model at different poses.

Invariance Represent a model with information that is independent of the transformation: Extended Gaussian Image, Horn: Translation invariant Shells Histograms, Ankerst: Rotation invariant D2 Shape Distributions, Osada: Translation/Rotation invariant The second method used for addressing the alignment problem is to extract information from the shape descriptor that does not change when the model is transformed. The advantages of these types of approaches is that any pose of the same model will give rise to the same representation, so that it is not necessary to explicitly solve for the best alignment. In the past, there have been a number of different shape descriptors proposed that address the alignment problem by designing the shape descriptor so that it is invariant to some class of transformations. For example, the Extended Gaussian Image, which bins the surface normals of a model, does not use information about the location of points on the surface of the model, and is therefore invariant to translation. The Shells Histogram, which bins the distance of points from the center of mass, leverages the fact that the distance of a point from the center of mass remains unchanged if the model is rotated to design a rotation invariant representation. And the D2 distribution, which bins the distance between pairs of points uses the fact that the distance between two vectors is unchanged when the model is translated or rotated to design a shape representation that is simultaneously translation and rotation invariant. Rather than focusing on any particular implementation of invariance, we will describe a general framework for extracting transformation invariant information from a shape descriptor. The methods that we describe are sufficiently general to be applicable to a broad class of shape descriptors, and it turns out that most previous descriptors that are designed to be transformation invariant implicitly use some of these methods. Shells Histogram EGI D2 Distribution

Invariance Represent a model with information that is independent of the transformation Power spectrum representation Fourier Transform for translation and 2D rotations Spherical Harmonic Transform for 3D rotations The general approach that we consider utilizes the fact that the power spectrum of a function, that is the amplitudes of its frequency components, discards phase information and results in a representation that is invariant to some class of transformations. I will first review the Fourier decomposition which can be used to obtain a translation invariant representation and then go on to describe Spherical Harmonic Transform which can be used to obtain a rotation invariant representation. Energy Energy Frequency Frequency Circular Power Spectrum Spherical Power Spectrum

Translation Invariance
1D Function In the one-dimensional case, our input is a function defined on an interval. By using the Fourier decomposition, we can express the initial function in terms of its cosine and sine expansion:

… = + + + + 1D Function The important fact about this expansion is that we can combine the cosine and sine terms to obtain a frequency decomposition, expressing the initial function as a sum of: Cosine/Sine Decomposition

… = + + + + 1D Function A constant term, = Constant Frequency Decomposition

… = + + + + + 1D Function A first order term, + = Constant 1st Order Frequency Decomposition

… = + + + + + 1D Function A second order term, + + = Constant 1st Order 2nd Order Frequency Decomposition

… = + + + + + 1D Function And so on. This is important for us because we know that the frequency of a function is unchanged by phase shift. For example, if a function can be expressed as the linear sum of cosine 2 theta and sine 2 theta, then any shift of the function can also be expressed as the linear sum of cosine 2 theta and sine 2 theta. … + + + + = Constant 1st Order 2nd Order 3rd Order Frequency Decomposition

Amplitudes invariant to translation … = + + + + + 1D Function So that in particular, this implies that the amplitude (or L2-norm) of each frequency component does not change when the initial function is translated. … + + + + = Constant 1st Order 2nd Order 3rd Order Frequency Decomposition

Rotation Invariance = + +
Represent each spherical function as a sum of harmonic frequencies (orders) + = In a similar fashion, given a spherical function we can obtain a rotation invariant representation using the amplitudes of its frequency components. In this image, I am representing a spherical function by scaling points on the surface of the sphere in proportion to their value. So that points on the sphere that have large value are pushed away from the center, and points with small value are pulled in. To obtain the decomposition of the spherical function into frequency components, we first express the function as a sum of its spherical harmonics. These are functions on the sphere that play the same role as the cosine and sine function do for a function on a circle. The only real difference is that as apposed to the cosine sine function, of which there are only two for each frequency, the spherical harmonics have the property that as the frequency increases, you need more and more of them. So that, for example, the 1st order frequency functions can be expressed as the sum of 3 harmonics, while the 2nd order frequency functions can be expressed as the sum of 5 harmonics, and so on. ----- Now just as with the cosine and sine function, we can take the sum of the harmonics within each frequency to obtain an expression of the initial spherical function as the sum of constant, first order, second order, and so on, frequency components. + Constant 1st Order 2nd Order 3rd Order

Rotation Invariance = + + +
Store “how much” (L2-norm) of the shape resides in each frequency to get a rotation invariant representation We then obtain a rotation invariant representation of the spherical function by storing only the energy of each frequency component. As with the circular power spectrum, this method gives rise to a rotation invariant representation because the different frequency components are fixed by rotation. Constant 1st Order 2nd Order 3rd Order = + + +

Power Spectrum Translation-invariance:
Represent the model in a Cartesian coordinate system Compute the 3D Fourier transform Store the amplitudes of the frequency components Cartesian Coordinates Using these approaches, we can extract transformation invariant information from the shape descriptor. Consider, for example the case that we have a shape descriptor that represents a model by a voxel grid. If we would like to make the shape descriptor invariant to translation, we compute the 3D Fourier transform of the descriptor. Then, we represent the model by its power spectrum, storing only the amplitude of the different frequency components, giving rise to a representation that does not change when the model is translated. y z Translation Invariant Representation x

Power Spectrum Single axis rotation-invariance:
Represent the model in a cylindrical coordinate system Compute the Fourier transform in the angular direction Store the amplitudes of the frequency components q Cylindrical Coordinates r Similarly, if we want to make the shape descriptor invariant to rotation about a fixed axis, we represent the model in cylindrical coordinates about that axis, getting a parameterization of the shape descriptor in terms of radius, height, and angle. If we fix a particular radius and height, we get a one-dimensional function giving the value of the shape descriptor within a fixed ring about the axis of rotation. By expressing each one of these one-dimensional functions in terms of its Fourier decomposition and storing only the amplitudes of the different frequency components, we get a representation of the initial shape descriptor that does not change when the model is rotated about the specified axis. h Rotation Invariant Representation

Power Spectrum Full rotation-invariance:
Represent the model in a spherical coordinate system Compute the spherical harmonic transform Store the amplitudes of the frequency components q Spherical Coordinates f Still more generally, if we want to make the shape descriptor invariant to all rotations, we represent the model in spherical coordinates, getting a parameterization of the shape descriptor in terms of radius and spherical angle. Again, if we fix a particular radius, we get a spherical function giving the value of the shape descriptor within a fixed shell. By expressing each one of these spherical functions in terms of its Spherical Harmonic Transform and storing only the amplitudes of the different frequency components, we get a representation of the initial shape descriptor that does not change when the model is rotated about the origin. r Rotation Invariant Representation

Power Spectrum Power spectrum representations
Are invariant to transformations Give a lower bound for the best match Tend to discard too much information Translation invariant: n3 data -> n3/2 data Single-axis rotation invariant: n3 data -> n3/2 data Full rotation invariant: n3 data -> n2 data The advantage of these types of approach is that obtained representation is always guaranteed to be transformation-invariant. Furthermore, the similarity between the transformation-invariant representations can often be used to bound the proximity of the initial shape descriptors at the best possible alignment. However, in practice we find that these approaches can be limited as a result of the fact that they discard too much information. That is, in discarding information that depends on the transformation of the model, they also tend to discard other information that is valuable for discriminating between different models. For example, if we compute a translation invariant or single-axis rotation invariant representation, we end up with half as much data as we started with, though in the translation case we only want to discard three degrees of freedom and in the single-axis rotation case we only want to discard one dimension of freedom. Similarly, if we compute a full rotation invariant representation we end up representing order n^3 data by order n^2 amount of information. Since the group of rotations is a three-dimensional manifold, we only really want to discard three degrees of freedom and as a result the rotation invariant representation must discard rotation-invariant information as well.

Power Spectrum Method Translation Rotation Shells Histogram EGI
Constant Order Crease Histograms Spherical: Constant Order D2 Square Shells Spherical Extent Cylindrical: Full Harmonic Descriptor Spherical: Full The second method used for addressing the alignment problem is to extract information from the shape descriptor that does not change when the model is transformed. The advantages of these types of approaches is that any pose of the same model will give rise to the same representation, so that it is not necessary to explicitly solve for the best alignment. In the past, there have been a number of different shape descriptors proposed that address the alignment problem by designing the shape descriptor so that it is invariant to some class of transformations. For example, the Extended Gaussian Image, which bins the surface normals of a model, does not use information about the location of points on the surface of the model, and is therefore invariant to translation. The Shells Histogram, which bins the distance of points from the center of mass, leverages the fact that the distance of a point from the center of mass remains unchanged if the model is rotated to design a rotation invariant representation. And the D2 distribution, which bins the distance between pairs of points uses the fact that the distance between two vectors is unchanged when the model is translated or rotated to design a shape representation that is simultaneously translation and rotation invariant. Rather than focusing on any particular implementation of invariance, we will describe a general framework for extracting transformation invariant information from a shape descriptor. The methods that we describe are sufficiently general to be applicable to a broad class of shape descriptors, and it turns out that most previous descriptors that are designed to be transformation invariant implicitly use some of these methods.

Normalization Place a model into a canonical coordinate frame by normalizing for: Translation Scale Rotation Translation Rotation As a result, there has been a third method proposed for addressing the alignment problem. In this approach, each model is placed into a canonical coordinate frame. The shape descriptor of the canonically aligned model is computed, and the assumption is made that if two models are both in their canonical alignment then they are optimally pair-wise aligned. In this approach we also do not need to try out all possible transformations as we already know what the best alignment is. Scale

Translation-Aligned Models
Normalization [Horn et al., 1988] Place a model into a canonical coordinate frame by normalizing for: Translation: Center of mass Scale Rotation Placing a model into a canonical coordinate frame proceeds in three steps. First the model is normalized for translation by moving the model so that its center of mass is at the origin. As shown in the work of Horn et al. when we are considering the task of whole object to whole object matching, this type of translation normalization guarantees that the two models are optimally translation aligned. Initial Models Translation-Aligned Models

Normalization [Horn et al., 1988] Place a model into a canonical coordinate frame by normalizing for: Translation Scale: Mean variance Rotation Next, the model is aligned for scale by rescaling the model so that its mean variance, or the average sum of square distances from the origin, is equal to one. As with translation alignment, Horn proves that in the case of whole object to whole object matching, if each model is scaled so that its mean variance is equal to one then the two models are guaranteed to be optimally scale aligned. Translation-Aligned Models Translation- and Scale-Aligned Models

Translation- and Scale-Aligned Models
Normalization Place a model into a canonical coordinate frame by normalizing for: Translation Scale Rotation: PCA alignment PCA Alignment In the final step, models are normalized for rotation by aligning each model so that its principal axes align with the x-, y- and z-axes. As apposed to the translation and scale normalizations, this type of approach is not guaranteed to give the optimal rotation alignment. However, since the principal axes of a model tend to be stable, using them to normalize for rotation can work well in practice. As is demonstrated by the example of the cow and the dog, we see that after applying these three normalization steps to each model independently, we get transformed models of the cow and dog that are near optimally aligned. Thus, we can now compare these models directly without having to search for the best alignment. Translation- and Scale-Aligned Models Fully Aligned Models

Directions of the axes are ambiguous
Rotation Properties: Translation and rotation normalization is guaranteed to give the best alignment Rotation normalization is ambiguous PCA Alignment While PCA can work well in practice, it is necessary to first resolve a fundamental ambiguity in the method. Because PCA alignment only uses the principal axes to align the model, it does not specify which direction of the axes to choose. As a result, there are generally eight different poses the model can be and still satisfy the condition that its principal axes are aligned with the x-, y-, and z-axes. For example, if we consider the case when we normalize the dog model, we find that the models on the right are in different poses, their covariance ellipsoids are exactly the same. As with the general class of transformations, this problem can be resolved in one of three ways: We can try to match at the eight different poses, we can try to normalize for the ambiguity by trying to consistently choose the direction of the axes, or we can make the shape descriptor invariant to this ambiguity. Directions of the axes are ambiguous

Translation Invariant Representation
Normalization (PCA) PCA defines a coordinate frame up to reflection in the coordinate axes. Make descriptor invariant to axial-reflections Reflections fix the cosine term Reflections multiply the sine term by -1 Because exhaustive search is expensive and because it is hard to find robust methods for normalizing for the axial direction, we focus instead on the invariance approach. In this normalization approach, we can use the fact that if we compute the Fourier decomposition of a one-dimensional function, reflections about the coordinate axis will not change the cosine terms but will multiply the sine terms by –1. So that by storing the cosine terms and the absolute values of the sine terms, we get a representation that is invariant to axial flips. This observation can be generalized directly to three-dimensions to give a representation of the shape descriptor that is guaranteed to be the same regardless of which directions of the principal axes are chosen. y Translation Invariant Representation x z

Retrieval Results (Rotation)
Size: Gaussian EDT Method Floats Exhaustive Search 8192 PCA + Flip Invariance PCA Cylindrical PS 4352 Spherical PS 512 Precision Time: As was mentioned earlier, translation and scale normalization are robust methods for addressing the alignment problem. However, since PCA alignment does not guarantee optimal pair-wise registration of two models, we would like to consider how the different approaches for resolving the rotation alignment problem compare. To test this we ran retrieval experiments on a database of classified models and evaluated how the different rotation alignment techniques compare to each other. The graph on the left shows the accuracy of the retrieval of several different methods: The plot in red, giving the best performance, gives the retrieval results when exhaustive search is used to find the best rotation. Since this approach explicitly solves for the optimal alignment, it is not surprising that it gives the best performance. However, even using fast signal processing techniques, we find that exhaustively searching for the best alignment takes too much time to be practical in settings where we want to match models at interactive rates. When models are normalized for rotation by aligning the principal axes of each model with the x-, y-, and z-axes, we only need to compare the models at a single alignment and the comparison time is substantially shorter. In this case we see that if we resolve the axial ambiguity by using an axial-flip-invariant representation we get results shown in yellow that are nearly comparable with the results obtaining using exhaustive search. However, if instead we use standard techniques for choosing the direction of the principal axes, the retrieval results are much worse and we get the white precision vs. recall plot. Finally, if we consider the single axis and full rotation invariant approaches, we find, as expected, that these approaches tend to discard too much information and as a result both the cylindrical power spectrum and spherical power spectrum representations give worse results. However, both these representations give rise to more compact shape descriptors which can be compared more efficiently. Thus, these methods are can still be used in retrieval applications where it is necessary to sacrifice a certain amount of discriminability in order to satisfy efficiency requirements. While the results shown in this slide only consider one specific type of shape descriptor, analogous experiments with other shape descriptors confirm these results. We should also point out that while these results indicate that PCA is effective in resolving the alignment problem for the test database, this may not be true for all databases. In particular, if the database consists of models that don’t have well defined principal axes, PCA will be less effective and rotation invariant representations may be more effective. Method Secs. Exhaustive Search 20.59 PCA + Flip Invariance .67 PCA Cylindrical PS .32 Spherical PS .03 Recall

Alignment Exhaustive search: Normalization: Invariance: Best results
Inefficient to match Normalization: Provably optimal for translation and scale Works well for rotation if models have well defined principal axes and the directional ambiguity is resolved Invariance: Compact Efficient Often less discriminating To summarize, we had discussed three different methods that for addressing the alignment problem in whole-object-to-whole-object matching. The most direct approach is to exhaustively search for the best alignment, though this approach tends to be too inefficient to be useful in practice. The second approach is to normalize for translation, scale and rotation by placing each model into a canonical coordinate frame. This type of approach is guaranteed to give the best translation and scale and we have seen that if the database models have well defined principal axes and the axial ambiguity of PCA-alignment can be resolved, then this approach also works well for rotation. Finally, while often less discriminating, we have seen that using transformations invariant representations can address the alignment problem, giving rise to shape descriptors that are more compact and can be compared more efficiently.

Normalization Part vs. Whole Conclusion While the methods that we have described can be used to match whole objects to whole objects, not all of them can be used in the case that we want to find the models containing a specific part.

Partial Shape Matching
Cannot use global normalization methods that depend on whole model information: Center of mass for translation Mean variance for scale Principal axes for rotation In particular, when we have only part of the model, the methods that we have described for normalization can no longer be used to solve for the optimal transformation alignment. Consider for the example the case when we would like to match the model of a cow to some subset of the model. If we were to shift both models so that their center of mass is at the origin the resulting models would not be optimally translation aligned. The reason for this is that these methods for transformation normalization use global shape information, making the assumption that we would like to match all of the points on one model to all of the points on the other. In the case that we are considering part-to-whole matching, we are often only interested in matching the part to some subset of the whole and this assumption is no longer satisfied. For example, when matching the whole cow with the front part of the cow, the points on the back of the cow are not matched to anything, and as a result should not play a role in determining the optimal translation. Normalized Whole Normalized Part (Mis-)Aligned Models

Partial Shape Matching
Cannot use global normalization methods that depend on whole model information: Exhaustively search for best alignment Normalize using local shape information Use transformation invariant representations Instead, methods for doing part-to-whole matching must resort to either using exhaustive search, locally-based alignment, or transformation invariance. Normalized Whole Normalized Part (Mis-)Aligned Models

Spin Images & Shape Contexts
Translation (Exhaustive Search): Represent each database model by many descriptors centered at different points on the surface. In practice, translation alignment is solved by using an exhaustive search approach: Rather than computing a single shape descriptor that is centered at the origin, partial shape matching techniques compute many different descriptors, centering at different points on the surface of the model. Model Multi-Centered Descriptors

Translation (Exhaustive Search): To match, center at a random point on the query and compare against the different descriptors of the target Query Part Randomly-Centered Descriptor Best Match Then, when a part of a model is presented as a query, a single shape descriptor is computed at some point on the surface of the query and matched against the collection of shape descriptors in the database. If the part is a subset of some whole model that is in the database, then there should be a shape descriptor for that database model that is centered at the same location as the shape descriptor from the query. If this is the case then these query will match the database descriptor and we will know both the matching model and the translation that aligns them. Target Descriptor

Rotation (Normalization): For each center, represent in cylindrical coordinates about the normal vector [Spin Images]: Store energy in each ring [Harmonic Shape Contexts]: Store power spectrum of each ring [3D Shape Contexts]: Search over all rotations about the normal for best match n n n In order to be able to match models, however, it is also necessary to address the rotational alignment problem. Traditional methods for part-to-whole matching use the surface normal to fix one direction of alignment and then use either rotation invariance or exhaustive search to address the other degree freedom. That is, for every shape descriptor center, these methods compute the surface normal and then represent the model in cylindrical coordinates about that normal. This gives rise to a collection of representations of the model in terms of radius, height, and normal angle. Then rotation invariant representations are obtained in one of two ways: In the Spin Image representation, the radius and height are fixed and the averages over the different normal angles is computed. Since the average remains the same when the model is rotated about the normal, this representation is invariant to normal rotation. In a signal processing framework, this is analagous to obtaining a rotation invariant representation by storing only the constant order term of the Fourier decomposition. Since the amplitudes of the higher order frequency terms are also rotation invariant, a more discriminating representation can be obtained by storing those amplitudes as well, which is the approach taken by the Harmonic Shape Contexts representation. And in the shape contexts representation, instead of making the descriptors invariant to rotation about the normal vector, rotation alignment can also be resolved by exhaustively searching over the space of rotations for the best alignment.

Image courtesy of Frome et al, 2003 Spin images and shape contexts allow for part-in-whole searches by exhaustively searching for translation and using the normal for rotation alignment [Spin Images]: Store energy in each ring [Harmonic Shape Contexts]: Store power spectrum of each ring [3D Shape Contexts]: Search over all rotations about the normal for best match When we look at the retrieval performance of these different methods we get the graph shown on the right, which gives the accuracy of retrieval as a function of the number of different centers used to resolve the translation alignment problem. As we would expect, using more centers makes it possible to solve the translation alignment problem more accurately, and as the plots indicate, the more centers are used to represent a model, the better the retrieval performance. We also see that since spin images use only the constant order component of the Fourier decomposition to achieve rotation invariance, they are less discriminating than the Harmonic Shape Contexts which use the amplitude information from all of the different frequencies. Finally, as we had seen before, the invariance approach tends suffer from too much information loss, and the 3D Shape Context, which explicitly finds the best rotation alignment, is better at retrieving models than either of the rotation-invariant approaches.

Conclusion Aligning Models: Partial Object Matching Exhaustive Search
Normalization Invariance Partial Object Matching Can’t use global normalization techniques Translation: Exhaustive Search Rotation: Normal + Exhaustive/Invariant In this part of the lecture, we have talked about how it is necessary to align models before matching them, and discussed three methods in which this can be done. Comparing the relative merits of exhaustive search, normalization and invariance. In considering the more difficult problem of partial shape matching, we have shown that standard alignment normalization methods can no longer be applied. Instead, we have demonstrated how existing methods get around this problem by using other alignment methods that only use local shape information in combination with exhaustive search and invariance methods to be able to perforrm partial shape matching across different translations and rotations.

Conclusion Shape Descriptors and Model Matching:
Decoupling representation from registration Can design and evaluate descriptors without having to solve the alignment problem Can develop methods for alignment without considering specific shape descriptors More generally, in this talk we have considered how shape descriptors can be used for designing systems for efficient and effective model retrieval. We have shown that while in the past, the questions of shape representation and shape alignment have been addressed simultaneously, these are in fact very different questions that can be decoupled. It is possible to focus on developing an effective shape descriptor without worrying about how to match across different transformations, And it’s possible to develop methods for matching models across different transformations without limiting oneself to a specific shape representation. Furthermore, in the past it was often difficult to compare different shape descriptors: For example, how would you compare the 3D Gaussian EDT descriptor that is normalized for translation, scale and rotation with the one-dimensional D2 distribution. It’s true that the Gaussian EDT gives better matching results, but it is also much larger. Perhaps if we reduced it down to a 1D representation we would get retrieval results that are worse than those of the D2 distribution. The ability to decouple representation from registration makes is possible to recognize that the D2 distribution gains its compactness and invariance through the use of general signal processing techniques. Applying these same techniques to the Gaussian EDT makes it possible to represent both shape descriptors as 1D translation and rotation invariant arrays, making the comparison of retrieval experiments more meaningful.

Alignment and Matching

Similar presentations

Presentation on theme: "Alignment and Matching"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Alignment and Matching

Similar presentations

Presentation on theme: "Alignment and Matching"— Presentation transcript:

Similar presentations

About project

Feedback