Projects:2017s1-100 Face Recognition using 3D Data - Revision history

A1670791 at 01:27, 30 October 2017

2017-10-30T01:27:37Z

A1670791 at 01:19, 30 October 2017

2017-10-30T01:19:39Z

A1670791 at 01:18, 30 October 2017

2017-10-30T01:18:36Z

A1670791 at 01:13, 30 October 2017

2017-10-30T01:13:09Z

A1670791 at 01:10, 30 October 2017

2017-10-30T01:10:55Z

A1667249: /* Facial recognition from 3D models */

2017-10-30T01:10:45Z

‎Facial recognition from 3D models

A1670791 at 01:09, 30 October 2017

2017-10-30T01:09:22Z

A1670791 at 01:08, 30 October 2017

2017-10-30T01:08:28Z

A1667249: /* Facial recognition from 3D models */

2017-10-30T01:03:03Z

‎Facial recognition from 3D models

A1670791 at 00:54, 30 October 2017

2017-10-30T00:54:40Z

← Older revision		Revision as of 01:27, 30 October 2017
Line 14:		Line 14:

	== 3D data from Polarimetry ==		== 3D data from Polarimetry ==
		+
		+	==== Method ====
		+
		+	The method used shall reflect the process that Gurton et al. used in their papers. Surface normals are calculated by applying Fresnel equations to the Kirchoff radiation law and then equating to degree of linear polarisation (DoLP). The surface normal are then integrated using the Frankot-Chellappa method to reconstruct the three-dimensional surface.
		+
		+	Firstly, known absorptivity and reflectivity as well as experimental directional spectral emissivity is input into the Kirchoff radiation law relationship pictured in Equation 1.
		+
		+	[[File:E1.png]]
		+
		+	Next, DoLP is calculated by applying Fresnel reflection coefficients to the DoLP formula in Equation 2 to get Equation 3.
		+
		+	[[File:E2.png]]
		+
		+	Where
		+
		+	[[File:E4.png]]
		+
		+	This is then equated to the experimentally-found values for the DoLP calculated from the Stokes images using Equation 6. The equated function can then be numerically solved by a range of equivalent methods but for computational speed and ease of use, Matlab’s fzero function is implemented. Also important is to take the modulo of theta with pi to ensure correct phase.
		+
		+	[[File:E6.png]]
		+
		+	The formula for the normal of a surface is given by Equation 7.
		+
		+	[[File:E7.png]]
		+
		+	From basic trigonometry it is then implied that for this application the partial derivatives are as shown in Equations 7.1 and 7.2.
		+
		+	[[File:E71.png]]
		+
		+	These partial derivatives show that the magnitude of the normal is given by the tangent of theta, or how close theta is to parallel to the camera’s line of sight and the direction is provided by the real and imaginary components of the azimuthal angle.
		+
		+	Phi is calculated as according to the conditions used in Equation 8. These conditions exist to correct the phase, which is falsely aligned, so that slopes are found in both directions. Again, like the calculations for theta, the modulo of phi with π must be taken to ensure correct values. The limitation of this calculation, is that phi values are only calculated in the range 0 to π whereas, phi, being any direction with a circle, should be of the range 0 to 2 π, meaning that some of the angles are π lower than they should be. There is no equation to calculate with certainty whether or not an angle phi should have π added to it, instead previous papers[13] have used the generally convex nature of the face to apply known values at the edges and then, assuming a relatively smooth surface, iterate inwards adjusting the nearby values if they are outside a threshold when compared to the corrected edge values.
		+
		+	[[File:E8.png]]
		+
		+	The final step is to integrate the normals using the Frankot-Chellapa method used by Gurton et al. for the reasons of ease and smoothing[13]. For this paper the open source Frankot-Chellappa Matlab code provided by Peter Kovesi[14] as shown in appendix a was used.
		+

	==== Initial Using US A.R.L. Database ====		==== Initial Using US A.R.L. Database ====

@@ Line 22: / Line 22: @@
 As expected, the theta values appear to be highest where the facets of the face are most oblique to the camera whereas flat portions have values close to zero.
 Phi values appear rather differently with parts of the face that are changing predominantly in the x axis being blue and those changing predominantly in the y axis being yellow.
 The x components of the derivative appear mostly as expected.  Note the lack of variation in the y axis as the image is cropped such that the top and bottom of the head, where there most slope is in that axis, are not present.  Also note that the values are inverted and there are outliers in the visibly low parts of Fig 7 but mathematically high parts.
 The y components of the derivative appear mostly as expected.  Note, again, the lack of variation in the y axis as the image is cropped such that the top and bottom of the head, where there most slope is in that axis, are not present.  The values are also inverted and there are outliers in the visibly low but mathematically high parts however they are notably less significant than in the x component.
 The figure below shows the integral calculated.  The face is not visible, instead there are rows and columns of spikes, a peculiar effect of the smoothing effect of the Frankot-Chellappa method.  This may be fixed by a combination of changes such as cropping the faces before integrating and doing pre-smoothing of dx and dy to get more consistent magnitudes to prevent smoothing of spikes in dx and dy propagating incorrectly and serving only to raise the magnitude of the column and row they are in.
@@ Line 55: / Line 42: @@
 As can be seen in the figure above, padding the boundary with zeros fixed the problem of linear surfaces not being constructed.  This works because adding the second plane now forces a single Fourier representation of the shape.
 Another issue identified with Frankot-Chellappa reconstructions was that large amplitudes towards the corners had a tendency to be mirrored in other corners, albeit with slightly lower amplitude.  Again, padding with zeros solved this.
 In fact, all surfaces were reconstructed in near-perfection once the zero padding was added, with only minor errors noticeable at the discontinuities between shapes and the zero padding plane.  The algorithm was even able to reconstruct relatively complex surfaces such as the two variable sinusoidal peak pattern shown below.
@@ Line 73: / Line 58: @@
 The theta values shown look like those calculated previously but have considerable noise, which should not be present for a relatively smooth human face so median filtering the values was tested, the result of which is illustrated below.
 [[File:Theta.png]]
 With reasonable values of theta calculated, phi could be found from them.  Phi values also looked acceptable, other than a significant outlier in the middle of Subject A’s face which was solved by median filtering as used to control noise in theta values.  The filtered values of phi are provided below.
 [[File:Phi.png]]
 From theta and phi, the partial derivatives were calculated and the integral done again, the results of which are shown below.
@@ Line 97: / Line 70: @@
 As is visible in above, a face shape was becoming visible, a major improvement from the spiky grid shown at the start that had resulted previously.  The partial reconstruction with a portion ‘missing’ or smoothed out immediately suggests a frequency or phase having been calculated incorrectly.  The spikes in the previous reconstruction are thought to have been due to theta and phi being of inverted forms what they should have been and the modulo bounding the magnitudes causing significant spike components.
- Working backwards in the analysis of where the code had gone wrong it was noticed that the partial derivatives were symmetric as they should not have been for a convex shape.  The partial derivative of a cylinder, a rough analogy for a face, in x and y in respect to one variable should be positive on one side and negative on the other.  Another concern was that downsampling may have been adding discontinuities affecting the frequency representation of the surface and so, to minimise this, images were used at their native resolution though cropped tighter to the facial area for both computational speed and ease of analysis.  Without this the drop to a flat background surrounding the faces in the databases’ images exhibited some bending and undulation that should not have been present.
+Working backwards in the analysis of where the code had gone wrong it was noticed that the partial derivatives were symmetric as they should not have been for a convex shape.  The partial derivative of a cylinder, a rough analogy for a face, in x and y in respect to one variable should be positive on one side and negative on the other.  Another concern was that downsampling may have been adding discontinuities affecting the frequency representation of the surface and so, to minimise this, images were used at their native resolution though cropped tighter to the facial area for both computational speed and ease of analysis.  Without this the drop to a flat background surrounding the faces in the databases’ images exhibited some bending and undulation that should not have been present.
 The angle phi determines what direction the derivative goes in and as such it was determined that the error must have come from there.  After going back to existing papers on the topic it was realised that the calculation for phi has an ambiguity of exactly pi.  The range of phi is from 0 to 2π however the calculation to get phi only provides answers from 0 to π.  It is therefore necessary to correct some values by applying known boundary conditions, e.g. that the right side of the face should have vectors pointing towards the right and thus be close to 0 to 2 π.  Similarly, the top should have vectors pointing upwards towards π/2, the left towards π and the bottom towards 3π/2.  The method used by Yuffa et al. was to apply these known conditions to the boundaries of the data and then correct nearby values to whichever of the phi or phi + π was closest to the known value next to it.  This is continued from the edges inwards towards the centre until all values have been corrected.
@@ Line 103: / Line 76: @@
 An attempt to implement a similar algorithm to correct phi was attempted but, due to time constraints, not completed.  Estimates of the correct partial derivatives were calculated by simply reversing the sign of half of each derivative, in the y axis for dx values and in the x axis for dy values, and then shifting them so they meet the original values.  This resulted in the far more accurate and detailed face reconstruction shown in above.
 This face is another large improvement on the previous reconstruction though still not ideal.  In particular, the noticeable effects of a discontinuity can be seen from the top to the bottom down the middle of the face.  It was hoped that this discontinuity would be improved by smoothing the partial derivatives however the integration is not significantly improved.  This is likely due to the filter window being too small to cover the indentation at the discontinuity.  While a larger windowed filter may be able to remove this valley, excessive use of post-processing would only serve to remove discriminative features present in the faces required for facial recognition applications.

@@ Line 41: / Line 41: @@
-Figures 9 and 10 show the integral calculated.  The face is not visible, instead there are rows and columns of spikes, a peculiar effect of the smoothing effect of the Frankot-Chellappa method.  This may be fixed by a combination of changes such as cropping the faces before integrating and doing pre-smoothing of dx and dy to get more consistent magnitudes to prevent smoothing of spikes in dx and dy propagating incorrectly and serving only to raise the magnitude of the column and row they are in.
+The figure below shows the integral calculated.  The face is not visible, instead there are rows and columns of spikes, a peculiar effect of the smoothing effect of the Frankot-Chellappa method.  This may be fixed by a combination of changes such as cropping the faces before integrating and doing pre-smoothing of dx and dy to get more consistent magnitudes to prevent smoothing of spikes in dx and dy propagating incorrectly and serving only to raise the magnitude of the column and row they are in.
 [[File:Old Result.png]]
@@ Line 54: / Line 54: @@
 [[File:Linear Fixed.png]]
-As can be seen in Figure 12, padding the boundary with zeros fixed the problem of linear surfaces not being constructed.  This works because adding the second plane now forces a single Fourier representation of the shape.
+As can be seen in the figure above, padding the boundary with zeros fixed the problem of linear surfaces not being constructed.  This works because adding the second plane now forces a single Fourier representation of the shape.
-Another issue identified with Frankot-Chellappa reconstructions is illustrated in Figure 13.  Large amplitudes towards the corners had a tendency to be mirrored in other corners, albeit with slightly lower amplitude.  Again, padding with zeros solved this as shown in Figure 14.
+Another issue identified with Frankot-Chellappa reconstructions was that large amplitudes towards the corners had a tendency to be mirrored in other corners, albeit with slightly lower amplitude.  Again, padding with zeros solved this.
-In fact, all surfaces were reconstructed in near-perfection once the zero padding was added, with only minor errors noticeable at the discontinuities between shapes and the zero padding plane.  The algorithm was even able to reconstruct relatively complex surfaces such as the two variable sinusoidal peak pattern shown in Figure 14.
+In fact, all surfaces were reconstructed in near-perfection once the zero padding was added, with only minor errors noticeable at the discontinuities between shapes and the zero padding plane.  The algorithm was even able to reconstruct relatively complex surfaces such as the two variable sinusoidal peak pattern shown below.
 [[File:Complex.png]]
@@ Line 71: / Line 71: @@
-The theta values shown above look like those calculated previously in Figure 3 but have considerable noise, which should not be present for a relatively smooth human face so median filtering the values was tested, the result of which is illustrated below in Figure 16.
+The theta values shown look like those calculated previously but have considerable noise, which should not be present for a relatively smooth human face so median filtering the values was tested, the result of which is illustrated below.
@@ Line 81: / Line 81: @@
-With reasonable values of theta calculated, phi could be found from them.  Phi values also looked acceptable, other than a significant outlier in the middle of Subject A’s face which was solved by median filtering as used to control noise in theta values.  The initial and filtered values of phi are provided below in Figures 17 and 18.
+With reasonable values of theta calculated, phi could be found from them.  Phi values also looked acceptable, other than a significant outlier in the middle of Subject A’s face which was solved by median filtering as used to control noise in theta values.  The filtered values of phi are provided below.
 [[File:Phi.png]]
@@ Line 91: / Line 91: @@
-From theta and phi, the partial derivatives were calculated and the integral done again, the results of which are shown in Figure 19.
+From theta and phi, the partial derivatives were calculated and the integral done again, the results of which are shown below.
 [[File:Result.png]]
-As is visible in Figure 19, a face shape was becoming visible, a major improvement from the spiky grid shown in Figures 9 and 10 that had resulted previously.  The partial reconstruction with a portion ‘missing’ or smoothed out immediately suggests a frequency or phase having been calculated incorrectly.  The spikes in the previous reconstruction are thought to have been due to theta and phi being of inverted forms what they should have been and the modulo bounding the magnitudes causing significant spike components.
+As is visible in above, a face shape was becoming visible, a major improvement from the spiky grid shown at the start that had resulted previously.  The partial reconstruction with a portion ‘missing’ or smoothed out immediately suggests a frequency or phase having been calculated incorrectly.  The spikes in the previous reconstruction are thought to have been due to theta and phi being of inverted forms what they should have been and the modulo bounding the magnitudes causing significant spike components.
   Working backwards in the analysis of where the code had gone wrong it was noticed that the partial derivatives were symmetric as they should not have been for a convex shape.  The partial derivative of a cylinder, a rough analogy for a face, in x and y in respect to one variable should be positive on one side and negative on the other.  Another concern was that downsampling may have been adding discontinuities affecting the frequency representation of the surface and so, to minimise this, images were used at their native resolution though cropped tighter to the facial area for both computational speed and ease of analysis.  Without this the drop to a flat background surrounding the faces in the databases’ images exhibited some bending and undulation that should not have been present.
@@ Line 101: / Line 101: @@
 The angle phi determines what direction the derivative goes in and as such it was determined that the error must have come from there.  After going back to existing papers on the topic it was realised that the calculation for phi has an ambiguity of exactly pi.  The range of phi is from 0 to 2π however the calculation to get phi only provides answers from 0 to π.  It is therefore necessary to correct some values by applying known boundary conditions, e.g. that the right side of the face should have vectors pointing towards the right and thus be close to 0 to 2 π.  Similarly, the top should have vectors pointing upwards towards π/2, the left towards π and the bottom towards 3π/2.  The method used by Yuffa et al. was to apply these known conditions to the boundaries of the data and then correct nearby values to whichever of the phi or phi + π was closest to the known value next to it.  This is continued from the edges inwards towards the centre until all values have been corrected.
-An attempt to implement a similar algorithm to correct phi was attempted but, due to time constraints, not completed.  Estimates of the correct partial derivatives were calculated by simply reversing the sign of half of each derivative, in the y axis for dx values and in the x axis for dy values, and then shifting them so they meet the original values.  This resulted in the far more accurate and detailed face reconstruction shown in Figure 20.
+An attempt to implement a similar algorithm to correct phi was attempted but, due to time constraints, not completed.  Estimates of the correct partial derivatives were calculated by simply reversing the sign of half of each derivative, in the y axis for dx values and in the x axis for dy values, and then shifting them so they meet the original values.  This resulted in the far more accurate and detailed face reconstruction shown in above.
-This face is another large improvement on the previous reconstruction though still not ideal.  In particular, the noticeable effects of a discontinuity can be seen from the top to the bottom down the middle of the face.  It was hoped that this discontinuity would be improved by smoothing the partial derivatives however the integration of that, shown in Figure 21, is not significantly improved.  This is likely due to the filter window being too small to cover the indentation at the discontinuity.  While a larger windowed filter may be able to remove this valley, excessive use of post-processing would only serve to remove discriminative features present in the faces required for facial recognition applications.
+This face is another large improvement on the previous reconstruction though still not ideal.  In particular, the noticeable effects of a discontinuity can be seen from the top to the bottom down the middle of the face.  It was hoped that this discontinuity would be improved by smoothing the partial derivatives however the integration is not significantly improved.  This is likely due to the filter window being too small to cover the indentation at the discontinuity.  While a larger windowed filter may be able to remove this valley, excessive use of post-processing would only serve to remove discriminative features present in the faces required for facial recognition applications.
 [[File:Stokes.png]]
 == 3D data from Xbox Kinect and Pre-processing ==

@@ Line 43: / Line 43: @@
 Figures 9 and 10 show the integral calculated.  The face is not visible, instead there are rows and columns of spikes, a peculiar effect of the smoothing effect of the Frankot-Chellappa method.  This may be fixed by a combination of changes such as cropping the faces before integrating and doing pre-smoothing of dx and dy to get more consistent magnitudes to prevent smoothing of spikes in dx and dy propagating incorrectly and serving only to raise the magnitude of the column and row they are in.
-   [[File:Old Result.png]]
+[[File:Old Result.png]]
 ==== Frankot-Chellappa Verification ====
 As previous Frankot-Chellappa integral results had provided unexpected outputs and investigation was done to test and verify that Peter Kovesi’s code functioned correctly.  This involved the construction of a range of three dimensional surfaces, calculation of their partial derivatives and putting them through the Frankot-Chellappa function.  These worked almost immediately, confirming the accuracy of the function though some experimentation was required to sort out some minor issues.
+[[File:Linear.png]]
 Shown above is the first shape tested with the Frankot-Chellappa function.  This shows one major limitation of the Frankot-Chellappa method, that it does not work well for surfaces with large linear components due to the lack of differences in Fourier representation.
-−
 As can be seen in Figure 12, padding the boundary with zeros fixed the problem of linear surfaces not being constructed.  This works because adding the second plane now forces a single Fourier representation of the shape.
@@ Line 62: / Line 62: @@
 In fact, all surfaces were reconstructed in near-perfection once the zero padding was added, with only minor errors noticeable at the discontinuities between shapes and the zero padding plane.  The algorithm was even able to reconstruct relatively complex surfaces such as the two variable sinusoidal peak pattern shown in Figure 14.
+[[File:Complex.png]]
@@ Line 74: / Line 74: @@
-   [[File:Theta.png]]
+[[File:Theta.png]]
@@ Line 83: / Line 83: @@
 With reasonable values of theta calculated, phi could be found from them.  Phi values also looked acceptable, other than a significant outlier in the middle of Subject A’s face which was solved by median filtering as used to control noise in theta values.  The initial and filtered values of phi are provided below in Figures 17 and 18.
-  [[File:Phi.png]]
+[[File:Phi.png]]
@@ Line 93: / Line 93: @@
 From theta and phi, the partial derivatives were calculated and the integral done again, the results of which are shown in Figure 19.
- [[File:Result.png]]
+[[File:Result.png]]
 As is visible in Figure 19, a face shape was becoming visible, a major improvement from the spiky grid shown in Figures 9 and 10 that had resulted previously.  The partial reconstruction with a portion ‘missing’ or smoothed out immediately suggests a frequency or phase having been calculated incorrectly.  The spikes in the previous reconstruction are thought to have been due to theta and phi being of inverted forms what they should have been and the modulo bounding the magnitudes causing significant spike components.
@@ Line 107: / Line 107: @@
 This face is another large improvement on the previous reconstruction though still not ideal.  In particular, the noticeable effects of a discontinuity can be seen from the top to the bottom down the middle of the face.  It was hoped that this discontinuity would be improved by smoothing the partial derivatives however the integration of that, shown in Figure 21, is not significantly improved.  This is likely due to the filter window being too small to cover the indentation at the discontinuity.  While a larger windowed filter may be able to remove this valley, excessive use of post-processing would only serve to remove discriminative features present in the faces required for facial recognition applications.
- [[File:Stokes.png]]
+[[File:Stokes.png]]

@@ Line 15: / Line 15: @@
 == 3D data from Polarimetry ==
- Initial Using US A.R.L. Database
+==== Initial Using US A.R.L. Database ====
 The results portrayed in this section are downsampled for purposes of illustration from original 250x200 images in the database to 50x40 arrays.  All images are of Subject A in the ARL database and oriented such that the forehead is at the top of the image and chin at the bottom.
+[[File:USARL.png]]
 As expected, the theta values appear to be highest where the facets of the face are most oblique to the camera whereas flat portions have values close to zero.
@@ Line 45: / Line 45: @@
    [[File:Old Result.png]]
-Frankot-Chellappa Verification
+==== Frankot-Chellappa Verification ====
 As previous Frankot-Chellappa integral results had provided unexpected outputs and investigation was done to test and verify that Peter Kovesi’s code functioned correctly.  This involved the construction of a range of three dimensional surfaces, calculation of their partial derivatives and putting them through the Frankot-Chellappa function.  These worked almost immediately, confirming the accuracy of the function though some experimentation was required to sort out some minor issues.
@@ Line 65: / Line 65: @@
-Re-application to US A.R.L. Database
+==== Re-application to US A.R.L. Database ====
 As tests with synthetic surfaces had proven successful with the Frankot-Chellappa code, it was re-integrated back into the program for reconstructing the US A.R.L. images.  At this point integrals started being calculated correctly, suggesting that there had been a simple error made somewhere in the code implementing Peter Kovesi’s function.  Despite the integrals now looking as though they were plausible, reconstructions of the provided information, the surfaces were not accurately depicting the subjects’ faces.  This demonstrated errors in the calculation of earlier components, namely the angles theta and phi.
@@ Line 107: / Line 107: @@
 This face is another large improvement on the previous reconstruction though still not ideal.  In particular, the noticeable effects of a discontinuity can be seen from the top to the bottom down the middle of the face.  It was hoped that this discontinuity would be improved by smoothing the partial derivatives however the integration of that, shown in Figure 21, is not significantly improved.  This is likely due to the filter window being too small to cover the indentation at the discontinuity.  While a larger windowed filter may be able to remove this valley, excessive use of post-processing would only serve to remove discriminative features present in the faces required for facial recognition applications.
-  [[File:Stokes.jpg]]
+  [[File:Stokes.png]]

@@ Line 193: / Line 193: @@
 ==== l1-minimisation ====
-Minimising the l2-norm is certainly feasible, however the solution is not sparse and therefore does not identify which of the training samples belong to the class of the test sample. As demonstrated in Equation (\ref{eq:x0}), $x_0$ should be sparse if there are sufficient training subjects, as the only non-zero coefficients in $x_0$ should be those that correspond to the class of the test sample. Therefore by seeking the sparsest possible solution to the linear set of equations, it should be possible to easily determine the class that the test sample belongs to. The sparsest possible solution can be found by minimising the $\ell^0$-norm:
+Minimising the l2-norm is certainly feasible, however the solution is not sparse and therefore does not identify which of the training samples belong to the class of the test sample. As demonstrated, x0  should be sparse if there are sufficient training subjects, as the only non-zero coefficients in x0 should be those that correspond to the class of the test sample. Therefore by seeking the sparsest possible solution to the linear set of equations, it should be possible to easily determine the class that the test sample belongs to. The sparsest possible solution can be found by minimising the l0-norm.
 === Results ===
-[[File:L1 differences2.PNG|thumbnail|upright=4]]
 [[File:Estimated sparse error.PNG|thumbnail|upright=4]]
 [[File:Overall block diagram 3dfacerecognition.PNG|thumbnail|upright=4]]

@@ Line 180: / Line 180: @@
 === Method ===
 A dictionary is built for each subject which is made up of that subject's training samples. These dictionaries can be utilised to identify a test sample by exploiting the assumption, that for each subject, these dictionaries will lie on a linear subspace in order to perform classification.
 === Results ===

@@ Line 13: / Line 13: @@
 * Facial recognition from 3D models
-= Project Team =
+== 3D data from Polarimetry ==
-Jesse Willsmore
-= Supervisors =
+From theta and phi, the partial derivatives were calculated and the integral done again, the results of which are shown in Figure 19.
-Dr Brian Ng
+ Working backwards in the analysis of where the code had gone wrong it was noticed that the partial derivatives were symmetric as they should not have been for a convex shape.  The partial derivative of a cylinder, a rough analogy for a face, in x and y in respect to one variable should be positive on one side and negative on the other.  Another concern was that downsampling may have been adding discontinuities affecting the frequency representation of the surface and so, to minimise this, images were used at their native resolution though cropped tighter to the facial area for both computational speed and ease of analysis.  Without this the drop to a flat background surrounding the faces in the databases’ images exhibited some bending and undulation that should not have been present.
-Dr David Booth (DST Group)
+The angle phi determines what direction the derivative goes in and as such it was determined that the error must have come from there.  After going back to existing papers on the topic it was realised that the calculation for phi has an ambiguity of exactly pi.  The range of phi is from 0 to 2π however the calculation to get phi only provides answers from 0 to π.  It is therefore necessary to correct some values by applying known boundary conditions, e.g. that the right side of the face should have vectors pointing towards the right and thus be close to 0 to 2 π.  Similarly, the top should have vectors pointing upwards towards π/2, the left towards π and the bottom towards 3π/2.  The method used by Yuffa et al. was to apply these known conditions to the boundaries of the data and then correct nearby values to whichever of the phi or phi + π was closest to the known value next to it.  This is continued from the edges inwards towards the centre until all values have been corrected.
-Sau-Yee Yiu (DST Group)
+An attempt to implement a similar algorithm to correct phi was attempted but, due to time constraints, not completed.  Estimates of the correct partial derivatives were calculated by simply reversing the sign of half of each derivative, in the y axis for dx values and in the x axis for dy values, and then shifting them so they meet the original values.  This resulted in the far more accurate and detailed face reconstruction shown in Figure 20.
-Philip Stephenson (DST Group)
 == 3D data from Xbox Kinect and Pre-processing ==
@@ Line 111: / Line 186: @@
 [[File:Estimated sparse error.PNG|thumbnail|upright=4]]
 [[File:Overall block diagram 3dfacerecognition.PNG|thumbnail|upright=4]]