GB2464453A

GB2464453A - Determining Surface Normals from Three Images

Info

Publication number: GB2464453A
Application number: GB0818664A
Authority: GB
Inventors: Carlos Hermandez; George Vogiazis; Roberto Cipolla
Original assignee: Toshiba Research Europe Ltd
Current assignee: Toshiba Europe Ltd
Priority date: 2008-10-10
Filing date: 2008-10-10
Publication date: 2010-04-21
Anticipated expiration: 2028-10-10
Also published as: WO2010041584A1; GB2464453A8; JP5586594B2; US20110292179A1; US8451322B2; GB2464453B; GB2464453B8; GB0818664D0; JP2012505368A

Abstract

Data is acquired of three images of an object, each image taken under illumination from a different direction S101, it is determined which pixels are in shadow such that data is only available from two images S103 and a range of possible solutions for the gradient of the surface normal of the pixel in shadow are calculated from the two images S105, then a solution is selected using the integrability of the gradient field over an area of the object as a constraint, and minimising a cost function S107. Thus photometric stereo may be used with just three images. The solution may be selected by using a least squares analysis and the constraint that the object is a continuous piecewise polynomial. The cost function may be expressed in terms of a regulisation parameter which is used to compensate for noise. Three spaced light sources may be used, a camera capturing images in sequence as each of the light sources are activated, or each light source may provide radiation of a different frequency or colour, the three images being captured simultaneously, allowing moving objects to be imaged. A three dimensional (3D) image may be generated S109.

Description

An Imaging System and Method The present invention is concerned with the field of imaging systems which may be used to collect and display data for production of 3D images. The present invention may also be used to generate data for 2D and 3D animation of complex objects.

The field of 3D image production has largely been hampered by the time which it takes to take the data to produce a 3D film. Previously, 3D films have generally been perceived as a novelty as opposed to a serious recording format. Now, 3D image generation is seen as being an important tool in the production of CG images.

Photometric stereo is a well established 3D reconstruction technique. A sequence of images (typically three or more) of a 3D scene are obtained from the same viewpoint and under illumination from different directions. From the intensity variation in each pixel it is possible to estimate the local orientation of the surface of the object that projects onto that pixel.

By integrating all these surface orientations a very detailed estimate of the surface geometry can be obtained. As any other reconstruction method, photometric stereo faces several difficulties when faced with real images. One of the most important of these difficulties is the frequent presence of shadows in an image. No matter how careful the arrangement of the light sources, shadows are an almost unavoidable phenomenon, especially in objects with complex geometries.

Previously, shadows have been dealt with using four or more images from four different illumination directions. This over-determines the local surface orientation and albedo which only requires three degrees of freedom. This implies that it is possible to use the residual of a least squares solution, to determine whether shadowing has occurred.

However when there are only three 2D images there are no spare constraints. Therefore the problem of detecting shadows becomes more difficult. Furthermore, when a pixel is in shadow in one of the three images most methods simply discard it.

The present invention seeks to address the above problems and in a first aspect provides a method for determining the gradients of the surface normals of an object, the method comprising: receiving data of three 2D images of said object, wherein each image is taken under illumination from a different direction; establishing which pixels of the image are in shadow such that there is only data available from two images from these pixels; determining a range of possible solutions for the gradient of the surface normal of a shadowed pixel using the data available for the two images; and selecting a solution for the gradient at a pixel using the integrability of the gradient field over an area of the object as a constraint and minimising a cost function.

The above estimates the surface geometry of a shadowed part of an object by enforcing

the integrability of the gradient field.

In a preferred embodiment, the method of selecting a solution for the gradient comprises using the constraint that the surface of the object is a continuous piecewise polynomial.

The cost function may be minimised as a least squares problem. The cost function may represent the difference between the real value for each gradient and the potential solutions for said gradient over said area. The cost function may also be configured to take into account noise.

The area of the object over which the solution is calculated will generally comprise both shadowed and non-shadowed pixels. In a further preferred embodiment, it will be assumed that there will be noise in the data for both shadowed and non-shadowed pixels.

Noise can cause unwanted artefacts in a generated three dimensional image. Therefore, in preferred embodiment, a regularisation constraint is applied to compensate for noise in the acquired data.

In an embodiment, to compensate for noise the cost function is expressed in terms of a regularisation parameter which is used to compensate for noise. A cost is then imposed on the regularisation parameter such that a pixel is forced to have a value which is close to the midpoint of its neighbours. The cost may be imposed in terms of the first order differential or higher order differential of the regularisation parameter. The regularisation parameter may be related to the length along the said straight line of possible solutions from a fixed point on said line.

The problem may be formulated by expressing the surface normals in terms of their gradients in a 2D gradient space such that the range of possible solutions lie along a straight line in said 2D gradient space. The constraint for the least squares analysis may then be expressed in terms of a regularisation parameter, the variation in the regularisation parameter between pixels to be less than a predetermined value and wherein said regularisation parameter is the length along the said straight line of possible solutions from a fixed point on said line.

For example, the range of possible solutions may be determined by expressing the normal as the albedo scaled surface normal b: b = Ii'c wherein c is [ci c2 C3] and c1, C2 and C3 represent the respective intensity of the pixel measured in each of the three 2D images and L=[11 12 13 T where Ii, 12 and 13 are the vectors which represent the direction of the light source which illuminates the object for each of the three 2D images; wherein when the pixel in the 1" 2D image is in shadow b is expressed as: b = LDc + pLe1 where e, is the th column of the 3x3 identity matrix, D is the identity matrix with a zero in the 1th position in the diagonal and p is a scalar parameter representing the value that c would have if it were not in shadow; projecting b into a 2D gradient space using the operator P[xi x2 x3](xi/x3, x2/x3)(p, q) such that for a point in shadow, the point (p, q) lies along the shadow line of the pixel defined as the line that joins P[L'D1c] and P[U'e].

The above method may further comprise minirnising the cost function is performed using a parameter w, where w is distance along the shadow line of point (p,q) from a fixed point on the shadow line, the method further comprising imposing a cost on Vw.

The step of determining if a pixel is in shadow in jth of said three images may be achieved by determining if the value of the intensity in the jth image divided by the magnitude of the intensity vector c for all three images is below a predetermined value.

The above method may further comprise establishing whether a pixel is in shadow is by introducing a further constraint to reduce the chance of a pixel being determined to be in shadow and a neighbouring pixel not in shadow.

The above method can be used when an object is imaged from just three directions.

Using photometric stereo on just three images may seem like an unreasonably hard restriction. There is however a particular situation when only three images are available.

This technique is known as colour photometric stereo and it uses three light sources with different light spectra. When the scene is photographed with a colour camera, the three colour channels capture three different photometric stereo images.

Since shape acquisition is performed on each frame independently, the method can be used on video sequences without having to change illumination between frames. In this it is possible to capture the 3D shape of deforming objects such as cloth, or human faces.

The method may further comprise acquiring the image data from the object to be imaged or it may operate on pre-recorded or remotely recorded data.

In a second aspect, the present invention provides an apparatus for determining the gradients of the surface normals of an object, the method comprising: means to receive data of three 2D images of said object, wherein each image is taken under illumination from a different direction; means to establish which pixels of the image are in shadow such that there is only data available from two images from these pixels; means to determine a range of possible solutions for the gradient of the surface normal of a shadowed pixel using the data available for the two images; and means to select a solution for the gradient using the integrability of the gradient field over an area of the object as a constraint and minimising a cost function.

The means may be a processor adapted to perform the above functions. Although the means receive data from 3 images, it may be configured to receive data from 4 or more images. However, the above method and apparatus is primarily intended for determining the gradient of the surface normal of a pixel when it is shadowed such that data is only available for that pixel from 2 images.

The apparatus may further comprise means to acquire the data of the three 2D images.

Such apparatus may be a colour photometric stereo apparatus, which further comprises: at least three light sources, irradiating the object from three different angles; a camera provided to collect radiation from said three radiation sources which has been reflected from said object; and an image processor configured to generate a depth map of the three dimensional object, wherein each radiation source emits radiation at a different frequency and said image processor is configured to distinguish between the reflected signal from the three different radiation sources.

However, the present invention may also be applied to standard photometric stereo techniques where multiple images are taken using light of the same frequency and the images from different illumination directions are captured in a sequential manner.

Although the apparatus can stand alone, it may be incorporated in part of a 3D image generation apparatus further comprising means for displaying a three dimensional moving image from said depth map.

As previously explained, the object can be moving and said camera may be a video camera.

The system may also be used in 2D or 3D animation where the system comprises means for moving a depth map generated from said determined gradients.

The system may also further comprise means for applying pattern to the depth map, the means configured to form a 3D template of the object from a frame of the depth map and determine the position of the pattern on said object of said frame and to deform said template with said pattern to match subsequent frames. The template may be deformed using a constraint that the deformations of the template must be compatible with the frame to frame optical flow of the original captured data. Preferably the template is deformed using the further constraint that the deformations be as rigid as the data will allow.

In summary, the present invention can be used for 3D reconstruction, medical imaging or cloth modelling.

The present invention will now be described with reference to the following preferred non-limiting embodiments in which: Figure 1 is a schematic of an apparatus configured to perform photometric stereo; Figure 2 is a schematic of a system in accordance with an embodiment of the present invention configured to perform photometric stereo; Figure 3 is a flow diagram illustrating a method in accordance with an embodiment of the present invention; Figure 4 is a plot showing a 2D gradient space which is used to describe a method in accordance with an embodiment of the present invention; Figure 5a is a schematic of a sphere having shaded regions which is to be imaged, figure 5b shows the sphere of figure 5a when imaged using a technique in accordance with an embodiment of the present invention; and figure 5c shows the sphere of figure 5a which imaged using a technique in accordance with a further embodiment of the present invention using a regularisation constraint; Figure 6a is a 2D image of a face and figure 6b is a shadow mask of the face of figure 6a produced in accordance with an embodiment of the present invention; Figure 7 is a schematic of a calibration board used in a method in accordance with an embodiment of the present invention; Figure 8a shows three 2D images of a frog obtained from different illumination directions, figure 8b shows the three images of figure 8a normalised in terms of their intensity; figure 8c shows the result of applying a technique to determine the position of shadows in accordance with an embodiment of the present invention; figure 8d shows a shadow mask determined from the data of figure 8c and figure 8e shows a 3D reconstruction of the frog of figure 8a performed in accordance with an embodiment of the present invention; and Figure 9a shows three 3D images generated ignoring shadows and figure 9b shows the corresponding results for images generated in accordance with an embodiment of the present invention.

Figure 1 is a schematic of a system in accordance with an embodiment of the present invention used to image object 1. The object is illuminated by three different light sources 3, 5 and 7.

In this embodiment, the system is either provided indoors or outside in the dark to minimise background radiation affecting the data. The three lights 3, 5 and 7 are arranged laterally around the object I and are vertically positioned at levels between floor level to the height of the object 1. The lights are directed towards the object 1.

The angular separation between the three light sources 3, 5 and 7 is approximately 30 degrees in the plane of rotation about the object 1. Greater angular separation can make orientation dependent colour changes more apparent. However, if the light sources are too far apart, concave shapes in the object 1 are more difficult to distinguish since shadows cast by such shapes will extend over larger portions of the object making data analysis more difficult. In a preferred arrangement each part of the object I is illuminated by all three light sources 3, 5 and 7.

Camera 9 which is positioned vertically below second light source 5 is used to record the object as it moves while being illuminated by the three lights 3, 5 and 7.

Figure 2 shows an overview of the whole system. An object which may be moving is illuminated by the three light source 3,5, and 7 described with reference to figure 1 and the image is captured by camera 9.

The output of camera 9 is then fed into processor 11 which processes the received data according to steps S103, S105, S107 and S109 of figure 3 to produce an image on display 13 of figure 2.

Figure 3 is a flow chart showing a method in accordance with an embodiment of the present invention. In step Si 01, data using a photometric stereo technique is acquired.

The data may be acquired real time and processed immediately or may be acquired by the system from previously recorded data.

As explained with reference to figure 1, a minimum of 3 lights is required to perform photometric stereo with no extra assumptions. If only 2 images are available, then it was previously believed that a constant albedo must be assumed in order to obtain meaningful data. In the present embodiment it is assumed that there is a smoothly varying albedo. Whenever more lights are available, the light visibility problem becomes a labelling problem where each point on the surface has to be assigned to the correct set of lights in order to successfully reconstruct the surface.

It is assumed that the camera used to capture the data is orthographic (with infinite focal length) for simplicity, even though the extension to the more realistic projective case is straightforward.

In the case of orthographic projection, it is possible to align the world coordinate system so that the xy plane coincides with the image plane while the z axis corresponds to the viewing direction. The surface in front of the camera can then be parameterized as a height function Z(x, y). If Z and are the two partial derivatives of Z, the normal can be defined as: that is locally normal to the surface at (x, y). For i = 1... 3 let c.(x, y) denote the pixel intensity of pixel (x, y) in the jth image. It is assumed that in the th image the surface point (xy Z(x, ))T is illuminated by a distant light source whose direction is denoted by the vector I and whose spectral distribution is E (X). It is also assumed that the surface point absorbs incoming light of various wavelengths according to the reflectance function R(x, y, X). Finally, it is assumed that the response of the camera sensor at each wavelength be given by S (2). Then the pixel intensity c1 (x, y) is given by c, (x, y)= (i "n)JE(i)R(x, y, (1) The value of this integral is known as the surface albedo of point (x y Z (x, y) )T This allows the albedo scaled normal vector to be defined as: b = u JE(A)R(x, y, so that (1) becomes a simple dot product°i 1b. (2) c, =Ib (2) Photometric stereo methods use the linear constraints of (2) to solve for b in a least squares sense. From this partial derivatives of the height function are integrated to produce the function itself In three-source photometric stereo, when the point is not in shadow with respect to all three lights, the three positive intensities c are measured and each of these gives a constraint on b. If L [Ii 12 131T and c= [ci c2 c3]T then the system has exactly one solution which is given by L'c (3) If the point however is in shadow, say in the th image, then the measurement of c1 cannot be used as a constraint. Since each equation (2) describes a 3D plane, the intersection of the two remaining constraints is a 3D line.

If e, is the th column of the 3 x 3 identity matrix and D, is the identity matrix with a zero in the ith position in the diagonal, the possible solutions for b are: b L'D,c + (4) where 1u is a scalar parameter. This parameter represents the value that c would have, had the point not been in shadow in the 1Ih image.

Projecting b into a 2D gradient space using the operator P[xi x2 X3](x1/X3, x2/x3)(p, q) such that for a point in shadow, the point (p, q) lies along the shadow line of the pixel defined as the line that joins P[LD1cJ and P[Le1J.

To provide a solution, the scaled normals b into 2D space with coordinates p and q.

The operator P[x1 X2 X3]=(XI/X3, x2/x3) is used to achieve this projection and the scaled normal b of surface point (x, y, Z(x, y)) projects to point P[b]=(p, q). Thus, the coordinates p and q are equal to the two partial derivatives Zx and Zy respectively.

According to the image constraints and assuming no noise in the data, one of the following three cases can occur: Case 1.The surface point is in shadow in two or more images. In this case there is no constraint in P[b] from the images.

Case 2.The surface point is not in shadow in any of the three images. In this case (p,q) coincides with P[V'c]=(P,Q).

Case 3 The surface point is in shadow in exactly one image, say the th* In this case (p, q) must lie on the line that joins P[V'D1c]=r(P,Q) and P[L'e1](Pj'Q) This line shall be referred to as the shadow line 57 of the shaded pixel.

The above three cases can be thought of as shadow labels for the pixels. Which shadow label should be applied to each pixel is determined in step S 103. How the shadow labels are assigned will be described later.

The description will now be concerned with case 3. For all pixelsj which are occluded under the i'd' image, the corresponding points in the 2D gradient space are (P, , Q) are x on the line 51 described the equation I y = 0, Also, the shadow lines of all of the pixels intersect at the point(PeQc°).

In the presence of noise in the data c, cases 2 and 3 above do not hold exactly as points (P, Q) and (P,Q) are corrupted: The point (p, q) is slightly different from (P, Q) for unoccluded pixels, and (P, Q) is not exactly on the line joining (p, q) and (p)Q).

Figure 4 shows the configuration for six pixels where pixel 1 is under shadow in the ith image while pixel 2 is not in shadow in any image. The points (p,, qj) (dark dots) represent the partial derivatives of the height function at pixelj. For each point (Pj, qj) there is a corresponding data point (white dot). Pixel 2 is unoccluded and hence (pa, q2) must be as close as possible to its data point (P2, Q2). Pixel 1 however is occluded so (ps, qj) must be as close as possible to its shadow line 57. This is the line 57 joining its data point (Pd., Q) and the intersection point (p)Q)).

Once a range of potential solutions has been determined in line with S 105, the method progresses to step S 107 where a solution is selected using a constraint. The constraint is in the form of an integrability condition.

The (p, q) coordinates are not independent for each pixel as they represent the partial derivatives of a scalar field and as a result they must satisfy an integrability condition.

By assuming that the height function is a continuous piecewise polynomial, the integrability condition takes the form of a convolution between p and q. If for example Z(x, y) is a linear interpolation between height values at pixel centres Z(i,j) integrability can be expressed as: p(i+ 1,j)-p(i,j)=q(i,j-i-1)-q(i,j) To obtain the solution, values (p, q) for each pixel that minimize the discrepancy between the noisy data and the model while satisfying the integrability condition are obtained.

As mentioned above, under noise in the image data c, the 2D points (F, Q) and (p, q) are not perfectly consistent with the model. For non-shadowed pixels, the difference between the model and data can be measured by the point-to-point square difference term e (p -+ (q -In the case of the shadowed pixels however there are a number of possible ways to quantify the non-colinearity of(p,q), (P,Q) and (pI)Q). The point (P,Q) contains noise and hence the distance which should be measured is from (P, Q) to the x intersection of the line 53 joining (p,q), and (eQ) with the line 51 I y = 0 which leads to the term: IK)-[(i -m.Ir)112 where m, is the 11h vector of V'. The above expression is non linear with respect top and q and thus mot suitable as a constraint for a least squares estimate.

Therefore, an alternative distance 55 expression is minimised which is the distance from (p,q) to the line 57 joining (P,Q) and (Pe°Q')* . The term this corresponds to is: e' (pp(I))2+(Q())2 where the quantity being squared is linear with respect to p and q.

Thus, once all pixels have been labelled with their shadow conditions in accordance with step S 103, the cost function which is to be minimised in order to solve the whole problems is: eJ + + + jeS jeS1 jS2 JES3 where S contains all of the non-shadowed pixels while S, contains pixels shaded in the 1th image.

The above cost function is quadratic terms in p and q, thus finding the minimum of this quantity subject to the integrability condition is a constrained linear least squares problem that can be solved by a variety of methods.

In step S 109, a 3D image is generated from the determined gradients. Knowledge of the gradients allows a depth map of the object to be determined. The depth map can be used to directly generate a 3D image or it can be used to generate data for animation.

For example, if the above method is used to capture data of a complex moving object such as cloth, hair or the like to produce data for a moving mesh. The moving mesh of can then be attached to an articulated skeleton.

Skinning algorithms are well known in the art of computer animation. For example, a smooth skinning algorithm may be used in which each vertex Vk is attached to one of more skeleton joints and a link to each jointj is weighted by WI,k. The weights control how much the movement of each joint affects the transformation of a vertex: WI,kSIVk, -1 The matrix S represents the transformation from the joint's local space to world space at time instant t.

The mesh was attached to the skeleton by first aligning a depth pattern of a fixed dress or other piece of clothing or the like with a fixed skeleton and for each mesh vertex a set of nearest neighbours on the skeleton. The weights are set inversely proportional to these distances. The skeleton is then animated using publicly available mocap data (Carnegie-mellon mocap database http://mocap.cs.crnu.edu). The mesh is animated by playing back a previously captured cloth sequence of cloth movement.

Figure 5 shows the above method of deriving gradients applied in practice on synthetic data. Figure 5a shows a synthetic sphere, the sphere is illuminated from three directions and the light regions indicate the regions where pixels are shadowed in one of the three images.

Figure Sb shows a 3D image of the sphere generated using the above method. The general shape of the shaded areas is well produced providing that the areas are surrounded by unshaded regions. The unshaded regions act as boundary conditions for the shadowed regions and give the problem a unique solution The presence of scratch artefacts are seen in the shadowed regions in figure 5b. These are caused by the point-to-line distances which do not introduce enough constraints in the cost function. The point (p, q) can move significantly in a direction parallel to the corresponding shadow line only to gain a slight decrease in the overall cost. This results in violent perturbations in the resulting height function that manifest themselves as deep scratches running perpendicular to the shadow lines.

Figure Sc shows an image generated from the data of figure 5a in accordance with an embodiment of the present invention where a regularisation scheme is used.

The regularizing criterion is consistent with the linear least squares framework described above such that no non-linear constraints can be enforced and it suppresses noise while preserving as much of the data as possible.

One possible choice for a regularization criterion is minimizing the Laplacian of the height field V 2 This is known to have good noise reduction properties and to produce smooth well behaved surfaces with low curvature. However, the Laplacian is isotropic so it tends to indiscriminately smooth along all possible directions.

There are other anisotropic alternatives to the Laplacian such as anisotropic filtering which is known for image processing.

An improved method of achieving regularisation in the specific case is to enforce regularization in (p, q) space along the direction of the shadow line for each shadowed pixel.

This is achieved by modifying the point to line distance term as follows: ( --(1 -W)Pe' )2 + (q -wQ -(1 -w)Q)2 (5) This introduces a new variable w per shaded pixel that specifies a location along the shadow line of that pixel. The term is still quadratic with respect top, q and w but this now allows regularizing of the solution in such a way that noise can be compensated.

The variable w is related to parameter jt of(4) by: eTL_ID.c eL1D1c + ,ueL'e1 (6) As explained previously p represents the value of c that would have been measured had the pixel not been in shadow in that image. The method puts a cost on the length of Vw inside the shaded regions, the cost may also be applied to a second or higher order term.

As w is a proxy for t, this corresponds to introducing smoothness in I1Tb. This allows for the elimination of scratch artefacts while letting b have variability in the directions perpendicular to I. the results form the regularisation procedure are shown in figure 5c.

Step S103 of figure 3 is concerned with determining a shadow mask which allows determination of which pixels are shadowed.

In photometric stereo with four or more images it is possible to detect shadows by computing the scaled normal that satisfies the constraints in a least squares sense. If the residual of this least squares calculation is high, this implies that the pixel is either in a shadow or in a highlight. However, with three images this becomes impossible as the three constraints can always be satisfied exactly, leaving a residual of zero.

Recently, Chandraker, M et al "Shadowcuts: Photometric stereo with shadows" Proceedings of Computer Vision and Pattern Recognition, 17-22 June 2007 p 1-8 proposed a graph-cut based scheme for labelling shadows in photometric stereo with four or more images. Based on the constraint residual, a cost for assigning a particular shadow label to each pixel was calculated. This cost was then regularized in an MRF framework where neighbouring pixels were encouraged to have similar shadow labels.

The basic characteristic of a shadow region is that pixel intensities inside it are dark.

However this can also occur because of dark surface albedo. To remove the albedo factor pixel intensities are divided with the magnitude of the intensity vector c. The cost for deciding that a pixel is occluded in the i-th image is c/ c. This still leaves the possibility that a pixel is mistakenly classified whose normal is nearly perpendicular to the jth illumination direction Ii. However in that case the pixel is close to being in a self shadow so the risk from misclassifying it is small. The cost for assigning a pixel to the non-shadowed set is given by -min--.

i Itch The costs are regularised in an MRF framework under a Potts model pairwise cost. This assigns a fixed penalty for two neighbouring pixels being given different shadow labels.

The MRF is optimized using the Tree Re-weighted message passing algorithm (Kolmogorov et al. "Convergent Tree-reweighted message passing for energy minimisation" IEEE Trans. Pattern Anal. Mach. Intell. 28 (2006) 1568-1583.) Figure 6a shows a face which is to be imaged and figure 6b shows the shadow mask determined using the above technique.

The above description has used three images to generate a 3D image. A particularly useful method of obtaining these three images is to use colour photometric stereo. Here, an object to be images is illuminated from 3 different directions with radiation of different wavelengths and a cameras is used which can distinguish between the three wavelengths so that the three images can be taken at the same time. Typically, the three colours of red, green and blue are used as these colours can be reflected from the object and detected with minimal mixing from a single RGB image.

As colour photometric stereo is applied on a single image, one can use it on a video sequence without having to change illumination between frames.

In colour photometric stereo each of the three camera sensors (R, G, B) can be seen as one of the three images of classic photometric stereo. The pixel intensity of pixel (x, y) for the i-th sensor is given by: c(x,y)= (7) Note that now the sensor sensitivity S, and spectral distribution E are different per sensor and per light source respectively. To be able to determine a unique mapping between RGB values and normal orientation a monochromatic surface is assumed.

Therefore R(x, y, X)cr(x,y)p(X) where a(x,y) is the monochromatic albedo of the surface point and pQ) is the characteristic chromaticity of the material. Let and

I

V, = V11VV1) The scaled normal is: b=a(x,y)n Then the vector of the three sensor responses at a pixel is given by c [vi V2 V3] [1 12 b.

Essentially each vector v provides the response measured by the three sensors when a unit of light from sourcej is received by the camera. If matrix [vi v2 v3] is known then: -t c = [vi V2 V2] C. The values of ô can be treated in exactly the same way as described above for standard photometric stereo.

The system may be calibrated in a number of ways. A calibration board of the type shown in figure 7 may be used. The calibration board 21 comprises a square of cloth 23 and a pattern of circles 25. Movement of the board 21 allows the homography between the camera 9 and the light sources 3, 5 and 7 to be calculated. Calculating the homography means calculating the light source directions relative to the camera. Once this has been done, zoom and focus can change during filming as these do not affect the colours or light directions. The cloth 23 also allows the association between colour and orientation to be measured.

By measuring the RGB response corresponding to each orientation of the material the following matrix may be estimated:

T

M = [v1 V2 v3] [i 1213] The results from the initial calibration routine where an image is captured for various known orientations of the board does not need to be performed for every possible board orientation as nearest neighbour interpolation can be used to determine suitable data for all orientations. It is possible to capture data from just 4 orientations in order to provide calibration data for a 3x4 matrix. Good calibration data is achieved from around 50 orientations. However, since calibration data is easily collected it is possibly to obtain data from thousands of orientations.

In another calibration method, a mirror sphere is used to estimate light directions I 2 and 13. Secondly, three sequences of the object moving in front of the camera are captured. In each sequence, only one of the three lights is switched on at a time. In the absence of noise and if the monochromatic assumption was satisfied, the RGB triplets acquired are multiples of v when lightj was switched on.

A least squares fit can then be performed to obtain the directions v1, v2 and v3. . To obtain the relative lengths of the three vectors the ratios of the lengths of the ROB vectors can be used. The length of v is set to the maximum length in ROB space, of all the triplets when lightj was switched on.

Figure 8 shows an image of a real object which in this case is a frog. Figure 8a shows three images taken from three different illumination directions. In the experiment, the frog was imaged from five different directions, but only data from three images were used to produce the final three-dimensional image.

Figure 8b shows the three images of figures 6a normalised in intensity using the expression: c/lie Figure 8c shows the calculated non-shadowing cost of i Ilcil Figure 8d is the shadow mask calculated using the data shown in figures 8b and 8c. The light regions of the shadow mask show the regions where one of the images is occluded.

These shaded regions are accommodated using the technique discussed above.

Figure 8e is the generated three dimensional image of the object.

Figure 9 shows video data of a white painted face illuminated by three coloured lights.

Three reconstructed frames from a video sequence where shadows are not taken into account are shown in column 9a. Column 9b shows the corresponding three frames where shadows are taken in account as explained above. It can be seen that the nose reconstruction is dramatically improved when correctly processing the shadows, even though only two lights are visible in the shadowed regions.

Claims

CLAIMS: 1. An apparatus for determining the gradients of the surface normals of an object, the method comprising: means to receive data of three 2D images of said object, wherein each image is captured under illumination from a different direction; means to establish which pixels of the image are in shadow such that there is only data available from two images from these pixels; means to determine a range of possible solutions for the gradient of the surface normal of a shadowed pixel using the data available for the two images; and means to select a solution for the gradient using the integrability of the gradient field over an area of the object as a constraint and minimising a cost function.
2. An apparatus according to claim 1, wherein said apparatus further comprises means to acquire data, said means to acquire data comprising: at least three light sources, irradiating the object from three different angles; a camera provided to collect radiation from said three radiation sources which has been reflected from said object; and an image processor configured to generate a depth map of the three dimensional object, wherein each radiation source emits radiation at a different frequency and said image processor is configured to distinguish between the reflected signal from the three different radiation sources.
3. An apparatus according to claim 2, wherein said object is moving and wherein said camera is a video camera.
4. An apparatus according to any preceding claim, further comprising means to generate an image from the determined surface normals.
5. A method for determining the gradients of the surface normals of an object, the method comprising: receiving data of three 2D images of said object, wherein each image is taken under illumination from a different direction; establishing which pixels of the image are in shadow such that there is only data available from two images from these pixels; determining a range of possible solutions for the gradient of the surface normal of a shadowed pixel using the data available for the two images; and selecting a solution for the gradient at a pixel using the integrability of the gradient field over an area of the object as a constraint and minimising a cost function.
6. A method according to claim 5, wherein said method of selecting a solution for the gradient comprises uses the constraint that the surface of the object is a continuous piecewise polynomial.
7. A method according to either of claims 5 or 6, wherein said method of selecting a solution is performed as a least squares analysis.
8. A method according to any of claims 5 to 7, wherein the cost function is expressed in terms of a regularisation parameter which is used to compensate for noise.
9. A method according to claim 8, wherein a cost is imposed on said regularisation parameter.
10. A method according to any of claims 5 to 9, wherein the range of possible solutions is determined by expressing the surface normals in terms of their gradients in a 2D gradient space such that the range of possible solutions lie along a straight line in said 2D gradient space.
11. A method according to claim 10, wherein the regularisation parameter is the length along the said straight line of possible solutions from a fixed point on said line.
12. A method according to claim 5, wherein the range of possible solutions is determined by expressing the normal as the albedo scaled surface normal b: b:L"c wherein c is [ci c2 c3] and CIC2 and c3 represent the respective intensity of the pixel measured in each of the three 2D images and L[1i 12 13 1T where I, I and 13 are the vectors which represent the direction of the light source which illuminates the object for each of the three 2D images; wherein when the pixel in the th 2D image is in shadow b is expressed as: b = LDc + where e is the 11h column of the 3x3 identity matrix, D* is the identity matrix with a zero in the position in the diagonal and 1u is a scalar parameter representing the value that c, would have if it were not in shadow; and projecting b into a 2D gradient space using the operator P[x1 x2 x3](x1/x3, x2/x3)=Q,, q) such that for a point in shadow, the point (p, q) lies along the shadow line of the pixel defined as the line that joins P[L'D,c] and P[E1e1].
13. A method according to claim 12, wherein minimising the cost function is performed using a parameter w, where w is distance along the shadow line of point (p,q) from a fixed point on the shadow line, the method further comprising imposing a cost on Vw.
14. A method according to any of claims 5 to 13 wherein establishing whether a pixel is in shadow in th of said three images is achieved by determining if the value of the intensity in the th image divided by the magnitude of the intensity vector c for all three images is below a predetermined value.
15. A method according to claim 14, wherein establishing whether a pixel is in shadow is further determined by introducing a further constraint to reduce the chance of a pixel being determined to be in shadow and a neighbouring pixel not in shadow.
16. A method according to any of claims 5 to 15, wherein the data for the three 2D images is acquired by using three radiation sources, irradiating the object from three different angles and wherein each radiation source emits radiation at a different frequency such that the data from said three radiation sources can be acquired at the same time.
17. A method of generating an image, the method comprising: determining the gradients of the surface normals according to any of claims 5 to 17; and generating an image from data of said surface normals.
18. A method according to claim 17, wherein the generated image is a three dimensional image.
19. A method of animating cloth, the method comprising: imaging cloth according to the method of any of claims 5 to 18 and animating a depth map generated from said determined gradients.
20. A computer running a computer program configured to perform the method of any of claims 5 to 19.