University of Houston • University of Houston-Clear Lake • ISSO Annual Report Y2002—pp. 37-46

Optical Tracking

Ioannis Kakadiaris (UH), Karolos Grigoriadis (UH), Darby Magruder (NASA-JSC), Kenneth Baker (NASA-JSC), and Carlos Barrón (UH)

Abstract
Recently, we developed a technique that allows semi-automatic estimation of anthropometry and pose from a single image. However, estimation was limited to a class of images for which an adequate number of human body segments were almost parallel to the image plane. In this paper, we present a generalization of that estimation algorithm that exploits pairwise geometric relationships of body segments to allow estimation from a broader class of images. In addition, we refine our search space by constructing a fully populated discrete hyper-ellipsoid of stick human body models in order to capture the variance of the statistical anthropometric information. As a result, a better initial estimate can be computed by our algorithm; thus, the number of iterations needed during minimization are reduced tenfold. We present our results over a variety of images to demonstrate the broad coverage of our algorithm.

MOST OF THE RESEARCH IN THE AREA OF HUMAN MOTION analysis is directed at tracking humans in monocular or multi-camera image sequences^1-4 because of the wide range of applications that could benefit from estimating and analyzing human motion. These applications include surveillance, performance measurement for athletes, the performance of patients with disabilities, motion capture for entertainment applications, and vision-based user interfaces.^5,6 The important problem of estimating an individual’s anthropometry and pose from a single uncalibrated image has, however, received considerably less attention even though it constitutes the first step in many human tracking (from monocular images) algorithms.

Recently, we developed a technique that allows semiautomatic estimation of anthropometry (up to a scale parameter) and pose from a single image.^7,8 We initially selected a set of image points that constituted the projection of selected landmarks. Using this information, along with a priori statistical information about the human body, we generated a set of plausible segment-length estimates. The third step produced a set of plausible poses based on joint limit constraints using a geometric method. In the fourth step, pose and anthropometric measurements were obtained by minimizing an appropriate cost function subject to the associated constraints. The novelty of that approach was the use of anthropometric statistics to constrain the estimation process that allowed the simultaneous estimation of both anthropometry and pose. Estimation was limited, however, to a class of images for which an adequate number of human body segments were almost parallel to the image plane. For example, that method could handle images like the one depicted in Fig. 1(a), but not images like the one depicted in Fig. 1(b).

Figure 1. (a) Instance of an image that can be handled by the algorithm described in Barrón and Kakadiaris⁷ and Yamamoto et al.¹⁹; (b) Instance of an image that could not be handled using the algorithm in Barrón and Kakadiaris⁷ and Yamamoto et al.,¹⁹ but can be handled using the algorithm described in this paper.

In this study, we present a generalization of that reconstruction algorithm to allow estimation from a broader class of images including the one depicted in Fig. 1(b). In particular, we first explore the projective properties of segments that have pairwise similar orientation with respect to the camera. Second, we extend our technique for exploiting prior statistical anthropometric information to allow us to obtain a better initial estimate of the human body model. As a result, the number of iterations needed by our algorithm to converge is reduced tenfold. Also, our algorithm can signal the absence of adequate information for producing a reliable estimate.

The following discussion describes our enhancements in more detail.⁹ In section 2, we review prior work in the area; in section 3 we provide our analysis and our proposed enhancements. In particular, we explore how to exploit additional geometric relationships of the human subject’s body segments by considering their foreshortening in the image. We describe our refinements of the prior statistical models to further constrain the selection of initial estimates for the minimization process. Finally, we illustrate results from our system.

Prior work
One of the challenges in model-based human tracking algorithms is the initialization of the model in the first frame of the image sequence. Tracking and posture estimation methods have been presented that use either one^10-19 or multiple cameras.^20-25 In most of the existing tracking approaches, the user specifies an approximate position and posture from the human model at the first frame of the image sequence.^13,22,26-28 For the initialization step of their human tracking method, Bregler and Malik minimize a cost function over position, angles, and body dimensions.¹² Unfortunately, no information is provided about the accuracy of their method nor for what class of postures their method worked.

Taylor²⁹ presented a method for recovering information about the configuration of articulated objects from a single image. Their effort is similar to our work in that both methods assume a scaled orthographic projection, and both use geometrical information as constraints. The difference is that our work combines the geometrical constraints with the prior statistical anthropometric information to drive the estimation process. Thus, our method begins and ends with a plausible human model. Rosales and Sclaroff³⁰ presented a method for pose estimation based on selecting the most likely pose given the learned probability distribution and the visual features’ similarities between hypothesis and input. For our algorithm, there is no need for a learning phase. Moeslund and Granum³¹ represented the human model in phase space spanned by its different degrees of freedom, and they used an analysis-bysynthesis approach to match the phase space model with real images for the case of a human-arm model. They currently examine what would be the impact of a significantly increased size of the phase space (as in the case of estimating the pose of a full human body model) in the efficiency of their algorithm. The contribution of our research is a systematic study and an improved technique that takes into consideration statistical anthropometric information to constrain the estimation process.

Analysis
The problem of anthropometry and pose estimation from a single image can be formulated as follows: Given a set of points in an image that corresponds to the projection of landmark points of a human subject, estimate both the anthropometric measurements (up to a scale) of the subject and his/her pose that best matches the observed image. In the following description, it is assumed that the selected landmarks at the image are the result of a scaled orthographic projection of three-dimensional landmark points of a human subject.

Stick human body model
For the purposes of this research, we have developed a generic stick human body model, SM, whose complete description can be found in Barrón and Kakadiaris.^7,8 Briefly, SM is a tree (s, S,A), where S is a set of sites/landmarks and A is a collection of edges (segments) with endpoints in S, and s Î S is the root. In our case, A = { HD, RY, LY, NK, UT, RC, LC, RUA, LUA, RLA, LLA, RHD, LHD, LT, RHP, LHP, RUL, LUL, RLL, LLL, RF, LF} as enumerated in Fig. 2b, and the set of landmarks consists of a set of joints J ={at, sp, la, lc, le, lh, lk, ls, lw, ra, rc, re, rh, rk, rs, rw, wt} (information about the SM’s joints is provided in Table 1), and other landmarks M = {ry (right eye), ly (left eye), rhd (base of the right middle finger), lhd (base of the left middle finger), rf (tip of the right foot), lf (tip of the left foot)} (S = J È M). Default data for the joints and the anthropometric measurements are extracted from Churchill et al.³²

Figure 2. (a) SM and associated coordinate systems; (b) Names of the SM’s segments (adapted from Barrón and Kakadiaris^7,8)

Table 1. Information Related to Joints of the SM

Exploiting geometric relationships
In this section, we examine the foreshortening of the body segments in the image, under the assumption of scaled orthographic projection. Let = [X_C, Y_C, Z_C]^T be the origin of the camera (see Fig. 3) and assume that the image plane is located at Z_IM on the z-axis of the camera. As known, under scaled orthographic projection the point  = [X₁, Y₁, Z₁] (see Fig. 3) projects to the point

     = [x₁, y₁, z₁] = [X_c + l₁(X₁ - X_c), Y_c + l₁(Y₁ - Y_c), Z_im]^T,

where l₁ = Z_IM - Z_C.

Similarly, the point  = [X₂, Y₂, Z₂] projects to the point

     = [x₂, y₂, z₂] = [X_c + l₂(X₂ - X_c), Y_c + l₂(Y₂ - Y_c), Z_im]^T,

where l₂ = Z_IM - Z_C/Z₂ - Z_C.

If we assume that this point lies on the same plane (normal to the camera z-axis) with point P₁ then l₁ = l₂. Thus, for any point on the line P₁P₂, its projection is given by the equation [x, y]^T = l₁S[X, Y, Z]^T, where

Similarly, for any point on the line , its projection is given by the equation [x,y]^T == l₃S[X, Y, Z]^T, where l₃ = (Z_IM - Z_C/Z₃ - Z_C) = (Z_IM - Z_C/Z₄ - Z_C) = l₄. If a_z is a real number such that

then Z₃ - Z_C = Z₁ - Z_C/1 + a_z
and l₃ = l₁(1 + a_z).

Therefore, the scaled orthographic projection for the points of  is given by:

     [x,y]^T = l₁(1 + a_z)S[X, Y, Z]^T.

Let L₁₂ = || - || and l₁₂ = || - ||, then
l₁₂ = [(x₂ - x₁)² + (y₂ - y₁)²]^1/2 = l₁[ (X₂ - X₁)² + (Y₂ - Y₁)²]^{1 / 2} = l₁L₁₂.

Also, we can obtain that l₃₄ = l₃L₃₄, where L₃₄ = || - || and l₃₄ = || - ||. Using the relation l₃ = l₁(1 + a_z), we can obtain that l₃₄ = l₁(1 + a_z)L₃₄. Finally, the ratio between l₁₂ and l₃₄ is given by l₁₂/l₃₄ = l₁L₁₂/(l₁(1 + a_z)L₃₄) = L₁₂/(1 + a_z)L₃₄ that implies the relation

which suggests Proposition 1.

Proposition 1. For segments that lie in planes almost parallel to the image plane, the ratio of segment lengths in 3D and the ratio of their corresponding projected lengths are similar if and only if a_z is very small.

Proof. The result is obtained from Eq. (2).

Our previous work^7,8 was based on the above proposition. In this work, we examine the relationship of segment lengths that lie in different planes using the distance between planes. For example, let’s examine the ratio of L₁₃ = || - || and L₂₄ = || - ||, and the ratio of l₁₃ = || - || and l₂₄ = || - ||.

Since, l₁₃ = [(x₃ - x₁)² + (y₃ - y₁)²]^1/2 = |l₁| {[(1 + a_z)X₃ - X₁]² + [(1 + a_z)Y₃ - Y₁]²}^1/2 = |l₁| {L₁₃² - (Z₃ - Z₁)² + 2a_z[X₃(X₃ - X₁) + Y₃(Y₃ - Y₁)] + a_z²(X₃² + Y₃²)}^1/2, then, l₁²L₁₃² = l₁₃² + l₁²{(Z₃ - Z₁)² - 2a_z[X₃(X₃ - X₁) + Y₃(Y₃ - Y₁)] - a_z²(X₃² + Y₃²)}.

Using Eq. (1), we obtain L₁₃ = l₁₃/ 1(1 + e_z), where 1 + e_z = {1 + l₂/l₁₃² {a_z²(Z₃ - Z_C)² - (X₃² + Y₃²)] - 2a_z[X₃(X₃ - X₁) + Y₃(Y₃ - Y₁)]}^1/2.

Similar arguments hold for L₂₄ and l₂₄. Thus,

where ij depends on the position of points .

Proposition 2. Assume that two segments have the following two properties: (a) their endpoints lie on planes perpendicular to the camera’s Z axis, and (b) their orientation with regard to the camera is almost similar. Then the ratio of lengths of these two segments is similar to the ratio of lengths of their corresponding projected segments.

Proof. According to our hypotheses, e₁₃ and e₂₄ must have similar values; thus, the conclusion follows from Eq. (3).

In summary, the following two observations hold: (a) For segments that are almost parallel to the image plane, a_z takes a small value, as per Eq. (2); (b) For segments that have orientation almost parallel to each other, the corresponding factors, 1 + e_z, for these segments must have similar value, as shown in Eq. (3). Based on these observations, during the initialization step of our algorithm, we mark the segments that have orientation almost parallel to the image plane (as in Barrón and Kakadiaris^7,8); in addition, we mark any pair of segments that have similar orientation with respect to the camera. These projected segments are employed to compute the ratios that will be used as input for the selection of the initial stick model.

Exploiting prior statistical information
Using prior statistical anthropometric information, we build our cadre family as a multivariate representation of the extremes of the population distribution, as described in Barrón and Kakadiaris.^7,8 The probability density function of the multivariate normal distribution is defined by

(4)

where k is the number of dimensions. In our case, the variables are the lengths of the 22 segments of our stick model, is a random vector, , and S are the mean and the covariance matrix of the population, and we use the quadratic form Q() = ( - )^TS( - ) whose shape depends on S. We compute the principal components of S, and we select the first seven (i = 1, . . . , 7) with large eigenvalues such that l₁ > l₂ > · · · > l₇. The following notation relates the SM and the eigenvectors of

where = [a1, . . . , a7]^T such that

(5)

That is, we are using linear combinations of the eigenvectors constrained by Eq. (5) to span on the hyper-ellipsoid of S. In particular, we build a family of q different models (SM(_q)), and we compute ratios of lengths r_k,q as

(6)

where m(l_n) is the mean length of the segment l_n.

Furthermore, we compute the covariance coefficients for these ratios as

We select an initial SM(_q*) that satisfies the equation

(8)

where r_k,q, r_j,q are ratios obtained from the SMs in our cadre family and s_k, s_j are the corresponding ratios computed from the input data as follows:

Our objective is to find a finite subset of

in order to build our cadre family SM(_q) that will provide a good initial estimate SM(_q*). The complete algorithm for the SM selection is described in Barrón and Kakadiaris.⁸

Intuitively, we build all the linear combinations of the eigenvectors with at least one nonzero coefficient under the constraint that all nonzero coefficients must be equal, and we minimize Eq. (8) using a discrete cadre family. Note that in Eq. (8) we can use instead of

The following proposition justifies this modification:

Proposition 3.

Proof. Indeed,

Therefore, we can restrict D to

To analyze the extended set of selected models {SM() | Î D^D, we introduce a metric space using the quadratic form Q, since D^D depends on the matrix S. Let êi = , i = 1, . . . ,7 where ||.|| is the Euclidean norm of R²². The following proposition presents the matrix S and an explanation of the bounds of our cadre family.

Proposition 4. For every (,) Î D^D and for every unit vector ê_i, the following properties hold:

1. The inner product induced by S, (,)_S = a^TSb , "a,b Î R²² is commutative.
2. (ê_i,ê_i)_S = l_i ¹ 0, and (ê_i,ê_j)_S = 0, i ¹ j (thus {ê_j} is an orthogonal set).
3. 

Proposition 3 explains why we restrict by the constraint

     .

The following equality holds:

Thus,

and by the previous proposition we obtain the inequality

This expression provides a theoretical bound for the extension of hyper-ellipsoid of Q(SM()) with the center in .

In the following, we describe our new full discretization D^D of D^D. Let I = {1, 2, . . . , 7} and k Í I. We select Î R⁷ such that S_iÎI a_i² = 1, a_j = 0, "j Ï I(k) and (a_i =a_j) " i, j Î k. For example, from this formulation, we can use a₁ = a₂ = . . . = a₇ = ± 1/Ö7, or a₁ = 0, and a₂ = . . . = a₇ = ± 1/Ö6, etc. This formulation extends our previous approach, and the number of SMs that we employ is

The coefficients and the number of elements for each SM are given in Table 2. The distribution of the SM(), Î D^D can be estimated in terms of S and {}.

Table 2. Factor and Its Number of Linear Combinations

Proposition 5. Given , Î D^D, then

and "k

Proof. The result follows from

However, this approach provides a large number of possible combinations for the initial anthropometric estimates. In particular, if Î D^D, then

where = [l_1m, l_2m,. . . ,l_22m] is the average human model, and = (dl_1i, dl_2i, . . . , dl_22i), i = 1, . . . , 7 are the eigenvectors of S. For example, for a particular that corresponds to the index q, the ratios r_k,q are given by the Eq. (6)

The terms and are used to compute the ratio, and they depend on the selection of a_i,q,q. The benefit of using a larger cadre family than the one used in our previous approach is the improvement of the initial estimation using only a few ratios. It enables finding a solution for the images that depict humans in complicated postures and pruning the user-selected ratios in order to compute a good initial SM which, in turn, improves the accuracy of our minimization process.

Algorithm
Our technique for simultaneously estimating the anthropometry and the pose from a single uncalibrated image has four basic steps.

Algorithm. Anthropometry and pose estimation
Step 1: Selection of projected landmarks and ratios.
Step 2: Ratio pruning and choice of initial Stick Model.
Step 3: Initial estimates for pose.
Step 4: Iterative minimization over lengths and angles.

In this work, we have modified the first and second step of our algorithm presented in Barrón and Kakadiaris,^7,8 as described in the following paragraphs.

Selection of projected landmarks and ratio computation. We have extended the user-interface developed for Barrón and Kakadiaris^7,8 to allow the user not only to select the projection of the visible landmarks of a subject’s body, but also to define the ratios to be used for the initial estimates using the properties described in Section 3.2. Each ratio is treated independently, since the properties described in Section 3.2 are local and are valid for at least two segments. Note that the selected segments must be in parallel planes or in a similar orientation.

Ratio pruning and choice of initial stick model. Our basic assumption is that there are a number of segments that have the properties described in Section 3.2. The user defines the input ratios from segments with one of these properties. In this step, our algorithm compares the selected ratios with the range of our cadre family and selects the ones that lie inside the range. For the case in which at least one selected ratio lies within the range of our cadre family, the algorithm selects an SM using the technique described in Barrón and Kakadiaris.⁸ This step weighs the ratios using the Mahalanobis distance in order to select a model that closely matches the input ratios obtained from the image. Otherwise, the user is informed that the selected ratios lie outside the range of the cadre family (which means that our algorithm cannot handle this image), and the user is asked to select additional ratios, if possible.

Experimental results
We have performed a number of experiments on synthetic and real data to assess the accuracy, limitations, and advantages of our approach. In the first experiment, we applied our technique to an image created using the virtual human modeling tool EAI Jack. Figures 4(a) and 4(b) depict the reconstructed 3D model from two views. Tables 4 and 5 contain statistical information related to the accuracy of the estimation process. In this experiment, the maximum estimated error of the anthropometry dimensions decreases from 3.1743% to 0.4782%. In the second experiment, we applied our technique to a real image from the subject Vanessa whose anthropometric dimensions were manually measured. Figure 4(c) depicts the selected points, and Figs. 4(d)–(f) depict the reconstructed model from a novel view. Table 3 captures the percentage errors (PE) of the estimation process. We observe that the estimation of anthropometric information is within 1% of the anthropometric dimensions of the subject, clearly an improvement of our previous method whose accuracy was within 3.2%. In general, we have performed numerous other experiments with a variety of subjects whose anthropometric dimensions are known with similar, very encouraging results. In the third experiment, we applied our algorithm to a variety of images from a variety of application domains where anthropometric information about the subjects was not available. Figure 4(g) depicts the input image along with the selected points. Figure 4(h) is the reconstructed SM from another point of view. Figures 5(a)-(c), 6(a)-(c), and 7 depict the input images and their reconstructed 3D models.

Figure 4. (a) Front and (b) side views of the virtual human along with our estimates. (c) Input image for the subject Vanessa, and (d) camera model and reconstructed 3D model. (e) and (f) novel view of the reconstructed 3D model. (g) landmarks selected by a user on this baseball player image; (h) anthropometry and pose estimates from our algorithm.

Table 3. Accuracy of the Length Estimates for the Subject Vanessa

Table 4. Accuracy of the Length Estimates for the Synthetic Experiment

Table 5. Accuracy of the Pose Estimates for the Synthetic Experiment

Figure 5. (a) Front input image depicting a basketball player along with the user-selected landmarks. (b) reconstructed model overlaid on the image. (c) novel view of the reconstructed model.

Figure 6. Examples from sports: (a) cyclist, (b) tennis player, and (c) golfer

Figure 7. Examples drawn from the automotive industry

Concluding Remarks
In this study, we investigated the problem of recovering the anthropometry (up to a scale parameter) and pose of the human figure from a single image. Specifically, we have to constrain the estimation process and allow the simultaneous estimation of both anthropometry and pose. Experimental results on a variety of images indicate that our method produces accurate results for a broad class of images.

Acknowledgments
This work was supported in part by the University of Houston Institute for Space Systems Operations.

References
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Publications
Barrón, C. and I. A. Kakadiaris. "Estimating Anthropometry and Pose from a Single Image," Computer Vision and Image Understanding 81.3 (2001): 269-84.
Barrón, C. and I. A. Kakadiaris. "On the Improvement of Anthropometry and Pose Estimation from a Single Uncalibrated Image," Machine Vision & Applications (2003). (In press.)

Presentations
Barrón, C. and I. A. Kakadiaris. "Estimating Anthropometry and Pose from a Single Image," IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Hilton Head Island, SC, June 13-15, 2000. 669-76.
Barrón, C. and I. A. Kakadiaris. "On the Improvement of Anthropometry and Pose Estimation from a Single Uncalibrated Image," IEEE Workshop on Human Motion, Austin, TX, Dec. 7-8 2000. 53-60.

Funding and proposals
"Control and Coordination for Teleoperation in Space Robotics." National Science Foundation, Co-PI: K. Grigoriadis (UH) and C. Layne (UH), June 2003-May 2006, $494,588 (not funded).

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