In this chapter, we will perform a more detailed analysis of the coloring algorithm given in section 1.1.1. To do this, we need to esh out the notions of environment, behavior, and behavioral equivalence. Throughout the paper, we will use a state space formalization of the environment. In this section, we will only be concerned with the environment states themselves, not with the possible transitions between them. We will also ignore the internal state of the agent. In section 7, we will add dynamics and internal state.
LetW be the set of possible world states. We will model environments as subsets of W (we will consider other state spaces in section 7.1). Thus E = 2W. Habitats, which we have dened as sets of environments, will then eectively just be (larger) regions of the state-space themselves. Habitat constraints, constraints over possible habitats, are then also eectively just subsets of W.
Since we are ignoring dynamics and internal state, we will consider only those per-ceptual systems that give information about the instantaneous world state. Thus a perceptual system is a mechanism that has an identiable output with an identiable set of possible states S such that the state of the output is causally determined by the state of the world. Eectively, the perceptual system computes a function from W to S. We will call that function the information type that the perceptual system computes. We will say that two perceptual systems are behaviorally equivalent if they compute the same information type. An information type is nite if its range is nite. Note that information types should not be confused with the concept of information as inverse probability used in classical information theory (see Hamming [43]). While the two are certainly compatible, classical information theory is con-cerned with measuring quantities of information, whereas our concern here is with distinguishing among dierent kinds of information.
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6.1 Derivability and equivalence
Often what is interesting about an information type is what other information types can be computed from it. We will say that one information typeI0:W!S0isderivable from another, I:W!S, if there exists a derivation function f for which I0=fI. I1
and I2 are equivalent (writtenI1 I2) if they are interderivable.
The range of an information type is irrelevant to derivability; We can arbitrarily rename the elements of its range without changing what can be derived from it. Thus what really matters is the partition PI it induces on the world states:
PI =fAW j x;y2A,I(x) = I(y)g
The elements of the partition are the maximal sets of world states that are indistin-guishable given onlyI.
Lemma 1
The following statements are equivalent:1. I1 and I2 are equivalent, that is, interderivable.
2. X is derivable fromI1 i it is derivable from I2, for all X. 3. The partitions PI1 and PI2 are identical.
4. I1 and I2 dier only by a bijection (a 1:1 onto mapping).
Proof: First, recall that derivability is transitive. Now suppose that (1) holds. Then if X is derivable from I2 and I2 is, by (1), derivable from I1, then X is derivable fromI1. Similarly, ifX is derivable from I1, then it must be derivable from I2. Thus (1) implies (2). Now assume (2). Since derivability is reexive, I1 and I2 must be interderivable, and so, equivalent. Now suppose that (3) is false. Then there must be a pair of world states w and w0 such that either I1(w)=I1(w0) but I2(w)6=I2(w0) or vice versa. Without loss of generality, we may assume that it is I2 which diers.
By interderivability, there is a derivation function f such that I2 =fI1 and so f(I1(w))6=f(I1(w0)), which contradicts the assumption thatI1(w)=I1(w0). Thus (3) must hold, and so (2) implies (3). Now note that for every information typeI:W!S, there is a trivial bijection betweenS and PI given by the rules7!I;1(s) for all s2S.
Since the inverses and compositions of bijections are themselves bijections, (4) must follow from (3). Finally, note that (1) follows trivially from (4) since the 1:1 mapping would itself be the derivation function. 2
We will say that I and I0 are conditionally identical given C (written I =C I0) if I(w) = I0(w) for all w 2 C. Note that I W= I and that I1 C=1 I2 and I2 C=2 I3
impliesI1 C1\=C2 I3. Finally, we will say that two perceptual systems are behaviorally equivalent if they compute the same information type and conditionally equivalent given C if their information types are conditionally identical given C.
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6.2 Unconditional equivalence transformations
We will use a single box labeled with an information type I
) I !
to represent a perceptual system that (somehow) computes I. The double arrow is meant to represent a connection to the environment. When we want to expose the internal structure in the system, we will use single arrows to represent connections wholly within the system. Thus
) I ! f ! g !:::
represents a system which rst computes I and then applies the transformations f, g, :::to it. Finally, we will denote predicates with a \?", thus
) I ! > T? !
denotes a system which outputs true when I(w) > T, and false otherwise. These diagrams inherit the associativity of function composition:
) f I ! g ! ) I ! f ! g ! ) I ! gf !
and so a simple optimization, which we might call \folding" (after constant-folding in compiler optimization), is the replacement of a series of computations with a single computation:
) I ! f ! ) fI !
While folding is formally trivial, the technique is quite useful in practice. For example, rather than rst computing pose information from an image and then run-ning a grasp planner to choose a grasping strategy, one might use the object's 2D appearance to index directly into a table of grasping strategies. To abuse our notation somewhat, we might say that
) recognize model ! compute pose ! plan strategy ! could be reduced to
) recognize view ! look up strategy ! 62
One example of an optimizing transformation is what might be called \decalibra-tion." Estimating precise parameters such as depth can be dicult and can require precise sensor calibration. Often all that is done with this information is to compare it to some empirical threshold. For example, we might estimate the distance to an obstacle to decide whether we should swerve around it, or whether it is too late and we must brake to avoid collision. Generally, the designer arbitrarily chooses a thresh-old or determines it experimentally. In such situations, we can use any mechanism that computes distance in any units, provided that we correct the threshold.
Lemma 2
(Decalibration) For any information type I:W!R (R is the set of real numbers) and any strictly increasing function f:R!R,) I ! > T? ! ) fI ! > f(T)? ! Proof: By associativity, the right hand side is equivalent to
) I ! (> f(T)?)f !
and for allx, f(x) > f(T) i x > T, thus (> f(T)?)f = (> T?).2
Decalibration allows a calibrated mechanism to be replaced with an uncalibrated mechanism, in certain cases.
6.3 Transformations over simple vision systems
The coloring algorithm used image plane height to discriminate depth and a texture detector to nd obstacles. In the remainder of this chapter, we will derive sucient conditions for the validity of these techniques. We will show that image plane height is a strictly increasing function of object depth, provided the object rests on the oor and its projection into the oor is contained within its region of contact with the oor.
We will also show that for oors whose surface markings have no spatial frequencies below! and which are viewed from a distance of at least d, any low pass lter with a passband in the region (0;d!) can be used to discriminate between objects and oor.
First, we need to dene our coordinate systems, one camera centered, in which the forward direction direction (
z
) is the axis of projection, and the other body-centered, in which the forward direction (Z
) is the direction of motion (see gure 6.1). We will assume that the camera faces forward, but somewhat down, and so the camera- and body-centered frames share their left/right axis, which we will callX
. We will call the up/down axes for the camera- and body-centered systemsy
andY
, respectively.We will assume that the ground plane lies at Y = 0. We will denote the image with range set X by I(X) so the b/w images are I(R) and the color images are I(R3).
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Ground plane center of body
coordinate system z y
Z Y
h
P camera
optic axis
Figure 6.1: A camera viewing a ground plane. The
X
axis (not shown) comes out of the page and is shared by the the camera and body coordinate frames. The body coordinate frame is formed byX
,Y
andZ
, the camera frame, byX
,y
andz
.z
is also the axis of projection, or optic axis, of the camera.h
is the height of the camera andP
is an arbitrary point on the ground plane.The projection process can be specied in either of these coordinate frames. In camera-centered coordinates, the projection process maps a point (X;y;z) in the world to a point (fX=z;fy=z) on the image plane, where f is the focal length of the lens. In the body-centered coordinate system, projection is best expressed with vector algebra. A point
P
in the world will be projected to the image plane pointp
= f(P
;h
)z
(P
;h
)(These are 3D coordinates; the 2D coordinates are obtained by projecting it onto the image plane axes