Contains research objectives and reports on one research project.National Institutes of Health (Grant MP-4737)National Science Foundation (Grant G-16526)Lincoln Laboratory, Purchase Order DDL B-00306U. S. ArmyU. S. NavyU. S. Air Force under Air Force Contract AF19(604)-740

Mason, Samuel J.

Troxel, Donald E.

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XXIX. SENSORY AIDS RESEARCH*Prof. S. J. Mason W. G. Kellner V. K. PrabhuD. A. Cahlander D. G. Kocher M. M. SchiffmanJ. Dupress W. B. Macurdy D. E. TroxelM. A. PillaRESEARCH OBJECTIVESThe long-range objective of this group is to develop useful artificial sensory mech-anisms for people afflicted with sensory deprivation - in particular, vision or vision-plus -audition deprivation.From our present viewpoint, one important long-range technical problem is todevelop "sensor" systems that extract small amounts of information from, for example,an optical image, and "display" systems that feed the processed information into theskin or the ear. Among the sensor-extractor systems that are being considered are:(a) edge tracers for the extraction of a "cartoon" from an image; (b) various two-dimensional autocorrelation schemes for extracting position-independent and size-independent qualities of an image; and (c) color sensors for the exploitation of colorinformation in an image. A multiplicity of different "filters" that simultaneously extractdifferent crude aspects of an image may turn out to be a useful sensor-system for amobility aid and perhaps for a reading machine.S. J. MasonA. BOUNDS ON INFORMATION TRANSFER FOR A PARTICULAR CLASSOF CHANNELSIn this report we discuss the derivation of bounds on information transfer for a modelchannel that is useful for characterizing many sensory-aid systems. Experimentalmeasurements have substantiated the validity of the model channel and the derivedbounds.This model channel is proposed in an attempt to make a first-order approximationto an actual sensory-aid channel. Upper and lower bounds have been derived in termsof two variables defined from the channel matrix. An experimentally measured channelwill be compared with the model channel, and the calculated information transfer com-pared with the derived upper and lower bounds.1. Description of the Class of ChannelsMost sensory aids can be characterized by information channels whose inputs arestimuli and whose outputs are the responses. The majority of the channels associatedwith useful sensory aids will have terms that are nearly unity on the major diagonal,This work was supported in part by the National Institutes of Health (GrantMP-4737); in part by the National Science Foundation (Grant G-16526); and in part byPurchase Order DDL B-00306 with Lincoln Laboratory, a center for research operatedby Massachusetts Institute of Technology, with the joint support of the U. S. Army, Navy,and Air Force under Air Force Contract AFl9(604)-7400.329(XXIX. SENSORY AIDS RESEARCH)P 0 r -PO P 0 r2OI I-PIIr PI-PO- - P rLFig. XXIX-1. Model channel.since the rather limited memory span of the human receiver precludes the use of com-plex codes. Furthermore, if the channel is large (32 stimuli, or larger), the dominantfeatures will be an abundance of zeros for the off-diagonal terms and a smattering ofrather small error terms.A first-order approximation to such a channel is given in Fig. XXIX-1. Each of themajor diagonal terms of this conditional probability matrix is taken to be equal to p.The off-diagonal terms in each row may be either zero or a constant. If r. is defined.thas the number of nonzero off-diagonal terms in the 1 row, then this constant is(1-p)/r . The order of the matrix is L, so that we have 1 r. < L - 1. Noassumptions have been made as to the probability that any off-diagonal term willbe zero. The zeros inserted in Fig. XXIX-1 are there only to serve as an example.Although a more general class of channels could be defined by allowing each majordiagonal term to be different, it is of interest to investigate this simpler channel,first.2. Derivation of the Lower BoundFanol has derived a lower bound on the equivocation, which isH(XIY) < H(e) + P(e) log (L-1) (1)whereH(e) = -P(e) log P(e) - (l-P(e)) log (1-P(e)),P(e) is the probability of error, and L is the order of the channel matrix. We shallassume that the input probability distribution p(x) is uniform for the derivation of bothbounds so that330(XXIX. SENSORY AIDS RESEARCH)H(X) = log L. (2)The information transfer can be expressed asI(X;Y) = H(X) - H(XIY). (3)Substituting (1) and (2) in (3), we obtain a lower bound ofI(X;Y) > log L - P(e) log (L-l) - H(e). (4)If L is large and P(e) reasonably small, the lower bound given in (4) is not decreasedsignificantly if we apply the inequality log L > log (L-l) to (4) to obtainI(X;Y) > (1-P(e)) log L - H(e). (5)The lower bound (5) is subject to the rather interesting interpretation that the transmittedinformation is at least as great as the percentage of the offered information that is iden-tified correctly, minus the information required to specify when an error was made.Note that the bounds (4) and (5) hold for all of the channels when the input probabilitydistribution is uniform; that is, they do not require the rather special properties of theproposed model channel. However, if the model channel is considered for the specialcase r i = L - 1, then the equality sign holds for the bound given by (4).3. Derivation of the Upper BoundSince the input p(x) is uniform, Eq. 2 still holds, and we always haveH(Y) < log L. (6)Pij PijH(XY) will now be evaluated as the negative sum over the whole matrix of -L log LConsider, first, the sum on row i.Pij PijS. 1  L log Lp p 1-p 1-p= log + r i Lr. log Lr.1 1p p 1-p 1-p 1-pL log + L log L - L log r iThen,p l-p l-p-H(XY) = S = p log L + (l-p) log L L log r. (7)i i331(XXIX. SENSORY AIDS RESEARCH)Since the joint probability of error and any input combined is (l-p)/L, the totalprobability of error is 1 - p = P(e). Substitution of P(e) in Eq. 7 yieldsP(e) (8)H(XY) = log L + H(e) - L log r . (8)iThe information transfer can be expressed asI(X;Y) = H(X) + H(Y) - H(XY). (9)Substituting (2), (6), and (8) in (9), we obtainP(e)I(X;Y) < log L - H(e) - L log r . (10)iThe equality sign holds when H(Y) = log L. We know that H(Y) does not vary rapidlyas p(y) departs from a uniform distribution, and we also know that p(y) is reasonablyuniform because of the dominance of the major diagonal terms. Thus we should expectthat the actual transmitted information will normally be rather close to this upper bound.If we consider the special case ri = L - 1, then the equality sign holds in (10), and wenote also that the upper and lower bounds are equal.4. Comparison of an Actual Sensory-Aid Channel with the Model ChannelHuman subjects were presented with input stimuli drawn with uniform probabilityfrom a population of 64. The actual equipment used has been described in detail in theauthor's thesis. 2 To compare this channel with the proposed channel, a series of testswas analyzed with the aid of the TX-0 computer. A major premise for the model channelis that all of the major diagonal terms are equal. The manner in which the actual channeldiffers from the model is illustrated by Fig. XXIX-2 in which the ordinate representsC,0 150n10-nO,, 5z 0 .04 .08 .12 .16 .20 .24 .281- Pi = P (eli)Fig. XXIX-2. Histogram of row P(e).332(XXIX. SENSORY AIDS RESEARCH)20 -15-10 -504 6IFig. XXIX-3.30252015105n' I I I I I0Fig. XXIX-4.12I 1 bDistribution of r..1.2 .4 .6Deviation from uniformityI I W U i.8 1,0of off-diagonal terms.the number of rows falling within the indicated range of probability of error. The dataof Fig. XXIX-2 are based on the analysis of responses to 3936 stimuli. The average,P(e), was calculated to be 0. 075.The most significant feature of the model, however, is the assumption that all of1 - piithe off-diagonal terms in a row are either zero or P. The test data indicate thatr.1there is, indeed, a large number of zeros, as indicated by Fig. XXIX-3 in which theordinate represents the number of rows having the indicated number of nonzeroterms (ri). th -Since each nonzero off-diagonal term in the it h row should equal r , the following1term can be taken as a measure of the difference between the actual and model channels.3331 1 _ I iIf.(XXIX. SENSORY AIDS RESEARCH)1 - piinonzero pij r.j, i j 1D. =1 -pIii(i)A plot of the frequency of occurrence of values of this measure 1. ,iven in 7ig. YXl,'-,i.5. Comparison of an Experimental Measurement of I(X;Y) with the U-p -*and Lower BoundsThe appropriate comparisons for a channel with L = 64Fig. XXIX-5.UPPER BOUNDLOWER BOUNDO UPPER BOUNDSCALCULATED I (XY)A LOWER BOUNDI I I I -:A. illustrated by0 .02 .04 .06 .08 .10 P(e)Fig. XXIX-5. Comparison of derived bounds and computedinformation transfers.The upper bound that is plotted is given by (10), and the lower bound by (4). As isexpected, the actual information transfer is very close to the upper bound.D. E. TroxeiReferencesI. R. M. Fano, Transmission of Information (The M. I. T.Mass., and John Wiley and Sons, Inc., New York, 1961), p. 187.Press, Cambridge,2. D. E. Troxel, Tactile Stimulation of the Fingers as an Information Input to theBlind, S. M. Thesis, Department of Electrical Engineering, M. I. T., September 1960.3346CC)- 5DC,Cr 4I-a ,

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