Method and apparatus for measurement, analysis, characterization, emulation, and translation of perception

• US 5,860,936 A
• Filed: 08/02/1996
• Issued: 01/19/1999
• Est. Priority Date: 08/02/1996
• Status: Expired due to Term

• Current Assignee: David N. Levin
• Original Assignee: David N. Levin
• Inventors: Levin, David N.
• Primary Examiner(s): Hindenburg, Max
• Assistant Examiner(s): WINGOOD, PAMELA LYNN
• Application Number:
  US08/691,615
• Time in Patent Office: 900 Days
• Field of Search: 600/558, 600/557, 600/559
• US Class Current: 600/558
• CPC Class Codes: A61B 5/16 Devices for psychotechnics using teaching or educational appliances G09B1/00 - G09B7/00; Testing reaction times ; Devices for evaluating the psychological state

Abstract

A method is used for measuring perception of an observer using a stimulus output device for presenting stimuli to the observer, and a stimulus manipulation device permitting the observer to modify the presented stimuli by selecting related stimuli from a database of stimuli. The method includes the steps of: a) selecting a sequence of stimuli from the database of stimuli, the stimuli represented as stimulus points contained in a stimulus space; b) determining a reference transformation of a beginning stimulus point of the sequence of stimuli, the reference transformation being presented to the observer by the stimulus output device; c) selecting a limited portion of the sequence of stimuli, the limited portion of the sequence defined between the beginning stimulus point and an ending stimulus point, the limited portion of the sequence being presented to the observer by the stimulus output device; d) determining an observer-defined transformation applied to the ending stimulus point of the limited portion of the sequence such that the observer perceives the observer-defined transformation applied to the ending stimulus point to be perceptually equivalent to the reference transformation applied to the beginning stimulus point; e) selecting a new limited portion of the sequence of stimuli, the new limited portion of the sequence defined by stimulus points selected from the database, at least a portion of the new limited portion having stimulus points different from the stimulus points defining the previously selected limited portion of the sequence; f) determining another observer-defined transformation applied to an ending stimulus point of the new limited portion of the sequence such that the observer perceives the observer-defined transformation to be equivalent to the previous observer-defined transformation; and g) performing steps (e) through (f) until a predetermined number of limited portions of the sequence of stimuli are processed such that the observer perceives each observer-defined transformation to be equivalent to previous observer-defined transformations.

Claims

1. A method for measuring perception of an observer, the method using a stimulus output device for presenting stimuli to the observer, and a stimulus manipulation device permitting the observer to modify the presented stimuli by selecting related stimuli from a database of stimuli, the method comprising the steps of:

a) selecting a sequence of stimuli from the database of stimuli, said stimuli represented as stimulus points contained in a stimulus space;

b) determining a reference transformation of a beginning stimulus point of the sequence of stimuli, the reference transformation being presented to the observer by the stimulus output device;

c) selecting a limited portion of the sequence of stimuli, the limited portion of the sequence defined between the beginning stimulus point and an ending stimulus point, the limited portion of the sequence being presented to the observer by the stimulus output device;

d) determining an observer-defined transformation applied to the ending stimulus point of the limited portion of the sequence such that the observer perceives the observer-defined transformation applied to the ending stimulus point to be perceptually equivalent to the reference transformation applied to the beginning stimulus point;

e) selecting a new limited portion of the sequence of stimuli, the new limited portion of the sequence defined by stimulus points selected from the database, at least a portion of the new limited portion having stimulus points different from the stimulus points defining the previously selected limited portion of the sequence;

f) determining another observer-defined transformation applied to an ending stimulus point of the new limited portion of the sequence such that the observer perceives the observer-defined transformation to be perceptually equivalent to the previous observer-defined transformation; and

g) performing steps (e) through (f) until a predetermined number of limited portions of the sequence of stimuli are processed such that the observer perceives each observer-defined transformation to be perceptually equivalent to previous observer-defined transformations.

2. The method according to claim 1 wherein at least one of the steps of selecting the sequence of stimuli from the database, determining a reference transformation of a beginning stimulus point, selecting a limited portion of the sequence of stimuli, and selecting a new limited portion of the sequence of stimuli, is performed by a computer.

3. The method according to claim 1 wherein at least one of the steps of selecting the sequence of stimuli from the database, determining a reference transformation of a beginning stimulus point, selecting a limited portion of the sequence of stimuli, and selecting a new limited portion of the sequence of stimuli, is performed by the observer using the stimulus manipulation device.

4. The method according to claim 1 wherein the limited portion of the sequence of stimuli is defined to be a single stimulus point such that the beginning stimulus point is the same as the ending stimulus point.

5. The method according to claim 1 wherein the step of determining the observer-defined transformation is performed by the observer using the stimulus manipulation device in response to observing the stimuli presented by the stimulus output device.

6. The method according to claim 1 wherein the step of determining the observer-defined transformation includes the step of recording each observer-defined transformation in a memory of a computer.

7. The method according to claim 1 wherein each new limited portion of the sequence of stimuli is a portion of sequential stimulus points, the new limited portion of the sequence defined to begin at a stimulus point located at the ending stimulus point of a prior portion of the sequence of stimuli.

8. The method according to claim 1 wherein the predetermined number of limited portions of the sequence of stimuli are processed until each stimulus point in the sequence of stimuli in the database of stimuli is processed.

9. The method according to claim 1 wherein the observer perceives a stimulus presented by the stimulus output device, the stimulus output device being at least one of a visual display device displaying the sequence of visual stimuli, and an audio output device producing the sequence of audio stimuli.

10. The method according to claim 9 wherein a computer stores the sequence of visual stimuli to be presented by the visual display device to the observer, the computer storing the sequence in the form of computer files, data comprising the computer files recorded from the group of devices consisting of video cameras, CCD cameras, optical photographic devices, optical microscopes, microwave detectors, x-ray radiography/tomography devices, radio frequency/MRI detecting devices, radioactivity detectors, ultrasonic detectors, and ultrasonic scanning devices.

11. The method according to claim 9 wherein a computer stores the sequence of audio stimuli to be presented by the audio output device to the observer, the computer storing the sequence in the form of computer files, data comprising the computer files recorded from microphones.

12. The method according to claim 1 wherein the step of determining the observer-defined transformation applied to at least one of the beginning stimulus point and the ending stimulus point includes the step of the observer using the stimulus manipulation device to modify the presented stimuli by selecting related stimuli from the database of stimuli, to create a transformation of the stimulus point, said transformation of the stimulus point displayed on the stimulus display device according to observer-defined preferences.

35. The method according to claim 1 further including the steps of:

a) providing a parallel transporter corresponding to a selected observer, said parallel transporter receiving the beginning stimulus point, the observer-defined transformations, and the transformations corresponding to each of the selected limited portions of the sequence of stimuli;

b) mapping the stimulus point at the beginning of each limited portion of the sequence, the observer-defined transformation at the beginning of said limited portion, and the transformation representing said limited portion, onto a resultant transformation of the stimulus point at an end of said limited portion, the parallel transporter performing the mapping, said resultant transformation being the observer-defined transformation perceived by the observer to be perceptually equivalent to the observer-defined transformation at the beginning of said limited portion, said parallel transporter defining an affine connection common to the observer-defined transformations, the transformations representing said limited portions, and the plurality of stimulus points contained in the stimulus space; and

c) storing a plurality of internal parameters of the parallel transporter in a memory of the computer, said internal parameters determining the mapping by the parallel transporter.

36. The method according to claim 35 wherein the parallel transporter is defined by an affine connection Γ

determined by substituting the values of the stimulus points at the beginning of said limited portions, the observer-defined transformations at the beginning and the end of said limited portions, and the transformations representing said limited portions, into the formula;

##EQU42## to derive constraints on said values of the internal parameters of the parallel transporter where x equals the coordinates of the stimulus point at the beginning of the limited portion, h^k is the observer-defined transformation at the beginning of the limited portion, h^k +δ

h^k is the observer-defined transformation at the end of the limited portion, dx_m is the transformation representing the limited portion, and Γ

is affine connection at the coordinate x that depends upon the internal parameters, such that at least one of a linear regression, non-linear regression, and back-propagation technique is used to obtain the values of the internal parameters to best fit the measurements of x, h^k, h^k +δ

h^k, and dx_m.

37. The method according to claim 36 wherein the function Γ

is defined by first J terms in a Taylor series expansion defined by the formula;

##EQU43## where the values of the coefficients Γ

_lm^k (x₀) and ∂

_n1 . . . ∂

_nj Γ

_lm^k (x₀) are the internal parameters of the parallel transporter and x₀ is a selected stimulus point in the database.

38. The method according to claim 36 wherein the function Γ

has a flat-space form according to the formula;

##EQU44## where the N functions ƒ

_a^k (x) for a=1, . . . , N are given by a Taylor series expansion with J terms according to the formula;

##EQU45## where ƒ

_ka (x) is the inverse of ƒ

_a^k (x) , determined by solving the linear equations ##EQU46## where x_0k are the coordinates of any stimulus point in the stimulus space, ƒ

_a^k (x₀) are any N linearly independent vectors with indices k and labels a=1, . . . , N, and where ∂

_n1 . . . ∂

_nj ƒ

_a^k (x₀) are the internal parameters of the parallel transporter.

39. The method according to claim 35 wherein the parallel transporter is formed from a neural network such that the neural network is at least one of a back-propagation model and a self-organizing model.

13. An apparatus for measuring perception of an observer comprising:

a) a stimulus output device for presenting stimuli to the observer;

b) a stimulus manipulation device permitting the observer to modify the presented stimuli by selecting related stimuli from a database of stimuli, said stimuli represented as stimulus points contained in a stimulus space;

c) a computer operatively coupled to the stimulus output device and operatively coupled to the stimulus manipulation device, the computer accessing the database of stimuli;

d) said stimulus manipulation device permitting observer selection of a sequence of stimuli from the database and permitting observer determination of a reference transformation of a beginning stimulus point of the sequence of stimuli, said reference transformation being presented to the observer by the stimulus output device;

e) said stimulus manipulation device permitting observer selection of a limited portion of the sequence of stimuli, the limited portion of the sequence defined between the beginning stimulus point and an ending stimulus point, the limited portion of the sequence being presented to the observer by the stimulus output device;

f) said observer determining an observer-defined transformation applied to the ending stimulus point of the limited portion of the sequence such that the observer perceives the observer-defined transformation applied to the ending stimulus point to be perceptually equivalent to the reference transformation applied to the beginning stimulus point; and

g) said computer recording the sequence of stimuli and each observer-defined transformation in a memory of the computer.

14. A method for measuring perception of an observer, the method using a stimulus output device for presenting stimuli to the observer, and a stimulus manipulation device permitting the observer to modify the presented stimuli by selecting related stimuli from a database of stimuli, the method comprising the steps of:

a) selecting a plurality of stimulus points from the database of stimuli, said stimulus points contained in a stimulus space;

b) presenting a selected stimulus point to the observer by the stimulus output device, the selected stimulus point selected from the plurality of stimulus points;

c) determining a first transformation of the selected stimulus point, the first transformation being presented to the observer by the stimulus output device;

d) determining a second transformation of the selected stimulus point, the second transformation being presented to the observer by the stimulus output device;

e) determining an observer-defined transformation applied to the stimulus point resulting from application of the second transformation to the selected stimulus point, such that the observer perceives the observer-defined transformation to be perceptually equivalent to the first transformation of the selected stimulus point;

f) providing another selected stimulus point, said another selected stimulus point selected from the plurality of stimulus points; and

g) performing steps (b) through (f) until a predetermined number of the plurality of stimulus points are processed.

15. The method according to claim 14 wherein at least one of the steps of selecting a plurality of stimulus points, determining a first transformation of the selected stimulus point, and determining a second transformation of the selected stimulus point, is performed by a computer.

16. The method according to claim 14 wherein at least one of the steps of selecting a plurality of stimulus points, determining a first transformation of the selected stimulus point, and determining a second transformation of the selected stimulus point, is performed by the observer using the stimulus manipulation device.

17. The method according to claim 14 wherein the selected stimulus point of the plurality of stimulus points is selected according to at least one of a random and a pseudo-random method.

18. The method according to claim 14 wherein the stimulus output device presents stimulus points and transformations of the stimulus points to the observer, the stimulus output device being at least one of a visual display device displaying a sequence of visual stimuli, and an audio output device producing a sequence of audio stimuli.

19. The method according to claim 14 wherein the step of determining the observer-defined transformation includes the step of recording each observer-defined transformation in a memory of a computer.

20. The method according to claim 14 wherein the step of determining the second transformation of the selected stimulus point is performed by the observer such that the second transformation is perceived by the observer to be perceptually different from the first transformation of the selected stimulus point.

21. The method according to claim 14 wherein the step of determining the observer-defined transformation applied to the stimulus point includes the step of the observer using the stimulus manipulation device to modify the presented stimuli by selecting related stimuli from the database of stimuli, to create a transformation of the stimulus point displayed on the stimulus display device, according to observer-defined perceptual preferences.

22. A method for measuring perception of an observer, the method using a stimulus output device for presenting stimuli to the observer, and a stimulus manipulation device permitting the observer to modify the presented stimuli by selecting related stimuli from a database of stimuli, the method comprising the steps of:

a) selecting a beginning stimulus point from the database of stimuli, the stimulus output device presenting the beginning stimulus point to the observer, the database containing a plurality of stimulus points defining a stimulus space;

b) specifying N-number of linearly independent vectors, each of the linearly independent vectors representing a transformation of the beginning stimulus point, the value of N equal to a predetermined the number of dimensions in a manifold representing the stimulus space;

c) determining M-number of transformation vectors, each transformation vector formed as a weighted combination of said N number of linearly independent vectors, each of said M-number of transformation vectors being transformations of the beginning stimulus point and having a common vertex located at the beginning stimulus point and diverging from the common vertex, each of said M-number of transformation vectors defined by a sequence of stimulus points starting at the beginning stimulus point and terminating at an ending stimulus point, each of said M-number of transformation vectors being presented to the observer by the stimulus output device; and

d) determining a sequence of P-number of transformation vectors forming a connected path, each transformation vector determined by the observer such that the observer perceives each transformation vector in said sequence of P-number of transformation vectors to be perceptually equivalent to at least one of the M-number of transformation vectors having a common vertex located at the beginning stimulus point.

23. The method according to claim 22 wherein at least one of the steps of selecting the beginning stimulus point, specifying the N-number of linearly independent vectors, and determining the M-number of transformation vectors having a common vertex located at the beginning stimulus point, is performed by a computer.

24. The method according to claim 22 wherein at least one of the steps of selecting the beginning stimulus point, specifying the N-number of linearly independent vectors, determining the M-number of transformation vectors having a common vertex located at the beginning stimulus point, and determining the sequence of P-number of transformation vectors, is performed by the observer using the stimulus manipulation device.

25. The method according to claim 22 wherein the step of determining each of the P-number of observer-defined transformation vectors includes the step of recording each transformation vector in a memory of a computer.

26. The method according to claim 22 wherein at least one of the steps of determining the M-number of transformation vectors and determining the sequence of P-number of transformation vectors includes the step of the observer using the stimulus manipulation device to modify the presented stimuli by selecting related stimuli from the database of stimuli, to create a transformation of the stimulus point displayed on the stimulus display device, according to observer-defined perceptual preferences.

27. The method according to claim 22 wherein each of the M-number of transformation vectors of the beginning stimulus point are equal to each other such that each transformation vector has a same length and a same direction.

28. The method according to claim 22 wherein each of the M-number of transformation vectors of the beginning stimulus point is proportional to the other of the M-number of transformation vectors such that each transformation vector has a same direction.

29. The method according to claim 22 wherein a sequential display of stimulus points along each of the P-number of observer-defined transformation vectors is adjusted by the observer using the stimulus manipulation device so that a rate of change of the stimulus points along each transformation vector is perceived by the observer to be constant, said rate of change being recorded in a memory of a computer.

30. The method according to claim 22 wherein the sum of the M-number of transformation vectors of the beginning stimulus point is equal to zero.

31. A method for measuring and characterizing perception of a selected observer, the method using a stimulus output device for presenting stimuli and transformations of the stimuli to the observer, and a stimulus manipulation device permitting the observer to modify the presented stimuli by selecting related stimuli from a database of stimuli, to create a plurality of transformations of selected stimulus points, the stimulus points contained in the database of stimulus points defining a stimulus space, the plurality of transformations recorded in a memory of a computer, the method including the steps of:

a) providing a parallel transporter corresponding to the selected observer, said parallel transporter receiving a beginning stimulus point, and first and second transformations of a beginning stimulus point,

b) mapping the beginning stimulus point and the first and second transformations onto a resultant transformation of a destination stimulus point, said parallel transporter performing the mapping, said destination stimulus point formed by application of the second transformation to the beginning stimulus point, said resultant transformation perceived by the observer to be perceptually equivalent to the first transformation of the beginning stimulus point, and said parallel transporter defining an affine connection common to the first transformation, the second transformation, the resultant transformation, and the plurality of stimulus points included in the stimulus space; and

c) storing a plurality of internal parameters of the parallel transporter in a memory of the computer, said internal parameters determining the mapping.

32. The method according to claim 31 wherein the parallel transporter is formed from a neural network such that the neural network is at least one of a back-propagation model and a self-organizing model.

33. The method according to claim 31 wherein the step of storing the internal parameters determining the mapping includes calculating an arithmetic mean and a standard deviation of a distribution of each of the internal parameters to characterize a perceptual performance of at least one of an individual observer and a group of observers, each set of internal parameters in said distribution corresponding to a mapping associated with an observer in at least one of said individual observer and a group of observers.

34. The method according to claim 31 wherein the step of storing the internal parameters is performed for a plurality of observers such that differences among the perceptions of the plurality of observers are represented by statistical measures of differences between the distributions of internal parameters corresponding to each of the observers.

40. The method according to claim 31 further including the steps of

(a) selecting a connected sequence of stimulus points defining a stimulus path;

(b) selecting a limited portion of the stimulus path defined between a beginning stimulus point and an ending stimulus point

(c) selecting N-number of standard linearly-independent transformations of the beginning stimulus point, the N-number of standard transformations representing a basic stimulus change for generating a description of the stimulus path;

(d) applying the parallel transporter corresponding to the observer to the beginning stimulus point, applying the parallel transporter corresponding to the observer to the transformation representing the limited portion of the stimulus path, and applying the parallel transporter corresponding to the observer to each of the N-number of standard transformations to create resultant standard transformations of the ending stimulus point;

(e) applying the parallel transporter corresponding to the observer to the ending stimulus point, applying the parallel transporter corresponding to the observer to each of said resultant standard transformations at the ending stimulus point, and applying the parallel transporter corresponding to the observer to a transformation representing another limited portion of the connected sequence of stimuli, said another limited portion beginning at the ending stimulus point of the previous limited portion, to generate resultant standard transformations of the ending stimulus point of said another limited portion;

(f) repeating step (e) until a predetermined number of stimulus points of the stimulus path are processed, to generate standard transformations at each stimulus point along the stimulus path, said resultant standard transformations being perceived by the observer to be perceptually equivalent to the N-number of standard transformations of the beginning stimulus point; and

(g) expressing each transformation representing each limited portion of the stimulus path as a numerically-weighted linear combination of the standard transformations of the stimulus point at the beginning of the limited portion, said numerically-weighted linear combination of the standard transformations having a plurality of numerical weights representing the observer'"'"'s description of the limited portion in terms of the application of the linear combination of the standard transformations.

41. The method according to claim 40 wherein the connected sequence of stimulus points and the N-number of standard transformations of the beginning stimulus point are selected by the computer.

42. The method according to claim 40 wherein the connected sequence of stimulus points and the N-number of standard transformations of the beginning stimulus point are selected by the operator using the stimulus manipulation device.

43. The method according to claim 40 wherein the numerical weights are the numbers defined by ds(i)_a that are computed according to the formula:

##EQU47## where ##EQU48## where x_k (i) are the coordinates of the stimulus points of the connected sequence of stimulus points, where h_a^k are the N-number of standard transformations of the beginning stimulus point x_rk, where h_ka is derived by solving the linear equations ##EQU49## where dx(i)_k are the transformations representing said limited portions, and where Γ

_lm^k (x) is the affine connection given by the equation;

##EQU50## where the values of the coefficients Γ

_lm^k (x₀) and ∂

_n1 . . . n_j Γ

_lm^k (x₀) are the internal parameters of the parallel transporter of the observer, x₀ is a selected stimulus point in the database, and where terms with j>

1 are set equal to zero.

44. The method according to claim 43 wherein Γ

_lm^k (x) is given by the formula ##EQU51## where the N functions ƒ

_a^k (x) for a=1, . . . , N are given by a Taylor series expansion with J terms according to the formula;

##EQU52## where ƒ

_ka (x) is the inverse of ƒ

_a^k (x), determined by solving the linear equations ##EQU53## where x_0k are the coordinates of any stimulus point in the stimulus space, ƒ

_a^k (x₀) are any N linearly independent vectors with indices k and labels a=1, . . . , N, and where ∂

_n1 . . . ∂

_nj ƒ

_a^k (x₀) are the internal parameters of the parallel transporter of the observer where terms with j>

2 are set equal to zero.

45. The method according to claim 40 in which the parallel transporter is given by a neural network implemented in at least one of a hardware configuration and a software configuration.

46. The method according to claim 31 further including the steps of

(a) selecting a stimulus point in the stimulus space, selecting N-number of linearly-independent vectors representing N-number of standard transformations of the selected stimulus point, and selecting an ordered sequence of Q-number of sets of transformation vector weights, each set in the sequence having N-number of weights;

(b) applying a transformation to the selected stimulus point, said transformation being a first linear combination of the N-number of standard transformations, said N-number of weights in the linear combination being equal to a first set of the ordered sequence of Q-number sets of weights, said transformation resulting in a first destination stimulus point;

(c) applying the parallel transporter corresponding to the selected observer to the selected stimulus point, applying the parallel transporter corresponding to the selected observer to each of the N-number of standard transformations, and applying the parallel transporter corresponding to the selected observer to the transformation equal to said linear combination of N-number of standard transformations, to produce N-number of standard transformations at the first destination stimulus point;

(d) applying a next transformation to said first destination stimulus point, said next transformation being the next linear combination of said N-number of standard transformations at said first destination stimulus point, the N-number of weights in said the next linear combination being equal to the next set of the ordered sequence of sets of weights, said next transformation resulting in another destination stimulus point;

(e) applying the parallel transporter corresponding to the selected observer to said first destination stimulus point, applying the parallel transporter corresponding to the selected observer to each of said N-number of standard transformations at said first destination stimulus point, and applying the parallel transporter corresponding to the selected observer to the transformation equal to said next linear combination of N-number of standard transformations at said first destination stimulus point, to produce N-number of standard transformations at said another destination stimulus point;

(f) repeating steps (d) and (e) using said another destination stimulus point in place of said first destination stimulus point to generate an ordered set of Q-number of transformations, each said transformation being equal to a linear combination of the N-number of standard transformations at the previously produced destination stimulus points, the linear combination having weights equal to at least one of the ordered set of Q-number of selected sets of weights, said Q-number of transformations forming a connected path in the stimulus space beginning at the selected stimulus point, the order of the Q-number of transformations being the same as the order of the Q-number of sets of weights; and

(g) said connected path in the stimulus space being described by the selected observer in terms of the successive application of the Q-number of transformations equal to the linear combinations of the N-number of standard transformations, and the sets of weights in said linear combinations being equal to the sets of weights in the Q-number of sets of selected transformation vector weights.

47. The method according to claim 46 wherein at least one of the selected stimulus point in the stimulus space, the N-number of vectors representing the N-number of standard transformations of the selected stimulus point, and the ordered sequence of Q-number of sets of transformation vector weights, are chosen by the computer.

48. The method according to claim 46 wherein at least one of the selected stimulus point in the stimulus space, the N-number of vectors representing the N-number of standard transformations of the selected stimulus point, and the ordered sequence of the Q-number of sets of transformation vector weights, are chosen by the selected observer.

49. The method according to claim 46 wherein the sequence of destination stimulus points y_k (i) is determined by substituting the numbers ds(i)_a, into the formula:

##EQU54## where y_k (0) is the selected stimulus point, g_a^k are vectors representing the selected N-number of standard transformations of the selected stimulus point, ##EQU55## ds(j)_a are the ordered sequence of Q-number of sets of vector transformation weights, where Γ

_lm^k (x) is the affine connection given by the formula;

##EQU56## where the values of the coefficients Γ

_lm^k (x₀) and ∂

_n1 . . . ∂

_nj Γ

_lm^k (x₀) are the internal parameters of the parallel transporter corresponding the selected observer and x₀ =y₀ is a selected stimulus point in the database, and where terms with j>

1 are set equal to zero.

50. The method in claim 49 in which Γ

_lm^k (x) is the affine connection given by the formula;

##EQU57## where the N functions ƒ

_a^k (x) for a=1, . . . , N are given by a Taylor series expansion with J terms according to the formula;

##EQU58## where ƒ

_ka (x) is the inverse of ƒ

_a^k (x), determined by solving the linear equations ##EQU59## where x_0k =y_0k is a stimulus point in the stimulus space, ƒ

_a^k (x₀) are any N linearly independent vectors with indices k and labels a=1, . . . , N, and where ∂

_n1 . . . ∂

_nj ƒ

_a^k (x₀) are the internal parameters of the parallel transporter corresponding the selected observer where terms with j>

2 are set equal to zero.

51. The method in claim 46 wherein the parallel transporter is implemented in at least one of a hardware configuration and a software configuration, as a neural network.

52. The method in claim 46 including the steps of

(a) selecting another stimulus space having a plurality of stimulus points, the stimulus points of the another stimulus space being at least one of identical to the stimulus points of the stimulus space and different from the stimulus points of the stimulus space;

(b) designating an another observer, the another observer being one of at least the selected observer and a new observer;

(c) deriving the ordered set of Q-number of sets of transformation vector weights from a selected sequence of stimulus points in said another stimulus space by performing the steps of;

(c1) selecting a connected sequence of stimulus points defining a stimulus path in the another stimulus space;

(c2) selecting a limited portion of the stimulus path defined between a beginning stimulus point and an ending stimulus point;

(c3) selecting N-number of standard linearly-independent transformations of the beginning stimulus point, the N-number of standard transformations representing basic stimulus changes for generating a description of the stimulus path;

(c4) applying the parallel transporter corresponding to the another observer to the beginning stimulus point, applying the parallel transporter corresponding to the another observer to the transformation representing the limited portion of the stimulus path, and applying the parallel transporter corresponding to the another observer to each of the N-number of standard transformations to create resultant standard transformations of the ending stimulus point;

(c5) applying the parallel transporter corresponding to the another observer to the ending stimulus point, applying the parallel transporter corresponding to the another observer to each of said resultant standard transformations at the ending stimulus point, and applying the parallel transporter corresponding to the another observer to a transformation representing another limited portion of the connected sequence of stimuli, said another limited portion beginning at the ending stimulus point of the previous limited portion, to generate resultant standard transformations of the ending stimulus point of said another limited portion;

(c6) repeating step (c5) until a predetermined number of stimulus points of the stimulus path are processed, to generate standard transformations at each stimulus point along the stimulus path, said resultant standard transformations being perceived by the another observer to be perceptually equivalent to the N-number of standard transformations of the beginning stimulus point;

(c7) expressing each transformation representing each limited portion of the stimulus path as a numerically-weighted linear combination of the standard transformations of the stimulus point at the beginning stimulus point of the limited portion, said numerically-weighted linear combinations of the standard transformations having a plurality of numerical weights representing the another observer'"'"'s description of the connected sequence of stimuli in terms of the successive application of the standard transformations and

(d) said ordered sequence of the Q-number sets of transformation vector weights generating a sequence of stimulus points in the stimulus space such that the selected observer of the generated sequence of stimulus points perceives the generated stimulus points in the same way as the another observer of the selected sequence of stimulus points in the another stimulus space perceives the selected stimulus points in the another stimulus space.

53. The method according to claim 52 wherein the plurality of stimulus points in the another stimulus space is at least one of visual and audio stimulus points, and the plurality of stimulus points in the stimulus space is at least one of visual and audio stimulus points, said stimulus points generated in the stimulus space so that the selected observer describes the at least one of the generated visual and audio stimulus points in the same way as the another observer describes the at least one of the visual and audio stimulus points in the another stimulus space.