Peter Meier et al.

Patent Trials and Appeals Board

Jan 3, 2020

15273591 - (D) (P.T.A.B. Jan. 3, 2020)

UNITED STATES PATENT AND TRADEMARK OFFICE
UNITED STATES DEPARTMENT OF COMMERCE
United States Patent and Trademark Office
Address: COMMISSIONER FOR PATENTS
P.O. Box 1450
Alexandria, Virginia 22313-1450
www.uspto.gov
APPLICATION NO. FILING DATE FIRST NAMED INVENTOR ATTORNEY DOCKET NO. CONFIRMATION NO.
15/273,591 09/22/2016 Peter Meier P28504USC1
(119-1119USC1)
2478
61947 7590 01/03/2020
Apple - Blank Rome
c/o Blank Rome LLP
717 Texas Avenue, Suite 1400
HOUSTON, TX 77002
EXAMINER
GILES, NICHOLAS G
ART UNIT PAPER NUMBER
2699
NOTIFICATION DATE DELIVERY MODE
01/03/2020 ELECTRONIC
Please find below and/or attached an Office communication concerning this application or proceeding.
The time period for reply, if any, is set in the attached communication.
Notice of the Office communication was sent electronically on above-indicated "Notification Date" to the
following e-mail address(es):
houstonpatents@blankrome.com
mbrininger@blankrome.com
PTOL-90A (Rev. 04/07)

UNITED STATES PATENT AND TRADEMARK OFFICE
BEFORE THE PATENT TRIAL AND APPEAL BOARD
Ex parte PETER MEIER and THOMAS SEVERIN
Appeal 2018-007426
Application 15/273,591
Technology Center 2600
BEFORE ELENI MANTIS MERCADER, DENISE M. POTHIER, and
STEVEN M. AMUNDSON, Administrative Patent Judges.
POTHIER, Administrative Patent Judge.
DECISION ON APPEAL

Appeal 2018-007426
Application 15/273,591

2
STATEMENT OF THE CASE
Pursuant to 35 U.S.C. § 134(a), Appellant1,2 appeals from the
Examiner’s decision to reject claims 22–41. Appeal Br. 7. Claims 1–21
have been canceled. Id. at 14 (Claims App.). We have jurisdiction under
35 U.S.C. § 6(b).
We AFFIRM.
CLAIMED SUBJECT MATTER
The claims are directed to “a method and of system capable of
providing multimedia information to a user at reduced battery consumption.”
Spec. ¶ 1.
Claim 22 is reproduced below:
22. An information system, comprising:
a camera;
a processor operatively coupled to the camera;
a device operatively coupled to the processor; and
a memory device operatively coupled to the camera, the
processor and the device, the memory device comprising
instructions executable by the processor to:
obtain, in a low-power mode of the information
system, an image captured by the camera;
extract, in the low-power mode, a first feature of
an object in the image;
generate, in the low-power mode, a higher level
descriptor of the first feature;

1 We use the word Appellant to refer to “applicant” as defined in 37 C.F.R.
§ 1.42(a) (2016). Appellant identifies the real party in interest as Apple, Inc.
Appeal Br. 3.
2 Throughout this opinion, we refer to the Final Action (Final Act.) mailed
August 14, 2017, the Appeal Brief (Appeal Br.) filed April 16, 2018, the
Examiner’s Answer (Ans.) mailed May 10, 2018, and the Reply Brief
(Reply Br.) filed July 10, 2018.
Appeal 2018-007426
Application 15/273,591

3
determine, in the low-power mode, that the higher
level descriptor matches a reference object feature
descriptor; and
activate, in response to the determination, a high-
power mode of the information system.

Appeal Br. 14 (Claims App.).
We have reviewed the Examiner’s rejections in light of Appellant’s
arguments presented in this appeal. Arguments which Appellant could have
made, but did not make in the briefs are deemed to be waived. See 37
C.F.R. § 41.37(c)(1)(iv). On the record before us, we are unpersuaded the
Examiner has erred. Except as noted, we adopt as our own the findings and
reasons set forth in the rejections from which the appeal is taken and in the
Examiner’s Answer.
THE DOUBLE-PATENTING REJECTIONS
Claims 22, 24–31, 33–38, 40, and 41 are rejected on the ground of
nonstatutory double patenting as being unpatentable over claim 9–12 of U.S.
Patent No. 9,560,273 and Holz. Final Act. 6–8. Claim 23, 32, and 39 are
rejected on the ground of nonstatutory double patenting as being
unpatentable over claim 9 of U.S. Patent No. 9,560,273, Holz, and Lord. Id.
at 8–10.
Appellant does not challenge the above rejections. See generally
Appeal Br.3 Accordingly, we summarily sustain these rejections. See Hyatt
v. Dudas, 551 F.3d 1307, 1314 (Fed. Cir. 2008) (explaining that when

3 These rejections were requested to be held in abeyance on page 6 of Reply
to Final Office Action mailed August 14, 2017, submitted November 6,
2017.

Appeal 2018-007426
Application 15/273,591

4
appellant fails to contest a ground of rejection, the Board may affirm the
rejection without considering its substantive merits); see also 37 C.F.R.
§ 41.37(c)(1)(iv); MANUAL OF PATENT EXAMINING PROCEDURE (MPEP)
§ 1205.02 (9th ed. rev. 08.2017 Jan. 2018) (“If a ground of rejection stated
by the examiner is not addressed in the appellant’s brief, appellant has
waived any challenge to that ground of rejection and the Board may
summarily sustain it.”).
REFERENCES
The prior art relied upon by the Examiner is:
Name Reference Date
Lord US 2011/0034176 Al Feb. 10, 2011
Holz US 2014/0192206 Al July 10, 2014

THE OBVIOUSNESS REJECTION
Claims 22–41 are rejected under 35 U.S.C. 103 as being unpatentable
over Holz and Lord. Final Act. 10–14. The Examiner determined Holz
teaches many of the features in claim 22, including a memory device
comprising instructions executable on a processor to obtain an image
captured by a camera in lower-power mode and activating a high-power
mode of the camera in response to detecting an object. Final Act. 10–11
(citing Holz ¶¶ 3–4, 18, 21, 25–29); Ans. 3–4, 7–8. The Examiner stated
Holz is silent related to the “extract,” “generate,” and “determine” elements
in claim 22 but turns to Lord in combination with Holz to teach these
limitations (Final Act. 12), namely “to show that the [captured] object of
interest can be detected” as claimed. Ans. 4.
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Application 15/273,591

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Appellant argues claims 22, 24–31, 33–38, 40, and 41 as a group.
Appeal Br. 7–11. We select claim 22 as representative. See 37 C.F.R.
§ 41.37(c)(1)(iv).
Claims 22, 24–31, 33–38, 40, and 41
When addressing the recited “extract, in the low-power mode, a first
feature of an object in the image,” “generate, in the low-power mode, a
higher level descriptor of the first feature,” and “determine, in the low-power
mode, that the higher level descriptor matches a reference object feature
descriptor” elements in claim 22, the Examiner discussed Lord and proposed
combining Lord with Holz. See Final Act. 12–13; see Ans. 3–5, 8. The
Examiner stated Lord teaches identifying local features in an unknown
image, generating keypoint data from the local features, comparing data with
a reference database, and determining the closest matching image in the
database using a Euclidian distance-like measurement to determine the
detected object. Final Act. 12 (citing Lord ¶¶ 253, 739, 900–910);
Ans. 4–5, 8.
Appellant argues Holz and Lord collectively do not teach extracting a
feature of an object in the image, generating a higher level descriptor of the
feature, and determining the higher level descriptor matches a reference
object feature descriptor in a low-power mode. Appeal Br. 7–11; Reply Br.
2–4. More specifically, Appellant argues Lord identifies keypoints, which
are local image features, and screened keypoints, which are a subset of the
keypoints, but that the screened keypoints are not a descriptor of an image’s
feature as claimed and as mapped by the Examiner. Appeal Br. 8.
Appellant contends the screened keypoints generate a subset of keypoints,
but none of these keypoints are a descriptor of a particular keypoint and thus
Appeal 2018-007426
Application 15/273,591

6
none is “a higher level descriptor of the first feature” as claimed. Id.; Reply
Br. 3–4. Appellant further asserts the Examiner is mapping the recited “first
feature” and the “higher level descriptor” to the same components in Lord.
Appeal Br. 8–9; Reply Br. 4. Appellant also argues Holz detects brightness
or reflectivity changes in the environment rather than detecting a particular
object or its features and extracting an object’s features. Appeal Br. 9–10;
Reply Br. 2–3. Appellant even further asserts Lord teaches away from
performing a SIFT (scale invariant feature transform) in a low-power mode.
Appeal Br. 10 (citing Lord ¶ 915); Reply Br. 3 (citing Lord ¶ 918).

ISSUES
Under § 103, has the Examiner erred by determining that Holz and
Lord collectively would have taught or suggested a memory device
comprising instructions executable by the processor in low-power mode to:
(1) “extract . . . a first feature of an object in the image”;
(2) “generate . . . a higher level descriptor of the first feature”; and
(3) “determine . . . that the higher level descriptor matches a
reference object feature descriptor”?
ANALYSIS
Based on the record before us, we sustain the Examiner’s rejection.
We begin by construing the key disputed limitation of claim 22, “a higher
level descriptor of the first feature [of an object in the captured image].”
Appeal Br. 14 (Claims App.). Turning to the Specification, as did the
Examiner (Ans. 4), the disclosure describes “at least one of the current
feature descriptor or the reference feature descriptor is a higher level
description of an object, making it invariant to scale and/or rotation and/or
Appeal 2018-007426
Application 15/273,591

7
light” (Spec. ¶ 49) and provides an example of “local feature descriptors as
SIFT” (id. ¶ 11). Based on the Specification, some SIFTs, which are local or
current feature descriptors (see id. ¶ 11), can be “a higher level descriptor of
the first feature” of an object in the image as claim 22 recites. See id. ¶ 49.
The Specification also describes “a feature descriptor [that] is
determined in order to enable the comparison and matching of features . . .
might be an n-dimensional binary vector” (id. ¶ 12) and “a descriptor can
advantageously be a vector, which is derived from a 2D image or a part of a
2D image or 3D data, which is created by not just transforming pixels into a
different color space or normalizing their values” (id. ¶ 89). Given the
broadest reasonable construction of “a higher level descriptor of the first
feature” in light of the Specification as it would be interpreted by one of
ordinary skill in the art, this “descriptor” is data or data structure, including
but not limited to a SIFT or vector, that describes at least one feature of an
object captured in an image.
Turning now to the prior art, Holz’s summary discloses known
cameras, operating in a low-power mode, capturing images and switching to
high-power mode “[o]nce an object of interest has been detected in the field
of view of the cameras” (e.g., a person enters the field of view). Holz ¶ 3.
Holz further describes “motion detection that triggers a ‘wake-up’ of the
motion-capture system can be accomplished in several ways.” Id. ¶ 4.
Although Holz describes using brightness to indicate an object’s presence
(id. ¶¶ 4, 18, 26), Holz discloses other existing detection techniques,
including “[i]n some implementations, images captured by the camera(s) at a
very low frame rate are analyzed for the presence or movement of objects of
interest,” while in a low power mode. Id. ¶ 4, quoted in Ans. 4; see Final
Appeal 2018-007426
Application 15/273,591

8
Act. 13 (stating “Holz was shown to, while in a lower power mode, detect
the moving object by image analysis of captured images.”). Thus, contrary
to Appellant’s assertion (Appeal Br. 9–10), Holz is not completely silent
regarding detecting a particular object and at least suggests some known
implementations where features of an object can be extracted to detect the
objection’s presence (e.g., analyzing the captured image as Holz describes)
in a low-power mode.
This suggestion in Holz further leads an ordinary skilled artisan to
look to Lord for known teachings related how to analyze for the presence of
an object within an image and thus detect an object of interest in an image.
Ans. 4 (stating “that Lord was used to show that the object of interest can be
detected” as claimed). For example, Lord teaches an image analysis/object
recognition technique (Lord ¶ 897; see Ans. 9) that searches an image,
extracts features from the image, and recognizes an object within an image,
including using a SIFT image recognition technique. See Lord, Title “Image
Search, Feature Extraction, Pattern Matching, Etc.” above ¶ 879; id. ¶ 896.
Lord discloses the SIFT strategy “works by identification and description—
and subsequent detection—of local image features.” Id. ¶ 900.
Lord specifically discloses SIFT involves extracting local image
features in a reference image (e.g., keypoints) (id. ¶¶ 900–901) and
computing a keypoint descriptor (e.g., a histogram) (id. ¶ 909), which also
leads to describing objects by a set of SIFT features (e.g., a SIFT feature
vector) (id. ¶¶ 900, 909). See Final Act. 12 (citing Lord ¶¶ 900–910). Lord
thus describes not only extracting object features (e.g., keypoints) of a
reference image but also creating a reference object feature descriptor (e.g.,
keypoint descriptors or a histogram), which describes reference object
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Application 15/273,591

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features (e.g., a SIFT feature vector). As such, Lord teaches creating “a
reference object feature descriptor” (e.g., SIFT features for reference image)
as recited in claim 22. Lord further states the image analysis/object
recognition technique (Lord ¶ 897) involves processing an unknown image
in a similar manner, including extracting local image features in the
unknown image (e.g., generate keypoint data) (see id. ¶¶ 900–901, 910). As
such, Lord teaches “extract[ing] a first feature of an object in the image”
(e.g., extracting local image features in the unknown image) as recited in
claim 22.
We therefore disagree with Appellant that Lord’s extracting features
of the unknown image mapped to the “first feature” in claim 22 (see Final
Act. 12) fails to teach the recited “instructions executable by the processor
to: . . . extract . . . a first feature of an object in the image” as recited.
Appeal Br. 7. Moreover, when combined with Holz, the combination
teaches to perform this “extract” function “in low-power mode” as claim 22
also recites.
Lord further describes the above process compiles a reference
database, suggesting that not only is keypoint data stored in the database but
also reference keypoint descriptors and SIFT feature vectors for subsequent
detection. Lord ¶¶ 900, 910. For example, Lord also discusses identifying a
closest-matching image in the database by using “a Euclidean distance
calculation.” Id. ¶ 910. Although a Euclidean distance calculation may
involve one dimension calculation,4 existing sources describe this distance

4 Euclidean Distance in ‘n’-Dimensional Space, available at
https://hlab.stanford.edu/brian/euclidean_distance_in.html (last visited
December 11, 2019).
Appeal 2018-007426
Application 15/273,591

10
calculation involves two or three dimensions.5 Moreover, Lord states as few
as three SIFT features from an object are enough to describe an object (Lord
¶ 900), suggesting the disclosed closest-matching image technique involving
Euclidean distance calculation requiring at least three SIFT features (e.g., a
vector) from the unknown image to detect a match. As such, Lord suggests
to one skilled in the art “generat[ing] . . . a higher level descriptor of the first
feature” of the captured image and “determin[ing] . . . that the higher level
descriptor matches a reference object feature descriptor” (e.g., matching the
SIFT feature vector of the reference image to the SIFT feature vector of the
unknown image) as claim 22 recites.
We recognize that the Examiner discusses and maps keypoint data to
“a higher level descriptor” and “a reference object feature descriptor” in
claim 22 at one point in the rejection. Final Act. 12; Ans. 5. But the
rejection further states the recited “generate . . . a higher level descriptor of
the first feature” and “determine . . . that the higher level descriptor matches
a reference object feature descriptor” is taught by paragraphs 900 through
910 in Lord, which addresses the above-discussed “closest matching image
technique” in Lord. Additionally, the Examiner specifically discusses
Lord’s teaching related to the “closest-matching image” (Final Act. 12),
“where the closest-matching image in the database is identified by a
Euclidian distance-like measure” (Lord ¶ 910) when addressing the
“generate” and “determine” elements in claim 22. Final Act. 12. Moreover,

5 Glossary - Euclidean distance, available at
http://rosalind.info/glossary/euclidean-distance/ (last visited December 11,
2019).

Appeal 2018-007426
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11
the Examiner relies on claim 22’s discussion for rejecting claim 23, which
further limits the “determine” function to include the “determine a distance
measure between the higher level descriptor and each of the plurality of
reference object feature descriptors.” See id. at 13.
Given the foregoing explanation, we are not persuaded by Appellant’s
arguments related to Lord’s keypoint data or that Lord’s disclosed keypoint
data are not “a higher level descriptor of the first feature” or “a reference
object feature descriptor” in claim 22. Appeal Br. 7–9; Reply Br. 3–4. That
is, these arguments do not address the Examiner’s additional findings related
to SIFTs and the Euclidean distance-like measure. That Appellant chooses
to focus on paragraphs 900 through 903 of Lord in the briefing (see Appeal
Br. 7–10) without addressing cited paragraphs 909 and 910 (see generally
Appeal Br.) is telling in this regard.
Regarding the “low-power mode” limitation, Appellant contends Lord
teaches away from performing a SIFT in a low-power mode because Lord
“indicates that a customized FPGA chip, presumably running at full power,
can barely perform SIFT at a full frame rate in the first place, much less on a
mobile phone while in a lower-power mode.” Appeal Br. 10 (citing Lord
¶ 915); Reply Br. 3 (citing Lord ¶¶ 918–919). We are not persuaded.
First, the combination proposes to use Lord’s technique with Holz’s
device (e.g., computer system 300), not specifically a cell phone device. See
Final Act. 11–12. Second, Lord teaches at least one implementation (e.g.,
Bonato’s) operating at a rate faster than the cell phone implementation.
Lord ¶ 915; see id. ¶ 912. Third, Lord teaches the implementations “can be
adapted for use on a cell phone platform.” Id. ¶ 918 (emphasis added).
Thus, Lord does not limit its technique to cell phones. Nor is claim 22
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12
limited to a cell phone. Appeal Br. 14 (Claims App.). Fourth, there is
insufficient persuasive evidence on this record to substantiate Appellant’s
contention that Lord is operating at “full power” or “can barely perform
SIFT at a full frame rate” as argued. Appeal Br. 10.
For the foregoing reasons, Appellant has not persuaded us of error in
the rejection of independent claim 22 and claims 24–31, 33–38, 40, and 41,
which are not argued separately.
Claims 23, 32, and 39
Appellant separately argues claims 23, 32, and 39 as a group. Appeal
Br. 7, 11–12. We select claim 23 as representative. See 37 C.F.R.
§ 41.37(c)(1)(iv).
Appellant asserts Holz and Lord fail to teach the “determine a
distance measure between the higher level descriptor and each of the
plurality of reference object feature descriptors” limitation in claim 23
because Lord discusses using a “‘best-bin-first’ [BBF] algorithm.” Appeal
Br. 11 (quoting Lord ¶ 910). According to Appellant, this algorithm is an
approximate algorithm that cuts off its approach “after a specific number of
nearest bins have been explored.” Id. at 12 (quoting Lowe, David G.,
Distinctive Image Features from Scale-Invariant Keypoints 20-216).
Appellant thus asserts the BBF algorithm does not perform the distance
calculation “for each neighbor” or “each of the plurality of reference object
feature descriptors” as claim 23 recites. Id.
This argument is unavailing. Lord states “[a] ‘best-bin-first’
algorithm is typically used instead of a pure Euclidean distance calculation.

6 Available at https://www.cs.ubc.ca/-lowe/papers/ijcv04.pdf.
Appeal 2018-007426
Application 15/273,591

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. . .” Lord ¶ 910 (emphasis added). Thus, although Lord teaches a typical
approach, Lord does not rule out using the other described “pure Euclidean
distance calculation.” Id. (emphasis added). Lord thus teaches and suggests
algorithms other than BBF, including an approach where the “distance
measure between the higher level descriptor and each of the plurality of
reference object feature descriptors” as recited claim 23 is performed.
For the foregoing reasons, Appellant has not persuaded us of error in
the rejection of independent claim 23 and claims 32 and 39, which are not
argued separately.
CONCLUSION
In summary:
Claims
Rejected 35 U.S.C. § Reference(s)/Basis Affirmed Reversed
22–41 Nonstatutory
double patenting
22–41
22–41 103 Holz, Lord 22–41
Overall
Outcome
22–41

TIME PERIOD FOR RESPONSE
No time period for taking any subsequent action in connection with
this appeal may be extended under 37 C.F.R. § 1.136(a). See 37 C.F.R.
§ 1.136(a)(1)(iv).

AFFIRMED

Notice of References Cited

Application/Control No.

Applicant(s)/Patent Under Patent
Appeal No.
Examiner

Art Unit

Page 1 of 1

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*A copy of this reference is not being furnished with this Office action. (See MPEP § 707.05(a).)
Dates in MM-YYYY format are publication dates. Classifications may be US or foreign.

U.S. Patent and Trademark Office
PTO-892 (Rev. 01-2001) Notice of References Cited Part of Paper No.
12/19/2019 'n'-Dimensional Euclidean Distance
Euclidean Distance In 'n'-Dimensional SRace
Euclidean distance is a measure of the true straight line distance between two points in Euclidean
space.
One Dimension
In an example where there is only 1 variable describing each cell (or case) there is only 1 Dimensional
space. The Euclidean distance between 2 cells would be the simple arithmetic difference: Xcell1 - Xcell2
(eg. APHWcell1 = 1.11603 ms and APHWc81110 = 0.97034 ms; they are (1.11603 - 0.97034) = 0.14569
ms apart).
Two Dimensions
This has already been described here.
Three Dimensions
With 3 variables the distance can be visualized in 30 space such as that seen below.
0.
E
ct:
(_)
(f)
a..
w
(l)
APHW(ms)
2
~
·#' I
C
-----
Distance 'e' would be the distance between cell 1 & cell 2. We could determine it using Pythagora's
theorem as seen previously, but we first need to find the value of 'd' using values 'a' and 'b'.
1. a= APHWcell1 -APHWcell2
2. b;;;; mTaucell1 - mTaucell2
3. d2 ;;;; a2 + b2 and
https://hlab.stanford.edu/brian/euclidean_distance_in.html 1/3
12/19/2019 'n'-Dimensional Euclidean Distance
4. c = eEPSCcell1 - eEPSCcell2
5. and e2 = d2 + c2
substituting a2 + b2 for c2 from line 3 you get
6. e2 = a2 + b2 + c2
substituting the values for 'a', 'b' & 'c' from 1, 2 & 4
7. e2 = (APHWcell1 - APHWce112)2 + (mTaucell 1 - mTauce112)2 + (eEPSCcell1 - eEPSCce112)2
This way each of the dimensions (APHW, mTau & eEPSC amplitude) are added together to get the
Euclidean distance in one equation.
More Than 3 Dimensions ('n'-dimensions)
While pretty much impossible to visualize Pythagora's principle can be applied to more than 3
dimensions.
To generalize this we could say that 'e', seen above, is the distance 'D' between any 2 cells (cell i and
cell j): Dij.
The number of dimensions being worked in depends on the number of variables each cell (case) is
described by. If each cell is described by 3 variables then it is 30 space, if there are 20 variables then
it is 200 space. Therefore 'n' variables is represented in 'n'-dimensional space. Each cell (case) will
have a value 'x' for each variable (the variables will be represented from the first 'v' = 1 to the last 'v' =
'n' so that 'v' = 1 to 'n'). So distance in each dimension (like in 1, 2 & 4 above) could be represented
by:
xvi - Xvj which in English would be: The value of vth variable for cell i minus the value of the vth variable
for cell j.
Which, remembering Pythagora's Theorem in English: The square of the hypotenuse is equal to the
sum of the squares of the other two sides. (P-erfectly nerdy_Eythagora's joke here), itself in English
would be: The square of the Distance between 2 cells (i & j; Dij) is equal to the sum p::), from the first
variable (v = 1) to the last variable (n), of the squares of the distances in each dimension (which is
found by finding the differences for each cell's value for each variable (cell i's value for the vth variable
being Xvi and cell j's value being Xvj)).
Back To Measuring
Back To Main Pag~
https://hlab.stanford.edu/brian/euclidean_distance_in.html 2/3
12/19/2019
Glossary
Euclidean distance
ROSALIND I Glossary I Euclidean distance
The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment
connecting the two points. It is the most obvious way of representing distance between two points.
The Pythagorean Theorem can be used to calculate the distance between two points, as shown in the figure below. If the
points (xi, yi) and (x2, y2) are in 2-dimensional space, then the Euclidean distance between them is
J(x2 - x1) 2 + (y2 - Y1) 2.
Y2 -y1
For points (xi, y1, z1) and ( x2, y2, z2) in 3-dimensional space, the Euclidean distance between them is
J(x2 - xi)2 + (y2 -yi)2 + (z2 - z1)2. For example, the Euclidean distance between (-1, 2, 3) and (4, 0, -3) is
J25+4+36 = y'65.
Wikipedia
rosalind.info/glossary/euclidean-distance/ 1/1
12/19/2019 'n'-Dimensional Euclidean Distance
Euclidean Distance In 'n'-Dimensional SRace
Euclidean distance is a measure of the true straight line distance between two points in Euclidean
space.
One Dimension
In an example where there is only 1 variable describing each cell (or case) there is only 1 Dimensional
space. The Euclidean distance between 2 cells would be the simple arithmetic difference: Xcell1 - Xcell2
(eg. APHWcell1 = 1.11603 ms and APHWc81110 = 0.97034 ms; they are (1.11603 - 0.97034) = 0.14569
ms apart).
Two Dimensions
This has already been described here.
Three Dimensions
With 3 variables the distance can be visualized in 30 space such as that seen below.
0.
E
ct:
(_)
(f)
a..
w
(l)
APHW(ms)
2
~
·#' I
C
-----
Distance 'e' would be the distance between cell 1 & cell 2. We could determine it using Pythagora's
theorem as seen previously, but we first need to find the value of 'd' using values 'a' and 'b'.
1. a= APHWcell1 -APHWcell2
2. b;;;; mTaucell1 - mTaucell2
3. d2 ;;;; a2 + b2 and
https://hlab.stanford.edu/brian/euclidean_distance_in.html 1/3
12/19/2019 'n'-Dimensional Euclidean Distance
4. c = eEPSCcell1 - eEPSCcell2
5. and e2 = d2 + c2
substituting a2 + b2 for c2 from line 3 you get
6. e2 = a2 + b2 + c2
substituting the values for 'a', 'b' & 'c' from 1, 2 & 4
7. e2 = (APHWcell1 - APHWce112)2 + (mTaucell 1 - mTauce112)2 + (eEPSCcell1 - eEPSCce112)2
This way each of the dimensions (APHW, mTau & eEPSC amplitude) are added together to get the
Euclidean distance in one equation.
More Than 3 Dimensions ('n'-dimensions)
While pretty much impossible to visualize Pythagora's principle can be applied to more than 3
dimensions.
To generalize this we could say that 'e', seen above, is the distance 'D' between any 2 cells (cell i and
cell j): Dij.
The number of dimensions being worked in depends on the number of variables each cell (case) is
described by. If each cell is described by 3 variables then it is 30 space, if there are 20 variables then
it is 200 space. Therefore 'n' variables is represented in 'n'-dimensional space. Each cell (case) will
have a value 'x' for each variable (the variables will be represented from the first 'v' = 1 to the last 'v' =
'n' so that 'v' = 1 to 'n'). So distance in each dimension (like in 1, 2 & 4 above) could be represented
by:
xvi - Xvj which in English would be: The value of vth variable for cell i minus the value of the vth variable
for cell j.
Which, remembering Pythagora's Theorem in English: The square of the hypotenuse is equal to the
sum of the squares of the other two sides. (P-erfectly nerdy_Eythagora's joke here), itself in English
would be: The square of the Distance between 2 cells (i & j; Dij) is equal to the sum p::), from the first
variable (v = 1) to the last variable (n), of the squares of the distances in each dimension (which is
found by finding the differences for each cell's value for each variable (cell i's value for the vth variable
being Xvi and cell j's value being Xvj)).
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https://hlab.stanford.edu/brian/euclidean_distance_in.html 2/3