274 F.3d 1371 (Fed. Cir. 2001)
INTERACTIVE PICTURES CORP., (formerly known as Omniview, Inc.) Plaintiff-Appellee,v.INFINITE PICTURES, INC., Defendant-Appellant,andBILL TILLMAN, CRAIG JONES, and GARY WALTENBAUGH (doing business as Graphics Effects, also known as Graphics EFX), Defendants.
No. 01-1029
United States Court of Appeals for the Federal Circuit
December 20, 2001

Appealed from: United States District Court for the Eastern District of Tennessee, Judge Leon Jordan[Copyrighted Material Omitted]
Robert F. Altherr, Jr., Banner & Witcoff, LTD, of Washington, DC, argued for plaintiff-appellee.  With him on the brief were Nina L. Medlock, and Bradley C. Wright. Of counsel was Thomas H. Jackson, Banner & Witcoff, LTD.
J. Michael Jakes, Finnegan, Henderson, Farabow, Garrett & Dunner, L.L.P., of Washington, DC, argued for defendant-appellant.  With him on the brief was Christine E. Lehman.
Before LOURIE, Circuit Judge, ARCHER, Senior Circuit Judge, and GAJARSA, Circuit Judge.
LOURIE, Circuit Judge.


1
Infinite Pictures, Inc. appeals from the decision of the United States District Court for the Eastern District of Tennessee denying its various motions following a jury verdict finding Infinite and the other defendants liable for infringement of Interactive Pictures Corp.'s U.S. Patent 5,185,667 under the doctrine of equivalents and awarding $1 million in damages.  Because the district court did not err as a matter of law by refusing to preclude infringement by equivalents, and because the jury's finding of equivalence and award of damages were supported by substantial evidence, we affirm.

BACKGROUND

2
A.  The '667 Patent and its Prosecution History


3
The '667 patent describes an image viewing system whereby a user can view a specified portion of a hemispherical field of view in corrected perspective.  '667 patent, col. 2, ll. 35-60.  The image viewing system utilizes, inter alia, a camera equipped with a fisheye lens, an input memory, an image transform processor, and an output.  Id. at col. 3, ll. 25-48, Figure 1.  In operation, the fisheye lens camera produces a circular image of an entire hemispherical field of view.  Id., Abstract.  As the hemispherical field of view is mapped into a circle, the fisheye lens introduces distortion - fisheye distortion - such that straight lines appear curved in the circular image.  Id. at col. 4, ll. 22-47.  The circular image data is stored in the input memory.  Id. at col. 3, ll. 25-48.  The image transform processor reads the circular image data from the input memory, accepts user input of pan and tilt angles to specify a region of interest in the image, mathematically transforms the image data in that region so as to correct the fisheye distortion, and outputs a perspective corrected view of that region to the system's output.  Id. at col. 2, ll. 35-60.  The user may also specify desired rotation and magnification of the region of interest.  Id.  The system performs the mathematical transformations electronically, without moving parts, id. at col. 3, ll. 10-24, preferably at real time rates, id. at col. 4, ll. 44-47.


4
Claim 1, the only claim at issue, reads as follows:


5
1. A device for providing perspective corrected views of a selected portion of a hemispherical view in a desired format that utilizes no moving parts, which comprises:


6
a camera imaging system for receiving optical images and for producing output signals corresponding to said optical images; fisheye lens means attached to said camera imaging system for producing said optical images, throughout said hemispherical field-of-view, for optical conveyance to said camera imaging system;


7
image capture means for receiving said output signals from said camera imaging system and for digitizing said output signals from said camera imaging system;


8
input image memory means for receiving said digitized signals;


9
image transform processor means for processing said digitized signals in said input image memory means according to selected viewing angles and magnification, and for producing output transform calculation signals according to a combination of said digitized signals, said selected viewing angles and said selected magnification;


10
output image memory means for receiving said output signals from said image transform processor means;


11
input means for selecting said viewing angles and magnification;


12
microprocessor means for receiving said selected viewing angles and magnification from said input means and for converting said selected viewing angles and magnification for input to said image transform processor means to control said processing of said transform processor means; and


13
output means connected to said output image memory means for recording said perspective corrected view according to said selected viewing angles and magnification.


14
Id. at col. 8, ll. 27-66 (emphasis added).


15
Claim 1 was initially rejected by an Examiner at the U.S. Patent and Trademark Office under, inter alia, 35 U.S.C.  103, as being obvious over U.S. Patent 5,067,019, issued to Juday et al., in light of U.S. Patent 5,068,735, issued to Tuchiya et al.  The applicant challenged the propriety of the rejection by noting several deficiencies in the Juday reference vis--vis claim 1 and disputing the relevance of Tuchiya.  Additionally, the applicant amended claim 1 in several ways, including inserting the words "transform calculation" to the section reciting the "image transform processor means," so that "output signals" became "output transform calculation signals."  That change was made without any explanation.  After receipt of the applicant's claim amendments and arguments, the Examiner allowed claim 1 without stating in the record any reasons for the allowance.

B.  The Accused Product

16
Infinite makes and markets a software package known as the "SmoothMove(r) Panorama Web Builder."  The product has two components:  a "PhotoSeamer" program and a "Viewer" program.  The PhotoSeamer is designed to accept a set of three images from a 180 fisheye lens camera or other sources, align and seam the images together into a 360 panorama, and save the panorama image in an equirectangular format (a ".pan" file).  The collection and preparation of an equirectangular panorama file is an "off-line" operation, meaning that it cannot be accomplished in real time with fast changing images, such as motion video.  Next, the Viewer reads the equirectangular panorama file and displays a portion of the image, as determined by the user's input of "navigation controls" (e.g., left/right, up/down, and zoom in/out) in a "SmoothMove(r) Pan" window.  Infinite Pictures, SmoothMove(r) Panorama Web Builder Developer Manual 24, version 2 (1996).  The Viewer reacts in real time to the user's input, producing the selected image without fisheye distortion.

C.  The District Court Proceedings

17
Interactive brought suit against Infinite alleging that a computer equipped with a fisheye camera and executing the SmoothMove(r) software infringed the '667 patent.  Interactive conceded that such a computer did not literally infringe the patent because SmoothMove's(r) equirectangular panorama file was different from the claimed "digitized signals" input to the "image transform processor means."  However, Interactive presented a theory of infringement based on the doctrine of equivalents, alleging that SmoothMove's(r) equirectangular panorama file, though not an image obtained directly from a fisheye lens camera, was substantially similar to it.  The case went to trial before a jury on the issues of infringement under the doctrine of equivalents, validity, and damages.  The jury returned a verdict finding claims 1-8 infringed and not invalid, and awarded Interactive $1 million in damages based on evidence of a reasonable royalty on Infinite's sales determined in a hypothetical negotiation between Interactive and Infinite.


18
The parties also submitted to the jury the question whether a hypothetical version of claim 1 would ensnare the prior art.  The hypothetical claim differed from claim 1 in that the term "said digitized signal" was replaced by "an equirectangular panorama file," as Infinite argued that the hypothetical claim would be unpatentable over certain prior art, but the jury determined that hypothetical claim 1 would not ensnare that prior art.


19
Following the trial, Infinite made a number of motions, including:  (1) a motion for judgment based on the defense of prosecution history estoppel; (2) a motion for judgment of noninfringement under the doctrine of equivalents as a matter of law ("JMOL") on the grounds that SmoothMove(r) lacks an equivalent to the structure corresponding to a means-plus-function limitation and that there was a lack of substantial evidence to support the verdict of equivalence; and (3) a motion to set aside the jury's damage award in favor of a remittitur or new trial on the issue of damages.  The district court denied all three motions.  In deciding the prosecution history estoppel issue, the district judge performed a "flexible bar" analysis, Interactive Pictures, Corp. v. Infinite Pictures, Inc., No. 3:96-CV-849, slip op. at 4-10 (E.D. Tenn. Mar. 31, 1999), as this court had not yet issued its decision in Festo Corp. v. Shoketsu Kinzoku Kogyo Kabushiki Co., 234 F.3d 558, 56 USPQ2d 1865 (Fed. Cir. 2000) (en banc), cert. granted, 69 U.S.L.W. 3779 (U.S. June 18, 2001) (No. 00-1543), which discarded that approach in favor of an absolute bar, id. at 563-64, 56 USPQ2d at 1868.


20
Infinite appeals from the judgment of the district court.  We have jurisdiction pursuant to 28 U.S.C.  1295(a)(1) (1994).

DISCUSSION

21
We review a district court's denial of a motion for JMOL de novo, reapplying the JMOL standard used by the district court.  Sextant Avionique, S.A. v. Analog Devices, Inc., 172 F.3d 817, 824, 49 USPQ2d 1865, 1869 (Fed. Cir. 1999).  JMOL is appropriate when "there is no legally sufficient evidentiary basis for a reasonable jury to find for that party on that issue."  Fed. R. Civ. P. 50(a)(1).  On appeal, this court must consider the record evidence in the light most favorable to the non-movant, "drawing all reasonable inferences in its favor, without disturbing the jury's credibility determinations or substituting our resolutions of conflicting evidence for those of the jury."  Applied Med. Res. Corp. v. U.S. Surgical Corp., 147 F.3d 1374, 1377, 47 USPQ2d 1289, 1290 (Fed. Cir. 1998).  To prevail, "an appellant must show that the jury's findings, presumed or express, are not supported by substantial evidence or, if they were, that the legal conclusion(s) implied from the jury's verdict cannot in law be supported by those findings."  Celeritas Techs., Ltd. v. Rockwell Int'l Corp., 150 F.3d 1354, 1358, 47 USPQ2d 1516, 1519 (Fed. Cir. 1998) (internal quotation marks omitted).


22
A determination of infringement requires a two-step analysis.  "First, the court determines the scope and meaning of the patent claims asserted, and then the properly construed claims are compared to the allegedly infringing device."  Cybor Corp. v. FAS Techs., Inc., 138 F.3d 1448, 1454, 46 USPQ2d 1169, 1172 (Fed. Cir. 1998) (en banc) (citations omitted).  Comparison of the claim to the accused device requires a determination that every limitation of the claim or its substantial equivalent be found in the accused device.  See Warner-Jenkinson Co. v. Hilton Davis Chem. Co., 520 U.S. 17, 29 (1997).  Whether a patentee can avail itself of the doctrine of equivalents implicates certain questions of law, which we review de novo.  See, e.g., id. at 39 n.8 (whether a patent's prosecution history estops the patentee from arguing an equivalent is an issue of law).  Determination of infringement by equivalents is an issue of fact, Bai v. L & L Wings, Inc., 160 F.3d 1350, 1353, 48 USPQ2d 1674, 1676 (Fed. Cir. 1998), which after a jury trial we review for substantial evidence, Gen. Electro Music Corp. v. Samick Music Corp., 19 F.3d 1405, 1409, 30 USPQ2d 1149, 1152 (Fed. Cir. 1994).  We will not disturb a jury finding of infringement by equivalents unless "the evidence so favors [the accused infringer] that reasonable jurors could not arrive at a contrary verdict."  Id.


23
An award of damages by a jury is upheld on appellate review unless it is clearly not supported by evidence, grossly excessive, or based only on speculation and guesswork.  Oiness v. Walgreen Co., 88 F.3d 1025, 1031, 39 USPQ2d 1304, 1308 (Fed. Cir. 1996).  We review a decision denying a grant of remittitur or a new trial because of an excessive damages award for an abuse of discretion.  See id. at 1029, 39 USPQ2d at 1306 (applying tenth circuit law); see also Skalka v. Fernald Envtl. Restoration Mgmt. Corp., 178 F.3d 414, 424 (6th Cir. 1999), cert. denied, 120 S. Ct. 2687 (2000) (applying same standard under sixth circuit law).


24
Infinite argues that devices containing its SmoothMove(r) software cannot infringe claim 1 of the '667 patent as a matter of law for three reasons:  (1) amendments to the "image transform processor means" limitations and prosecution arguments relating to the same limitation estop Infinite from relying on the doctrine of equivalents for that limitation; (2) the asserted range of equivalents would ensnare certain prior art; and (3) the asserted equivalent of the "image transform processor means" limitation - a limitation construed under 35 U.S.C.  112, paragraph 6, and not literally present in SmoothMove(r) - predates the patent, thereby precluding Infinite from relying on the doctrine of equivalents for that limitation.  Additionally, Infinite argues that there are substantial differences under both the known interchangeability test and the function-way-result test.  Finally, Infinite argues that Interactive's testimony concerning damages was flawed.  Interactive responds that neither prosecution history estoppel, prior art, nor the predating nature of SmoothMove's(r) technology precludes it from prevailing on its claim of infringement under the doctrine of equivalents.  Interactive also responds that substantial evidence supports both the jury's findings of infringement by equivalence and the jury's award of damages.


25
A.  Legal Preclusion of the Doctrine of Equivalents

1.  Prosecution History Estoppel

26
Infinite argues that Interactive is estopped from asserting infringement by equivalents for two reasons.  First, Infinite contends that Interactive narrowed the "image transform processor means" limitation during prosecution by replacing the words "output signals" with the words "output transform calculation signals," and Infinite is thus barred from asserting any equivalents to that limitation, under Festo.  In Infinite's view, the amendment changed the nature of the claimed output signals.  Infinite further asserts that a more particular recitation of the output signals was necessary to define the invention over the prior art cited prior to the amendment.  Second, Infinite argues that Interactive's arguments that were made during prosecution in order to traverse the outstanding prior art rejections estop Interactive from asserting infringement by equivalents of the "image transform processor means" limitation.  Infinite contends that Interactive's remarks to the Examiner characterized the invention as a one-step image transformation system while characterizing Juday as disclosing a two-step image transformation process - first creating a look-up table off-line and then transforming the image using the look-up table.  Infinite points out that SmoothMove(r) also involves a two-step process - first creating the equirectangular panorama file with the PhotoSeamer, an off-line operation, and then viewing selected portions using the Viewer.


27
Interactive responds that the amendment changing "output signals" to "output transform calculation signals" did not narrow the claim, but merely clarified it by relabeling the signals without changing their identity or qualities.  Interactive also responds that the arguments made during prosecution did not disclaim coverage of the SmoothMove(r) software.  Interactive characterizes its remarks to the Examiner as not drawing a one-step versus two-step distinction, but rather pointing out that Juday's system cannot provide continuous changes in viewing angle, rotation, and magnification of the perspective corrected output image portion in real time relative to the user's input of different viewing angle, rotation, and magnification.


28
We agree with Interactive that the claim amendment and arguments do not estop Infinite from asserting infringement on the ground that the "image transform processor means" limitation is met by equivalence.  As to the amendment-based estoppel issue, we conclude that the addition of the words "transform calculation" was not a narrowing amendment because that addition did nothing more than make express what had been implicit in the claim as originally worded.  That interpretation flows from the original claim as a whole and in light of the specification.  First, the claim names the processor a "transform processor," and thus "transform" describes what the processor does. Secondly, the specification refers to the corresponding structure as "X-MAP and Y-MAP transform processors," '667 patent, col. 4, ll. 1-2 (emphasis added), and derives equations by which the transform is calculated, id. at col. 7, ll. 3-54.  Given such context, one skilled in the art could reach no other conclusion than that the "output signals" are the result of a transform calculation.  Because the amendment merely made explicit what had been implicit in the claim, the amendment was not made for a "substantial reason related to patentability" and thus does not create prosecution history estoppel.  Warner-Jenkinson, 520 U.S. at 33, 41 USPQ2d at 1873; see also Turbocare Div. of Deman Delaval Turbomach. Corp. v. Gen. Elec. Co., 264 F.3d 1111, 1125-26, 60 USPQ2d 1017, 1028 (Fed. Cir. 2001) (holding that addition of the word "contact" to a claim did not narrow the claim and therefore did not estop application of the doctrine of equivalents for that element because another claim phrase, "small diameter position," when read in light of the specification, necessarily implied "contact.").


29
Infinite also argues that Interactive narrowed the definition of the phrase "output signals" in order to avoid the prior art.  Infinite's argument is based on the following applicant remark made during prosecution concerning "[t]he present invention, as defined most broadly in Claim 1":


30
The [digitized fisheye image] information in this input memory is processed with a transform processor to produce output signals corresponding to the perspective corrected views according to selected viewing angles and magnification, with these output signals being produced according to a combination of the digitized signals, the selected viewing angles and the selected magnification.


31
Although that remark does describe an aspect of the invention not found in the prior art, it did not alter what the phrase "output transform calculation signals," or more simply the phrase "output signals," meant in the claim and the patent as a whole.  On the contrary, the remark is consistent with the plain language of the claim and the portion of the specification concerning the output signals.  That remark reinforces our conclusion that no difference exists between the two phrases.  The remark uses the shorthand "output signals" even though claim 1 contemporaneously had been amended to recite "output transform calculation signals."  The applicant's interchanging of the two phrases therefore indicates that the phrases have the same meaning.


32
As to the argument-based estoppel issue, we agree with Interactive that the arguments made during prosecution to distinguish Juday's '019 patent do not create estoppel.  As illustrated in Figure 1 of the '019 patent, Juday discloses an image processing system, including a camera 4, an analog-to-digital converter 6, a programmable remapper 10, a digital-to-analog converter 14, and a monitor 18, connected serially in that order.  '019 patent, Figure 1.  The programmable remapper 10 transforms a digitized camera image into another form according to an operator selected look-up table.  Id. at col. 3, ll. 49-59.  By specifying a different look-up table, a different transform may be applied.  Id.  Juday discloses specific transforms to accommodate particular vision impairments, viz., remapping an image away from the central portion and into the periphery to accommodate maculopathy, and compressing an image including its periphery into just the central portion to accommodate tunnel vision.  Id. at col. 4, ll. 4-19.  Each look-up table is determined off-line and then stored in the memory of the remapper 10.  Id. at col. 6, ll. 1-25.  Juday's only mention of a fisheye lens is to note that the remapper 10 could introduce fisheye distortion when the image processing system is used as a prosthesis for retinitis pigmentosa.  Id. at col. 14, ll. 40-58.


33
The Examiner rejected claim 1 as being obvious over Juday because, according to the Examiner, Juday's image processing system read on claim 1 except that Juday's system lacked the capability to accept viewing angle and magnification selections by the user.  The Examiner considered that capability to be well known in the art and concluded that one of ordinary skill in the art would have been motivated to add those features to Juday's device so as to result in the invention of claim 1.


34
The applicant responded to the obviousness rejection by noting, inter alia, three deficiencies in the Juday system:  (1) the lack of a fisheye lens; (2) the lack of "pan and tilt implementation," image rotation, and magnification; and (3) the lack of real time capability.  Concerning the latter, the applicant stated:


35
Juday et al. utilizes a group of pre-selected 'look-up' tables for remapping an image into the pre-selected configurations.  A specific look-up table is then used by the researcher to achieve a given remapping of the input data.  Due to this functioning, the resultant 'corrected image' is created in real-time at video rates relative to the input to the system for only a limited number of image orientations and magnifications.  Continuous change in orientation and magnification is not feasible in real time at video rates with the Juday et al. device.  Hence, perspective corrected pan and tilt is not possible in real time with that device (in contrast to the action with the device of the Applicant). . . .  Further, Juday et al. fails to provide either magnification or image rotation.  These cannot be accomplished by Juday et al. without continuous redetermination of the transformal mapping, with this not being possible with the look-up table type of transformations.


36
We do not read those remarks as drawing a distinction between a one-step viewing system and a two-step viewing system.  Infinite's reading appears to misapprehend Juday's look-up tables by assuming that they are intermediate forms of the image.  That is not the case.  Juday's look-up tables represent the transformations themselves, not the output images.  '019 patent at col. 3, ll. 49-59.  Rather than drawing a one-step versus a two-step distinction, the remarks point out that Juday's system cannot produce a view with a different orientation, rotation, or magnification, unless a different look-up table is utilized, and widespread changing of look-up tables is not practical in real time.  In contrast, the invention's transform calculation can provide a perspective corrected output image portion with variable viewing angle, rotation, and magnification.  Because the viewing angle, rotation, and magnification are variables built into the transformation, the output can change in real time relative to the user's input of changes to those parameters.  (That is also how the Viewer module of the accused software operates, albeit on an intermediate form of the image.)  Accordingly, because the applicant's remarks pertain only to the invention's capability to respond in real time to user parameters, Interactive is not estopped from asserting that Infinite's two-step viewing software infringes the '667 patent under the doctrine of equivalents.


37
Because neither the claim amendment nor the applicant's arguments during prosecution estop Interactive from asserting that the SmoothMove(r) software contains an element equivalent to the "image transform processor means," we affirm the district court's denial of Infinite's post-trial motion on the issue of prosecution history estoppel.


38
2.  Prior Art Limitations on the Doctrine of Equivalents


39
Infinite next argues that the asserted scope of equivalents would not have been patentable over the following prior art articles:  G. David Ripley, DVI - A Digital Multimedia Technology, 32 Communications of the ACM 811 (1989) ("Ripley"); Ned Greene, Environment Mapping and Other Applications of World Projections, 6 IEEE Computer Graphics and Applications 21 (1986) ("Greene"); and Mario Onoe & Yoshinori Kuno, Digital Processing of Images Taken by Fish-Eye Lens, IEEE: Proceedings, New York 105 (1982) ("Onoe").  According to Infinite, Ripley and Greene disclose fisheye lens cameras capturing hemispherical views, which are digitized and transformed in their entirety to equirectangular projections, from which a user can choose to view a certain portion, while Onoe discloses transformation algorithms for converting a fisheye image to a perspectively correct rectangular image.  Interactive responds that substantial evidence supports the jury's finding that the hypothetical claim does not encompass what Ripley, Greene, and Onoe teach.  More particularly, Interactive states that the articles individually and cumulatively do not disclose or suggest perspective corrective transformation of only a portion of an image with user selection of magnification and rotation.  Infinite counters that the jury's finding is merely advisory on that issue of law.


40
It is well settled law that a patentee cannot assert a range of equivalents that encompasses the prior art.  Wilson Sporting Goods Co. v. David Geoffrey & Assocs., 904 F.2d 677, 683, 14 USPQ2d 1942, 1948 (Fed. Cir. 1990).  To test this limit, the notion of a hypothetical claim may be useful.  Id. at 684, 14 USPQ2d at 1948.  A hypothetical claim may be constructed to literally cover the accused device.  Id.  If such a claim would be unpatentable under 35 U.S.C.  102 or 103, then the patentee has overreached, and the accused device is noninfringing as a matter of law.  Id. at 683-84, 14 USPQ2d at 1948.  The burden of producing evidence of prior art to challenge a hypothetical claim rests with an accused infringer, but the burden of proving patentability of the hypothetical claim rests with the patentee.  Streamfeeder, LLC v. Sure-Feed Sys., Inc., 175 F.3d 974, 984, 50 USPQ2d 1515, 1521 (Fed. Cir. 1999).


41
We conclude that Ripley, Greene, and Onoe collectively fail to disclose or suggest the hypothetical version of claim 1 presented to the jury.  The Ripley article, written by Infinite's founder, describes a system called "Palenque," Ripley at 881-19, which was a precursor of SmoothMove(r).  Like SmoothMove(r), Palenque produced an equirectangular panorama file from a photographic image captured by a fisheye lens camera.  Id.  Although the Ripley article does not address whether Palenque provided perspective correction, magnification, or rotation of the selected portion of the image, the testimony of Ripley and others firmly establishes that Palenque lacked those capabilities; Infinite does not indicate otherwise.  In other words, Palenque merely copied a user-specified portion of the panorama for display on an output device, without perspective correction and without the capability for magnification or rotation.


42
The Greene article describes a technique for projecting a fisheye image onto a half cube and thus providing perspective correction.  Greene at 27-28.  The Onoe article describes three computational methods for correcting distortion in fisheye images and thereby accurately measuring parameters (e.g., distance) from fisheye images.  Onoe at 105.  One method employs a single fisheye image with partial a priori knowledge of parameters.  Id.  The other two methods employ stereo fisheye images without a priori knowledge of parameters.  Id.


43
While Greene or Onoe may have taught one of ordinary skill in the art to add a perspective correction capability to the Palenque system described by Ripley, it is not clear that perspective correction is applied only to the user-selected portion of the image, rather than to the entire image.  Even assuming perspective correction of only a portion of the image, the combination taught by the cited articles would lack at least the capability of the user to magnify or rotate the selected, perspective-corrected portion of the image, and the capability to do so in real time relative to the user's input would certainly not have been obvious.  Because claim 1 requires that the perspective correction processing performed by the "image transform processor means" be done "according to a combination of said digitized [input image] signals, said selected viewing angles and said selected magnification," we conclude that hypothetical claim 1 would not have been unpatentable over Ripley, Greene, and Onoe.  Furthermore, because claim 5 additionally recites user input of a rotation angle, we conclude that a hypothetical version of claim 5, depending from and further limiting hypothetical claim 1, would also not have been unpatentable over the prior art of record.


44
We reach our conclusion on this issue independently of the jury's verdict.  Because we reach the same conclusion as the jury, we need not address the level of deference to which a jury's verdict is entitled in conducting a hypothetical claim analysis.  Compare Wilson Sporting Goods, 904 F.2d at 683, 14 USPQ2d at 1948 (holding ensnarement to be an issue of law reviewed de novo) with Tegal Corp. v. Tokyo Electron Am., Inc., 257 F.3d 1331, 1348, 59 USPQ2d 1385, 1398 (Fed. Cir. 2001) (holding that underlying questions of fact in the context of obviousness are reviewed for clear error).


45
3.  Equivalent of a Means-Plus-Function Limitation


46
Finally, Infinite argues that Interactive cannot assert infringement by equivalents of the "image transform processor means" limitation, because that limitation is not literally present in the SmoothMove(r) software, and the equivalent feature that is present in the SmoothMove(r) software predates the '667 patent.  Infinite cites Chiuminatta Concrete Concepts, Inc. v. Cardinal Indus., Inc., 145 F.3d 1303, 46 USPQ2d 1752 (Fed. Cir. 1998), as the basis for its argument.


47
Interactive responds that this case is distinguishable from Chiuminatta; we agree.  In Chiuminatta, we held that a finding that a component of an accused product is not a structure "equivalent" to the corresponding structure of a means-plus-function limitation for purposes of literal infringement analysis precludes a finding that the same structure is equivalent for purposes of the doctrine of equivalents, unless the component constitutes technology arising after the issuance of the patent.  Id. at 1311, 46 USPQ2d at 1758.  However, when a finding of noninfringement under 35 U.S.C.  112, paragraph 6, is premised on an absence of identical function, then infringement under the doctrine of equivalents is not thereby automatically precluded.  WMS Gaming Inc. v. Int'l Game Tech., 184 F.3d 1339, 1353, 51 USPQ2d 1385, 1395 (Fed. Cir. 1999).  That is because infringement under the doctrine of equivalents may be premised on the accused and the patented component having substantially the same function, whereas structure corresponding to the disclosed limitation in a means-plus-function clause must perform the identical function.  Id.


48
In this case, as in WMS Gaming, the absence of literal infringement is due to a lack of identical function of the claimed means, not a lack of equivalent structure, as the district judge aptly recognized.  Interactive Pictures, Corp. v. Infinite Pictures, Inc., No. 3:96-CV-849, slip op. at 3-5 (E.D. Tenn. Sept. 25, 2000).  More precisely, the difference is that a different signal is input to the "image transform processor means."  The claim recites that the input signal constitutes "said digitized signals," which refer to a digitized form of the image signals produced by a fisheye lens camera imaging system, whereas the asserted equivalent's input signal is an equirectangular panorama file.  A difference of inputs to a signal processing element resulting in the same output necessarily implies a difference in function performed by the element.  In fact, the jury in this case concluded that the difference was insubstantial.  We will not reverse that fact-based decision as a matter of law.  Accordingly, we affirm the district court's denial of Infinite's post-trial motion for JMOL on this issue.

B.  The Factual Finding of Equivalents

49
Having concluded that resort to the doctrine of equivalents is not barred as a matter of law, we turn to the issue of equivalence in fact.  The precise issue before us is whether substantial evidence supports the jury's finding that SmoothMove(r), with its equirectangular panorama file, contains an element substantially the same as the "image transform processor means" limitation.


50
Infinite argues that there are substantial differences according to both the known interchangeability test and the function-way-result test.  Concerning the former, Infinite argues that the equirectangular panorama .pan file is not interchangeable with the claimed digitized fisheye image data because the SmoothMove(r) Viewer is compatible with only the .pan file format.  Concerning the latter, Infinite argues that the "image transform processor means" and the SmoothMove(r) program differ in function, way, and result because the claimed "means" can transform images in real time relative to motion in the image, whereas SmoothMove(r) can transform only static images.


51
Interactive responds that substantial evidence supports the jury's finding of equivalence.  In particular, Interactive cites the testimony of its expert, Dr. Birdwell, that notwithstanding the incompatibility of data formats, the equirectangular panorama format is a known substitute for digitized fisheye signals and that the function of transforming a portion of an image in equirectangular panorama format is substantially the same as the function of transforming a portion of an image represented by digitized fisheye signals.  Furthermore, Birdwell testified that the SmoothMove(r) Viewer contains structure the same as or equivalent to the structure corresponding to the "image transform processor means."


52
We agree that substantial evidence supports the finding of equivalence.  Although Infinite attacks Birdwell's testimony as being conclusory, the testimony is more than that.  Birdwell showed the jury both a fisheye image and an image in an equirectangular panorama format, while testifying that one can be derived from the other.  He also explained that perspective-correction processing of one of the two image forms is substantially the same as that of the other, and he illustrated the similarity with reference to Infinite's own block diagrams of the processing steps.


53
Infinite's arguments do not persuade us to the contrary.  First, incompatibilities in computer data files do not necessarily rise to the level of substantial differences.  Even the slightest difference in file formats often creates an incompatibility, because computers are exact machines.  However, the law of patent infringement is not so limiting.  A patent claim may be infringed if an element of the infringing device is only substantially the same as a limitation of the patent claim.  Warner-Jenkinson, 520 U.S. at 29.  Indeed, we have upheld determinations of equivalence on the ground that hardware and software implementations of a component of an invention are interchangeable substitutes, Overhead Door Corp. v. Chamberlain Group, Inc., 194 F.3d 1261, 1269-70, 52 USPQ2d 1321, 1326 (Fed. Cir. 1999), even though such a substitution would require ancillary changes in affected circuitry and packaging.  Rather than focusing on physical or electronic compatibility, the known interchangeability test looks to the knowledge of a skilled artisan to see whether that artisan would contemplate the interchange as a design choice.  Id.  Viewed in that light, Birdwell's testimony that the claimed digital fisheye image and SmoothMove's(r) equirectangular panorama file are "interchangeable alternatives" is substantial evidence supporting equivalence.


54
Secondly, we reject Infinite's function-way-result argument because the ability to transform moving images in real time relative to the motion is not part of either the function, way, or result of the "image transform processor means."  As a preliminary matter, we consider the entire "image transform processor means" with its two associated functions to be one limitation.  We do so because the two functions are accomplished by the same structure and are closely related.  In fact, the first function, "processing said digitized signals in said input image memory means according to selected viewing angles and magnification," and the second function, "producing output transform calculation signals according to a combination of said digitized signals, said selected viewing angles and said selected magnification," are practically inseparable.  Having identified the functions in the claim language itself, Lockheed Martin Corp. v. Space Sys./Loral, Inc., 249 F.3d 1314, 1324, 58 USPQ2d 1671, 1678 (Fed. Cir. 2001), we look to the way in which those functions are accomplished and the attendant results in the specification.  In this case, the specification discloses particular transform equations, '667 patent, ll. 30-54 (equations (17) and (18)), implemented in a pair of "X-map and Y-map transform processors," which are "application specific integrated circuits or other means as may be known to persons skilled in the art," id. at col. 4, ll. 1-5.  The results achieved by those functions are not limited to real time capabilities relative to movement in the image.  Instead, as the applicant's remarks to the Examiner indicated, the results include real time capabilities relative to user input of viewing angle and magnification.  Accordingly, Infinite's argument in terms of the function-way-result test misses the mark.


55
Thus, Infinite's arguments do not reveal any flaw underlying the jury's finding of equivalence so serious as to indicate that the jury's finding is unsupported by substantial evidence.  Essentially, Infinite invites us to reweigh the evidence of substantial similarity in its favor.  We cannot accept that invitation.  Nutrinova Nutrition Specialties & Food Ingredients GmbH v. Int'l Trade Comm'n, 224 F.3d 1356, 1359, 55 USPQ2d 1951, 1953 (Fed. Cir. 2000).  We therefore affirm the district court's denial of Infinite's post-trial motion for JMOL on the issue of equivalence in fact.

C.  Damages

56
Because we affirm the jury's verdict of infringement, we turn next to the jury's award of damages.  Interactive's theory of damages was based on a reasonable royalty that the parties might have agreed to during a hypothetical negotiation when infringement began in 1996.  Interactive's damages expert, Dr. Donald L. Martin, testified that the reasonable royalty would have been fully paid-up, covering a term of five years, at a rate of 10% of gross revenue.  Martin presented two figures - an upper range and a lower range.  They differed only in the royalty base.  The upper range, $3.2 million, assumed sales increasing from $0.9 million in 1997 to $53 million in 2001, according to a five-year annual projection of total sales stated in an Infinite business plan prepared in 1996.  The lower range, $1 million, assumed sales of $4 million in 1997, a figure also taken from the 1996 business plan, with annual increases at the rate of 2.68%, the average rate of growth in the economy over the last 20 years.  The jury presumably accepted Martin's lower range.  In both cases, Martin asserted as the royalty base a projection of Infinite's total future sales.


57
Infinite first challenges the royalty base asserted by Interactive.  Infinite argues that the royalty base was flawed because (1) Martin's projection of Infinite's future sales was speculative, as it was based on an outdated business plan and its optimistic assumptions of future revenue growth; and (2) Martin erroneously included sales of all of Infinite products, rather than just infringing products.  Secondly, Infinite argues that the jury's award was tainted by testimony that Infinite's infringement caused a decrease in Interactive's stock price.  Interactive defends the royalty base by responding that the 1996 business plan, dated two months before infringement began, was the best information at the time for a hypothetical negotiation and that Infinite's bundling and convoyed sales justified its inclusion of all Infinite products in the royalty base. Interactive also responds that Infinite failed to object to the testimony regarding its stock devaluation and that the jury was properly instructed to award damages on the basis of a reasonable royalty and not to award damages based on speculation or conjecture.


58
We agree with Interactive that Martin's testimony was not speculative by virtue of its reliance on Infinite's 1996 business plan.  We have previously upheld awards of damages premised on a lump sum royalty payment based on an infringer's expected sales.  E.g., Snellman v. Ricoh Co., 862 F.2d 283, 289, 8 USPQ2d 1996, 2001 (Fed. Cir. 1988).  We have also endorsed the conceptual framework of a hypothetical negotiation between the patentee and the infringer as a means for determining a reasonable royalty.  Rite-Hite Corp. v. Kelley Co., 56 F.3d 1538, 1554, 35 USPQ2d 1065, 1076-77 (Fed. Cir. 1995) (en banc).  When that framework is employed, the negotiation must be hypothesized as of the time infringement began.  Wang Labs., Inc. v. Toshiba Corp., 993 F.2d 858, 869-70, 26 USPQ2d 1767, 1777-78 (Fed. Cir. 1993).  Although "a trier of fact must have some factual basis for a determination of a reasonable royalty," consideration of a hypothetical negotiation "necessarily involves an element of approximation and uncertainty."  Unisplay, S.A. v. Am. Elec. Sign Co., 69 F.3d 512, 517, 36 USPQ2d 1540, 1544 (Fed. Cir. 1995).  In this case, the 1996 business plan and its projections for future sales were prepared by Infinite two months before infringement began.  Thus, rather than being outdated for purposes of the hypothetical negotiation, those projections would have been available to Infinite at the time of the hypothetical negotiation.  The fact that Infinite did not subsequently meet those projections is irrelevant to Infinite's state of mind at the time of the hypothetical negotiation.  Nor does Infinite's subsequent failure to meet its projections imply that they were grossly excessive or based only on speculation and guesswork.  Instead, Infinite's subsequent failure to meet its projections may simply illustrate the "element of approximation and uncertainty" inherent in future projections.  Id.  Because Martin's use of the sales projections from the 1996 business plan was supported by evidence, not grossly excessive, nor based only on speculation and guesswork, we will not the disturb the jury's damage award on that basis.


59
Interactive argues that Lindemann Maschinenfabrik GmbH v. Am. Hoist & Derrick Co., 895 F.2d 1403, 13 USPQ2d 1871 (Fed. Cir. 1990), compels a contrary result.  We disagree.  In that case we affirmed a district court's award of damages in derogation of the patentee's expert's testimony, which was based on estimates of the infringer's anticipated profits "that bore no relation to actual profits."  Id. at 1407, 13 USPQ2d at 1875.  Lindemann therefore supports the proposition that an actual infringer's profit margin can be relevant to the determination of a royalty rate in a hypothetical negotiation.  Id. at 1407-08, 13 USPQ2d at 1875 (citing Trans-World Mfg. Co. v. Al Nyman & Sons, Inc., 750 F.2d 1552, 1568, 224 USPQ 259, 269 (Fed. Cir. 1984)).  However, Lindemann does not require that estimates of sales revenues, as referenced in a hypothetical negotiation at the time infringement began, must later bear a close relation to actual sales revenue.  Such a proposition would essentially eviscerate the rule that recognizes sales expectations at the time when infringement begins as a basis for a royalty base as opposed to an after-the-fact counting of actual sales.


60
We also conclude that Martin permissibly included all Infinite products in the royalty base, and that Martin did not provide for an unfair double recovery by factoring the bundling and convoying sales into the royalty rate.  The jury was entitled to rely on evidence of bundling and convoyed sales in determining the proper scope of the royalty base.  Deere & Co. v. Int'l Harvester Co., 710 F.2d 1551, 1559, 218 USPQ 481, 487 (Fed. Cir. 1983) (authorizing such an approach as "eminently reasonable"); cf. Rite-Hite, 56 F.3d at 1548-49, 35 USPQ2d at 1072 (allowing recovery of lost profits from the patentee's unpatented components that compete directly with the infringing device and function together with the patented device).  The fact that bundling and convoyed sales affected Martin's estimate of both the royalty base and the royalty rate is thus not a sufficient reason to nullify the jury's award.  The "extent of . . . derivative or convoyed sales" is one of the often-cited Georgia-Pacific factors relevant to a determination of a reasonable royalty rate. Georgia-Pacific Corp. v. U.S. Plywood Corp., 318 F. Supp. 1116, 1120, 166 USPQ 235, 238 (S.D.N.Y. 1970); see also State Indus., Inc. v. Mor-Flo Indus., Inc., 883 F.2d 1573, 1580-81, 12 USPQ2d 1026, 1031 (Fed. Cir. 1989) ("The value of collateral sales could also be factored into the royalty rate.") (citations omitted).  Martin explained the application of that factor to the royalty rate in this case.  Yet, the bundling and convoyed sales were just one of thirteen Georgia-Pacific factors used by Martin to justify a royalty rate of 10%.  Even if we were to ignore the bundling and convoyed sales as a factor, the other twelve Georgia-Pacific factors provide ample evidentiary support for the resulting royalty rate, and we conclude that Martin's opinion, in its entirety, was not so excessive, speculative, the product of guesswork, or devoid of evidentiary support to warrant upsetting the jury's award.


61
We will also not reverse the jury's award of damages based on the testimony regarding stock devaluation in this case.  Interactive's CFO testified that the market value of the company dropped $14 million in 1997, apparently due to Infinite's infringement.  While it is true that a decrease in a company's stock value is generally too remotely related to patent infringement to be compensable under the patent laws, Rite-Hite, 56 F.3d at 1546, 35 USPQ2d at 1069, it is not clear that the testimony tainted the jury in this case.  Rather, the jury award appears to have been based solely on Martin's lower range estimate of a reasonable royalty.


62
We have considered Infinite's other arguments on the issue of damages and find them to be without merit.  Because substantial evidence supports the jury's award of damages, we conclude that the district judge did not abuse his discretion in denying Infinite's motion for remittitur or a new trial.

CONCLUSION

63
Because infringement by equivalents was not precluded as a matter of law and the jury's verdict of infringement and award of damages were supported by substantial evidence, we


64
AFFIRM.

