NIES, Judge.
This appeal is from the decision of the Patent and Trademark Office Board of Appeals (board) affirming the rejection of claims 5-7 and 33^471 in their application serial No. 850,892, filed November 15, 1977, for “Tomographic Scanner.” The claims stand rejected under 35 U.S.C. § 1012 as being drawn to nonstatutory subject matter. We affirm the rejection of claims 5 and 7 and reverse with respect to all remaining claims on appeal.
The Invention
Appellants’ invention is in the field of image processing particularly as applied to computerized axial tomography or CAT scans. Specifically, appellants’ invention is directed to an improvement in computed tomography whereby the exposure to X-ray is reduced while the reliability of the produced image is improved. Some understanding of tomography, both conventional and computed, is necessary as background for the subsequent analysis of the present invention.
Conventional tomography, also known as laminography, employs the simultaneous movement in opposite directions of an X-ray source and an X-ray film. The method produces a well-defined image of a plane through the body parallel to the plane of the X-ray film. In contrast to an ordinary X-ray, shadows of body structure which lie outside the plane of investigation are blurred so that they do not interfere with a focused image of the plane under investigation. Conventional tomographic systems are not practical when a visualized cross-section transverse to the body axis is desired. Computed tomography was developed to overcome this deficiency among others.
Basically, computed tomography provides an image representing a transverse slice of the body. This slicing is accomplished by rotating an X-ray source and a detection means around the perimeter of the section to be viewed. The source and detection means are placed 180° from each other to allow the detection means to measure the attenuation of the beam as it passes through the plane of interest. When enough measurements have been taken, a computer is implemented to mathematically interpret the data, which is then displayed as a reconstruction of the slice on, inter alia, a television screen for diagnostic purposes. Computed tomography is also useful for looking at only a section of a slice. Thus, if the plane of interest were in the abdomen region of a human, but only the liver were of concern, the computed tomography ma[904]*904chine would act in the manner of a conventional tomograph and blur the images outside of this “Region of Interest.” It has, however, been necessary that a spread of X-rays “S” be sufficiently wide to subtend the entire body in order to produce an image of the region of interest “R” by this blurring method, as illustrated by the following drawing:
Appellants have discovered that it is unnecessary to expose the body in the above fashion. Rather, they have discovered that the spread of X-ray, S, can be reduced so as to subtend only the region R thus:
Narrowing the beam is advantageous not only because the exposure of a body to X-ray is thereby reduced but also because computer calculation time to produce the image is shortened inasmuch as the amount of data to be processed is less. However, because fewer data can be collected due to the narrower beam, there is insufficient information to cancel out an object, n, such as a piece of rib, for example, which is in the beam bath. Thus, the resultant image shows the region R with an artifact3 appearing therein due to that object, n:
Appellants’ invention is directed to an improvement in CAT scan imaging technique whereby the body is exposed to less radiation and, through use of a weighting function in the calculations producing the image, the artifacts are eliminated.
The Rejection
The examiner rejected the claims on appeal under the authority of Parker v. Flook, 437 U.S. 584, 98 S.Ct. 2522, 57 L.Ed.2d 451 (1978). In the final rejection and in the examiner’s answer before the board, the examiner construed Flook as mandating the following test:
Taking each claim as a whole, it is assumed, for analysis purposes only, that any mathematical calculation in the claim is part of the prior art. If what is left is new and unobvious, then the claim, taken as a whole, protects more than a mathematical calculation and it is deemed statutory. But if the remainder of the claim is not novel nor unobvious, then the claim, taken as a whole, merely seeks to protect the mathematical calculation and, as such, does not comprise statutory subject matter.
Applying the above test, the examiner determined that, apart from the mathematical calculations, the remaining steps were well known or were “merely a necessary antecedent step to provide values for solving the mathematical equations,” and, thus, were directed to nonstatutory subject matter, citing In re Richman, 563 F.2d 1026, 195 USPQ 340 (Cust. & Pat.App.1977).
The Board’s Decision
The board did not address the examiner’s contentions, relying instead on In re Freeman, 573 F.2d 1237, 197 USPQ 464 (CCPA 1978), as modified by In re Walter, 618 F.2d 758, 205 USPQ 397 (Cust. & Pat.App.1980). Without resort to detailed claim language, the board affirmed the rejection under 35 U.S.C. § 101 as follows:
[905]*905When the claims are analyzed in [the manner dictated by Walter ], it is manifest that the mathematical algorithm is not implemented in a manner to define structural relationships between physical elements in the apparatus claims or to refine or limit claim steps in the process claims. The claims do no more than present and solve a mathematical algorithm and are manifestly nonstatutory.
One member dissented with respect to rejection of claims 6 and 33-47 concluding that these claims are directed to “producing a product, an improved tomographic X-ray image,” and are, therefore, directed to statutory subject matter citing Diamond v. Diehr, 450 U.S. 175, 101 S.Ct. 1048, 67 L.Ed.2d 155 (1981).
OPINION
I
A
We agree with the board that a two-part analysis 4 is the proper vehicle for resolution of issues here presented under 35 U.S.C. § 101. However, we agree with appellants that the second step of the analysis is not as limited as the board held it to be.
B
In Gottschalk v. Benson, 409 U.S. 63, 93 S.Ct. 253, 34 L.Ed.2d 273 (1972), the Supreme Court concluded that claims directed to a particular “algorithm,” conversion of binary coded decimal numbers to binary numbers, did not define patentable subject matter. In that case the Court defined the term “algorithm” as “[a] procedure for solving a given type of mathematical problem.” Id. at 65, 93 S.Ct. at 254. The Court’s holding in Benson became the basis for the first part of the two-part analysis set forth by this court in In re Freeman, 573 F.2d 1237, 197 USPQ 464 (Cust. & Pat.App.1978).
In Freeman, 573 F.2d at 1245, 197 USPQ at 470, this court concluded:
As a bare minimum, application of Benson in a particular case requires a careful analysis of the claims, to determine whether, as in Benson, they recite a “procedure for solving a given type of mathematical problem.” [Citation omitted. Emphasis in original.]
Hence, the first part of the analysis requires:
First, it must be determined whether the claim directly or indirectly recites an “algorithm” in the Benson sense of that term .... [Jdf. at 1245, 197 USPQ at 471.]
The second part of the Freeman analysis is derived from the further holding in Benson, 409 U.S. at 72, 93 S.Ct. at 257, that any patent issued in that case “would wholly pre-empt the mathematical formula and in practical effect would be a patent on the algorithm itself.” Thus, it was concluded that the presence of an “algorithm” in a claim would not render a claimed invention nonstatutory unless the invention claimed only the “algorithm.” Stating this conclusion in the language of Benson, this court declared:
Second, the claim must be further analyzed to ascertain whether in its entirety it wholly preempts that algorithm.
[Freeman, Id. at 1245, 197 USPQ at 471.]
This latter step in the Freeman analysis was not reached because of this court’s conclusion that the claims did not recite an “algorithm.” In In re Toma, 575 F.2d 872, 197 USPQ 852 (Cust. & Pat.App.1978), the same test was discussed but, again, the second part of the analysis was not reached. Subsequently, the Supreme Court handed down its decision in Parker v. Flook, 437 U.S. 584, 98 S.Ct. 2522, 57 L.Ed.2d 451 (1978), making clear that the second part of [906]*906the above analysis was erroneous. The Court held that the claim need “not ... cover every conceivable application of the formula” to be nonstatutory. Id. at 586, 98 S.Ct. at 2523.
In sum, the Court’s decisions have made clear that a claim does not present patentable subject matter if it would wholly preempt an algorithm, Benson, supra, or if it would preempt the algorithm but for limiting its use to a particular technological environment, Flook, supra. However, these decisions leave undefined what does constitute statutory subject matter.
In In re Johnson, 589 F.2d 1070, 1075, 200 USPQ 199, 205 (Cust. & Pat.App.1978), this court held that, while reciting an algorithm, the claims did not merely define a method of solving a mathematical equation because
any calculations which may be performed in practicing the process ... are but a part of the process which includes the other recited steps.
... [They] are incident to producing a noise-free signal trace from a reference trace. [Id. at 1080, 200 USPQ at 209 (emphasis added).]
This conclusion rests on the premise that an otherwise statutory process remains statutory when implemented by a computer, id. at 1078, 200 USPQ at 207 n.8, a premise subsequently approved by the Supreme Court in Diamond v. Diehr, 450 U.S. 175, 101 S.Ct. 1048, 67 L.Ed.2d 155 (1981). Accord, In re Bradley, 600 F.2d 807, 202 USPQ 480 (Cust. & Pat.App.1979), aff’d sub nom by equally divided court, Daimond v. Bradley, 450 U.S. 381, 101 S.Ct. 1495, 67 L.Ed.2d 311 (1981) (Burger, C. J., not participating).
In Johnson, supra, the interrelationship of the algorithm to the remaining limitations of a claim was held to be determinative of whether the claim defined statutory subject matter. Relying on the same reasoning, in In re Walter, 618 F.2d 758, 205 USPQ 397 (Cust. & Pat.App.1980), the second part of the two-step analysis5 was defined as follows:
If it appears that the mathematical algorithm is implemented in a specific manner to define structural relationships between the physical elements of the claim (in apparatus claims) or to refine or limit claim steps (in process claims), the claim being otherwise statutory, the claim passes muster under § 101. If, however, the mathematical algorithm is merely presented and solved by the claimed invention, as was the case in Benson and Flook, and is not applied in any manner to physical elements or process steps, no amount of post-solution activity will render the claim statutory; nor is it saved by a preamble merely reciting the field of use of the mathematical algorithm. [Id. at 767, 205 USPQ at 407 (emphasis added).]
In Walter, the claims were directed to a process6 for correlating and cross-correlating signals. All of the claims steps were algorithm steps for performing the correlation or cross-correlation. There were no limitations in the claims, other than a field of use set forth in the preamble of the claims which stated that the algorithm was for use in connection with seismic surveying. The court concluded that the claims were directed to claiming only the algorithm, were not applied in any manner to any process steps, and were, therefore, directed to nonstatutory subject matter. Id. at 769, 205 USPQ at 409.
Appellants summarize the Walter test as setting forth two ends of a spectrum: what is now clearly nonstatutory, i.e., claims in which an algorithm is merely presented and solved by the claimed invention (preemption), and what is clearly statutory, i.e., claims in which an algorithm is implemented in a specific manner to define structural relationships between the physical elements [907]*907of the claim (in an apparatus claim) or to refine or limit claim steps (in a process). Appellants urge that the statement of the test in Walter fails to provide a useful tool for analyzing claims in the “gray area” which falls between the two ends of that spectrum. We agree that the board’s understanding and application of the Walter analysis justifies appellant’s position. However, the Walter analysis quoted above does not limit patentable subject matter only to claims in which structural relationships or process steps are defined, limited or refined by the application of the algorithm.
Rather, Walter should be read as requiring no more than that the algorithm be “applied in any manner to physical elements or process steps,” provided that its application is circumscribed by more than a field of use limitation or non-essential post-solution activity. Thus, if the claim would be “otherwise statutory,” id., albeit inoperative or less useful without the algorithm, the claim likewise presents statutory subject matter when the algorithm is included. This broad reading of Walter, we conclude, is in accord with the Supreme Court decisions.
In Diamond v. Diehr, supra, the Court held that a process for curing synthetic rubber constituted patentable subject matter notwithstanding that the process used an equation for controlling the in-mold time which was constantly updated by a digital computer. In Diehr, were the claim to be read without the algorithm, the process would still be a process for curing rubber, although it might not work as well since the in-mold time would not be as accurately controlled. Hence, the Court concluded that the claimed invention fell within § 101 because it presented “an application of a law of nature or mathematical formula to a known structure or process.” Id. at 187, 101 S.Ct. at 1056 (emphasis in original).7 Accord, Mackay Radio & Telegraph Co. v. Radio Corp. of America, 306 U.S. 86, 59 S.Ct. 427, 83 L.Ed. 506 (1939).
Finally, the purpose of the two-part analysis supports the view taken here. The goal is to answer the question “What did applicants invent?” If the claimed invention is a mathematical algorithm, it is improper subject matter for patent protection, whereas if the claimed invention is an application of the algorithm, § 101 will not bar the grant of a patent.
In answering that question,
[ejach invention must be evaluated as claimed; yet semantogenic considerations preclude a determination based solely on words appearing in the claims. In the final analysis under § 101, the claimed invention, as a whole, must be evaluated for what it is. [In re Sarkar, 588 F.2d 1330, 1333, 200 USPQ 132, 137 (Cust. & Pat.App.1978) (footnote omitted).]
Hence, the analysis “requires careful interpretation of each claim in light of its supporting disclosure .... ” In re Johnson, 589 F.2d at 1079, 200 U.S.P.Q. at 208.
II
In this case, each of the independent claims and, necessarily, each of the dependent claims, includes the limitation “calculating . .. the difference” either as a step in a process or as a means in an apparatus. Accordingly, all of the claims may be directed to nonstatutory subject matter as each presents a mathematical formula or a sequence of mathematical operations. See, e.g., Diehr, supra, wherein the claims included a limitation requiring “calculating.” Cf. Johnson, 589 F.2d at 1078, 200 USPQ at 208 (use of term “compute” at least suggests the execution of a mathematical algorithm). In any event, appellants concede that their claims “implement a mathematical algorithm.”
III A
We now turn to the second part of our analysis to determine whether what is [908]*908claimed is a statutory process or apparatus or a nonstatutory algorithm.
We begin by contrasting the two broadest process claims, claims 5 and 6:
5. A method of displaying data in a field comprising the steps of
calculating the difference between the local value of the data at a data point in the field and the average value of the data in a region of the field which surrounds said point for each point in said field, and
displaying the value of said difference as a signed gray scale at a point in a picture which corresponds to said data point.
6. The method of claim 5 wherein said data is X-ray attenuation data produced in a two dimensional field by a computed tomography scanner.
We conclude that claim 5 is directed solely to the mathematical algorithm portion of appellants’ invention and is, thus, not statutory subject matter under § 101. We reach the opposite conclusion with respect to claim 6.
The method of claim 6, unlike that of claim 5, requires “X-ray attenuation data.” The specification indicates that such attenuation data is available only when an X-ray beam is produced by a CAT scanner, passed through an object, and detected upon its exit. Only after these steps have been completed is the algorithm performed,8 and the resultant modified data displayed in the required format.9
Were we to view the claim absent the algorithm, the production, detection and display steps would still be present and would result in a conventional CAT-scan process. Accordingly, production and detection cannot be considered mere antecedent steps to obtain values for solving the algorithm as in In re Richman, cited by the examiner. Indeed, claim 6 presents data gathering steps not dictated by the algorithm but by other limitations which require certain antecedent steps.10 It is these antecedent steps that dictate what type of data must be obtained. Compare In re Sarkar, 588 F.2d at 1336, 200 USPQ at 139 n.18, where we did not reach the effect of data gathering steps which are not dictated by the algorithm. In any event,, we view the production, detection, and display steps as manifestly statutory subject matter and are not swayed from this conclusion by the presence of an algorithm in the claimed method.
In Flook, supra, “[t]he patent application did not ‘explain how to select ... any of the variables’ ” used in the algorithm and, thus, no process other than the algorithm was present. Diehr, 450 U.S. at 186 n.10, 101 S.Ct. at 1056 n.10, quoting Flook, 437 U.S. at 586, 98 S.Ct. at 2523. A fortiori, no process steps to which the algorithm could be applied were present. Accord, Walter, supra. In the instant case, claim 6 defines the variables and places the algorithm in a particular relationship to a series of steps in a particular type of process, permitting the algorithm to be applied as a further process step. In re Taner, 681 F.2d 787, 790 (Cust. & Pat.App.1982).
The algorithm, when properly viewed, is merely applied to the “attenuation data” to eliminate what would otherwise appear as [909]*909artifacts upon display of the data in the manner claimed. The algorithm does not necessarily refine or limit the earlier steps of production and detection as would be required to achieve the status of patentable subject matter by the board’s narrow reading of Walter. What appellants have done is to discover an application of an algorithm to process steps which are themselves part of an overall process which is statutory. Hence, claim 6 cannot be construed as a mere procedure for solving a given mathematical problem. As was the case in Diehr and Johnson, both supra, the algorithm is but a part of the overall claimed process.
We are faced simply with an improved CAT-scan process comparable to the improved process for curing synthetic rubber in Diehr, supra. The improvement in either ease resides in the application of a mathematical formula within the context of a process which encompasses significantly more than the algorithm alone.
We do not reach the same conclusion with respect to claim 5. This claim presents no more than the calculation of a number and display of the result, albeit in a particular format.
The specification provides no greater meaning to “data in a field” than a matrix of numbers regardless of by what method generated. Thus, the algorithm is neither explicitly nor implicitly applied to any certain process. Moreover, that the result is displayed as a shade of gray rather than as simply a number provides no greater or better information, considering the broad range of applications encompassed by the claim. Indeed, this claim does not even attempt to “limit the use of the formula to a particular technological environment,” Diehr, 450 U.S. at 191, 101 S.Ct. at 1058, as was done in Flook, supra. Hence, we view claim 5 as directed merely to a mathematical formula which is not proper subject matter under § 101.
C
Appellants do not argue and, in any event, we see no basis for treating their apparatus claims differently from their method claims.
If the functionally-defined disclosed means and their equivalents are so broad that they encompass any and every means for performing the recited functions, the apparatus claim is an attempt to exalt form over substance since the claim is really to the method or series of functions itself. In computer-related inventions, the recited means often perform the function of “number crunching” (solving mathematical algorithms and making calculations). In such cases the burden must be placed on the applicant to demonstrate that the claims are truly drawn to specific apparatus distinct from other apparatus capable of performing the identical functions.
If this burden has not been discharged, the apparatus claim will be treated as if it were drawn to the method or process which encompasses all of the claimed “means.” [In re Walter, 618 F.2d at 768, 205 USPQ at 408.]
Thus, claim 7, the apparatus counterpart to claim 5, suffers the same defects as does claim 5.
7. Apparatus for displaying data values representative of values at data points in a two dimensional field comprising:
means for calculating the differences between the local values at each data point and the average value at data points in a limited region of said field surrounding each said data point, and
means for displaying the value of said differences as signed gray scale values at points in a picture which correspond to said data points.
D
What was said about claim 6 applies equally to independent method claim 33 and its apparatus counterpart claim 36.
33. A method of computed tomography comprising the steps of:
[910]*910measuring the values of the line integrals of an incoherent propagation along a plurality of paths through a region of interest in a body;
calculating, from the values of said integrals at each of a number of reconstruction points in said region of interest, the difference between the local value of a characteristic at said point and the average value of the characteristic in a local region surrounding said point; and
reconstructing a representation of features in said region of interest by displaying the calculated value for each reconstruction point at a point in a picture which corresponds to said reconstruction point.
36. Computed tomography apparatus comprising:
means for measuring the values of the line integrals of an incoherent propagation along a plurality of paths through a region of interest in a body;
calculating means, connected to receive the values of said integrals from said means for measuring and to calculate, at each of a number of reconstruction points in said region of interest, the difference between the local value of a characteristic at said point and the average value of said characteristic in a local region surrounding said point; and
means for reconstructing a representation of features in said region which function to receive said calculated values from said calculating means and to display the calculated value for each reconstruction point at a point in a picture which corresponds to said reconstruction point.
Indeed, the step of “measuring ... line integrals of an incoherent progagation . . . through ... a body” explicitly requires the same steps implicit in claim 6, viz., production of a beam and detection of the beam after it is attenuated by passing through a body. Moreover, while the display of claims 33 and 36 is only of a number, we have already disposed of the contention that the last step of a claim may not be a number. See In re Taner, supra, overruling In re Christensen, 478 F.2d 1392, 178 USPQ 35 (Cust. & Pat.App.1973). See also n. 9, supra.
E
In view of our holding with respect to independent claims 33 and 36, the remaining claims, all of which depend from claim 33 or 36, must likewise be proper subject matter for patenting under 35 U.S.C. § 101. The remaining claims on appeal appear in the Appendix for reference purposes.
MODIFIED.
APPENDIX
34. The method of claim 33 wherein said characteristic is a radiation attenuation coefficient.
35. The method of claim 33 wherein said region of interest lies in the body plane and comprises less than the entire area of said body plane.
37. The apparatus of claim 36 wherein said means for measuring comprises tomographic scanner means which function to direct one or more beams of penetrating radiation through said body, to determine the attenuation of said beams within said body, and to sequentially redirect said beams with respect to said region of interest whereby each of the reconstruction points in said region of interest is scanned by said beams from a plurality of different orientations.
38. The apparatus of claim 37 wherein said radiation is X-radiation.
39. The apparatus of claim 37 wherein said region of interest is an area in a plane passing through said body which area includes less than all of a body plane and wherein said scanner beams function to direct and redirect said beams to scan all reconstruction points within said region of interest from all of said orientations and to scan points in said body plane outside of said region [sic of] interest from less than all of said orientations.
40. The apparatus of claim 39 wherein said calculating means function to calculate the values of said difference by first assign[911]*911ing assumed values of said characteristic to points which are outside said region of interest.
41. The apparatus of claim 40 wherein said assumed values correspond to the value of the characteristic at adjacent reconstruction points at the boundary of said region of interest.
42. The apparatus of claim 36 wherein said calculating means function to calculate the values of said difference in accordance with the formula
43; The apparatus of claim 42 wherein said calculating means further function to apply a weighting function in said calculations which reduces interpolation errors in said calculation.
44. The apparatus of claim 43 wherein said weighting function is a Gaussian function.
45. The apparatus of claim 36 wherein the calculating means include a general purpose digital computer which includes a stored program which effects the calculation.
46. The apparatus of claim 36 wherein the means for reconstructing includes a display device which displays said calculated values as gray scale values.
47. The method of claim 46 wherein neutral gray represents zero values of said calculated values and wherein shades lighter and darker than neutral gray represent non-zero positive or negative signed values.