Method of quantifying the topographic structure of a surface

• US 5,307,292 A
• Filed: 06/24/1992
• Issued: 04/26/1994
• Est. Priority Date: 06/24/1992
• Status: Expired due to Fees

• Current Assignee: Christopher A. Brown
• Original Assignee: Christopher A. Brown
• Inventors: Brown, Christopher A., Johnsen, William A., Charles, Patrick D.
• Primary Examiner(s): Teska, Kevin J.
• Assistant Examiner(s): CHOI, KYLE JAEHUN
• Application Number:
  US07/903,355
• Time in Patent Office: 671 Days
• Field of Search: 364/560, 364/562, 364/564, 364/474.24, 364/578, 364/577, 356/376, 356/379, 33/546, 33/547
• US Class Current: 702/156
• CPC Class Codes: B82Y 15/00 Nanotechnology for interacting, sensing or actuating, e.g. quantum dots as markers in protein assays or molecular motors
• CPC Class Codes: G06T 7/62 of area, perimeter, diameter or volume
• CPC Class Codes: Y10S 977/839 Mathematical algorithms, e.g. computer software, specifically adapted for modeling configurations or properties of nanostructure
• CPC Class Codes: Y10S 977/849 with scanning probe
• CPC Class Codes: Y10S 977/852 for detection of specific nanostructure sample or nanostructure-related property
• CPC Class Codes: Y10S 977/861 Scanning tunneling probe

Abstract

A fractal-based or "patchwork" method for analyzing topographic data simulates covering the surface of a specimen surface with triangular patches in order to determine the relative surface area which is a function of patch size or scale of observation or interaction. The specimen surface has X and Y horizontal axes and a vertical Z axis. Height data is obtained for each point of a grid network of points on the surface, the points being arranged in parallel rows and the rows being spaced. The specimen surface is defined with triangles having a surface area equal to a preselected patch area value. The total area of the planar triangles is calculated to obtain a total measured area of value. The area of the specimen surface in the X-Y plane that is defined by the triangles is calculated to obtain a total measured area value. The total measured area value is divided by the total projected area value to obtain a relative area value. The relative values for several patch area values are plotted to obtain a slope and a threshold, or crossover, point. The threshold is indicative of a point which separates the relatively large scales of observation or interaction which are best described by Euclidean geometry from those smaller ones which are best described by fractal geometry.

Claims

1. A method of quantifying the topographic structure of a specimen surface having a first horizontal dimension which is parallel with a horizontal X axis, a second horizontal dimension which is parallel to a horizontal Y axis and which is transverse to said X axis, and a variable vertical dimension which is parallel to a vertical Z axis, said method comprising:

(a) obtaining vertical height data for each point of a grid network of data points on said surface, said points being arranged in rows which are parallel to the X axis, said rows being spaced along the Y axis,

(b) selecting a patch area value,

(c) defining substantially the entire surface of said specimen surface with planar triangles which have an area substantially equal to said patch area value by using said data points as reference points for determining the positions of the three vertices of each triangle within said grid network of data points,

(d) calculating the total area of said planar triangles to obtain a measured area value,

(e) calculating the area of said specimen surface in the X-Y plane which is defined by said triangles to obtain a projected area value,

(f) dividing said measured area value by said projected area value to obtain a relative area value,

(g) repeating steps (b) through (f) for a plurality of patch area values to obtain a plurality of relative area value, and

(h) plotting said plurality of relative area values to obtain a slope value and a threshold value which is defined as the point where the relative area value departs significantly from the slope value.

2. A method of quantifying the topographic structure of a specimen surface as recited in claim 1, wherein each vertex of each triangle is located at one of said data points.

3. A method of quantifying the topographic structure of a specimen surface as recited in claim 1, wherein the vertical height data for each point is obtained by the use of electronic profilometry apparatus which produces electronic signals which are indicative of the vertical height for each point, wherein said height data is transmitted to the memory file of a computer, and wherein all of the mathematical calculations to obtain said relative area values are performed by said computer.

4. A method of quantifying the topographic structure of a specimen surface as recited in claim 1, wherein said triangles are equilateral triangles.

5. A method of quantifying the topographic structure of a specimen surface having a first horizontal dimension which is parallel to a horizontal X axis, a second horizontal dimension which is parallel to a horizontal Y axis which is transverse to said X axis and a variable vertical dimension which is parallel to a vertical Z axis, said method comprising:

(a) obtaining vertical height data for each point of a grid network of data points on said surface, said points being arranged in rows which extend parallel to the X axis, said rows being spaced along the Y axis,

(b) selecting a patch area value,

(c) selecting a first row of said data points as a first active data row,

(d) selecting a second row of said data points as a second active data row,

(e) selecting a first starting data point in said first active data row,

(f) selecting a second starting data point in said second active data row,

(g) selecting a third data point in one of said active data rows which defines with said first and second starting points a first planar triangle which has an area which is substantially equal to said patch area value,

(h) calculating the area of said triangle,

(i) selecting another third data point in one of said active data rows which defines with two of the points of said first planar triangle a second planar triangle which has an area which is substantially equal to said patch area value,

(j) calculating the area of said second planar triangle,

(k) selecting additional third data points from either of said first and second active data rows for successive triangles in accordance to steps (i) and (j) to define additional planar triangles, each of said additional planar triangles having an area which is substantially equal to said patch area value until the area between said first and second active rows is substantially defined by triangles having areas which are substantially equal to said patch area value,

(l) selecting a third row of points from said grid network of points to interrelate with said second row of points so that said second and third rows of points are the first and second active data rows for defining successive triangles in accordance with steps (e) through (k),

(m) selecting subsequent active data rows to interrelate with a previous active data row in accordance with steps (e) through (k) until substantially all of the topographic surface area of said specimen surface is defined by triangles having areas which are substantially equal to said patch area value,

(n) adding the surface areas of each of said triangles to obtain a measured area value,

(o) calculating the area of said specimen surface in the X-Y plane which is defined by said triangles to obtain a projected area value,

(p) dividing said measured area value by said projected area value to obtain a relative area value,

(q) repeating steps (b) through (p) for a plurality of patch area values to obtain a plurality of relative are values, and

(r) plotting said plurality of relative area values to obtain a slope value and a threshold value which is defined as the point where the relative area value departs significantly from the slope value.

6. A method of quantifying the topographic structure of a specimen surface as recited in claim 5, wherein the third point of successive triangles between two active data rows is selected alternately from the first and second active data rows.

7. A method of quantifying the topographic structure of a specimen surface as recited in claim 5, wherein the vertical height data for each point is obtained by the use of electronic profilometry apparatus which produces electronic signals which are indicative of the vertical height data for each point, wherein said height data is transmitted to the memory file of a computer, and wherein all of the mathematical calculations to obtain said relative area values are performed by said computer.

8. A method of quantifying the topographic structure of a specimen surface as recited in claim 5, wherein each additional third data point represents the data point which will define a triangle which has an area which is closest to the patch area value.

9. A method of quantifying the topographic structure of a specimen surface as recited in claim 5, wherein each additional third data point represents the closest data point to the last defined triangle which will define a triangle which has a area which is at least equal to the patch area value.

10. A method of quantifying the topographic structure of a specimen surface having a first horizontal dimension which is parallel to a horizontal X axis, a second horizontal dimension which is parallel to a horizontal Y axis which is transverse to said X axis and a variable vertical dimension which is parallel to a vertical Z axis, said method comprising:

(a) obtaining vertical height data for each point of a grid network of points on said surface, said points being arranged in rows which extend parallel with the X axis, said rows being spaced along the Y axis,

(b) selecting a patch area value,

(c) selecting a first row of said points as a first active data row,

(d) selecting a second row of said points as a second active data row,

(e) selecting a first starting point in said first active data row,

(f) selecting a second starting point in said second active data row,

(g) selecting a third point in one of said active rows which defines with said first and second points a preliminary planar triangle which has an area which is at least as great as said patch area value,

(h) interpolating between said third point and the point in the same row immediately preceding said third point to find a theoretical point which defines with said first and second points a first active planar triangle which has an area which is equal to said patch area value,

(i) selecting an additional point in one of said active data rows which defines with two of the points of said first active planar triangle a preliminary planar triangle which has an area which is at least as great as said patch area value,

(j) interpolating between said additional point and the point in the same row immediately preceding said additional point to find a theoretical third point which defines with two of the points of said first planar triangle a second active planar triangle which has an area which is equal to said patch area value,

(k) selecting additional third points from either of said first and second active data rows to define additional active planar triangles in accordance with steps (i) and (j), each of said additional active planar triangles having an area which is equal to said patch area value,

(l) selecting a third row of points from said grid network of points to interrelate with said second row of points so that said second and third rows of points are the first and second active data rows for defining successive active planar triangles in accordance with steps (e) through (k),

(m) selecting subsequent active data rows to interrelate with a previous active data row in accordance with steps (e) through (k) until substantially all of the topographic surface area of said specimen surface is defined by said active planar triangles,

(n) adding the measured areas of said active planar triangles to obtain a total measured area value,

(o) calculating the area of said specimen surface in the X-Y plane which is defined by said active planar triangles to obtain a total projected area value,

(p) dividing said total measured area value by said total projected area value to obtain a relative area value,

(q) repeating steps (b) through (p) for a plurality of patch area values to obtain a plurality of relative are values, and

(r) plotting said plurality of relative area values to obtain a slope value and a threshold value which is defined as the point where the relative area value departs significantly from the slope value.

11. A method of quantifying the topographic structure of a specimen surface as recited in claim 10, wherein the third point of successive active planar triangles between two active data rows is selected alternately from the first and second active data rows.

12. A method of quantifying the topographic structure of a specimen surface as recited in claim 10, wherein the vertical height data for each point is obtained by the use of electronic profilometry apparatus which produces electronic signals which are indicative of the vertical height data of said points, wherein said vertical height data is transmitted to the memory file of a computer, and wherein all of the mathematical calculations to obtain said relative area values are performed by said computer.

13. A method of quantifying the topographic structure of a specimen surface having a first horizontal dimension which is parallel to a horizontal X axis, a second horizontal dimension which is parallel to a horizontal Y axis and which is transverse to said X axis, and a variable vertical dimension which is parallel to a vertical Z axis, said method comprising:

(a) obtaining vertical height data for each point of a grid network of data points on said surface, said data points being arranged in rows which are parallel to the X axis, said rows being spaced along the Y axis,

(b) selecting a patch area value,

(c) randomly selecting a first point from one of said data points,

(d) selecting a second point from one of said data points,

(e) interpolating a third point which defines with said first and second points a first planar triangle which has an area which is substantially equal to said patch area value,

(f) calculating the area of said first planar triangle,

(g) selecting two vertices of the said first planar triangle,

(h) interpolating a third point which forms with the two vertices of said first planar triangle a second planar triangle which is substantially equal to the said patch area value,

(i) calculating the area of said second planar triangle,

(j) repeating steps (g) through (i) for all vertices of said first and second triangle and all subsequent triangles for defining subsequent triangles each of which;

(1) has an area which is substantially equal to said patch area value,

(2) lies within the X and Y bounds of said grid network of data points, and

(3) does not violate prescribed conditions for a valid triangle,

(k) calculating the total area of said planar triangles to obtain a total measured area value,

(l) calculating the area of said specimen surface in the X-Y plane which is defined by said triangles to obtain a total projected area value,

(m) dividing said total measured area value by said total projected area value to obtain a measured area value,

(n) repeating steps (b) through (m) for a plurality of patch area values to obtain a plurality of relative area values, and

(o) plotting said plurality of relative area values to obtain a slope value and a threshold value which is defined as the point where the relative area value departs significantly from the slope value.

14. A method of quantifying the topographic structure of a specimen surface as recited in claim 13, wherein said planar triangles are equilateral triangles so that each leg of the triangle has a specific equilateral leg length and the third point of each triangle which is generated is obtained by interpolation using said data points as reference points.