Method of kernel selection for image interpolation

• US 7,054,507 B1
• Filed: 12/17/1999
• Issued: 05/30/2006
• Est. Priority Date: 12/18/1998
• Status: Expired due to Fees

• Current Assignee: Canon Kabushiki Kaisha (Canon Inc.)
• Original Assignee: Canon Kabushiki Kaisha (Canon Inc.)
• Inventors: Bradley, Andrew Peter, Huang, Kai
• Primary Examiner(s): Bali, Vikkram
• Application Number:
  US09/466,178
• Time in Patent Office: 2,356 Days
• Field of Search: 382/300, 382/199, 382/205, 382/260, 382/263, 382/266, 382/308, 382/176, 382/299, 382/254, 382267-269, 382/298, 345/667, 345/698, 358/1.2, 358/525, 358/528, 358/451, 348/441
• US Class Current: 382/300
• CPC Class Codes: G06F 17/175 of multidimensional data
• CPC Class Codes: G06T 3/403 Edge-driven scaling; Edge-based scaling
• CPC Class Codes: H04N 1/3935 with modification of image resolution, i.e. determining the values of picture elements at new relative positions
• CPC Class Codes: H04N 1/40062 Discrimination between different image types, e.g. two-tone, continuous tone

Abstract

A method of interpolating image data is disclosed. The method accesses a first set of discrete sample values of the image data and calculates kernel values for each of the discrete sample values using one of a plurality of kernels. The kernel is selected depending upon an edge orientation indicator, an edge strength indicator, and an edge context indicator for each of the discrete sample values. The calculated kernel values are convolved with the discrete sample values to provide a second set of discrete sample values.

Claims

1. A method of interpolating a first set of discrete sample values to generate a second set of discrete sample values using one of a plurality of interpolation kernels, said method comprising the steps of:

identifying text regions in the first set of discrete sample values depending on a local contrast indicator for each of the discrete sample values of the first set;

identifying edge regions in the first set of discrete sample values depending on an edge strength indicator and an edge direction indicator for each of the discrete sample values of the first set;

combining the text regions and the edge regions to form a kernel selection map;

cleaning the kernel selection map by re-assigning orientations of any edge regions having isolated edge directions occurring in an otherwise uniformly directed local region of the first set of discrete sample values, according to the uniform direction; and

selecting the interpolation kernel using the cleaned kernel selection map for use in interpolating the first set of discrete sample values to generate the second set of discrete sample values.

2. The method according to claim 1, wherein the plurality of interpolation kernels are each derived from a universal interpolation kernel, h(s).

3. The method according to claim 1, wherein the plurality of kernels are given by:

${h<br><br>\left(s_x,s_y\right)}_{0\le <br><br>\mathrm{\theta <br><br>}\le <br><br>\mathrm{\pi <br><br>}/2}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\left(1-2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_y\right)}_{c=0.5}·<br><br>{h<br><br>\left(\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-1\right)<br><br>s_y\right)<br><br>w<br><br>\left(\mathrm{\theta <br><br>}\right)\right)}_{c=0}\right\}$ ${h<br><br>\left(s_x,s_y\right)}_{\mathrm{\pi <br><br>}/2<<br><br>\mathrm{\theta <br><br>}<<br><br>\mathrm{\pi <br><br>}}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-1\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-2\right)<br><br>s_y\right)}_{c=0.5}·<br><br>{h<br><br>\left(\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-2\right)<br><br>s_x+\left(1-2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_y\right)<br><br>w<br><br>\left(\mathrm{\theta <br><br>}\right)\right)}_{c=0}\right\},$

and wherein h(s) is a universal interpolation kernel, s_x=x/Δ

x and s_y=y/Δ

y are re-sampling distances in the horizontal and vertical directions, respectively, and ·

indicates matrix multiplication.

4. The method according to claim 1, wherein the plurality of kernels are given by:

$h<br><br>\left(s\right)=\{\begin{array}{cc}\left(2-\frac{3}{2}<br><br>b-c\right)<br><br>{<br><br>s<br><br>}^3+\left(-3+2<br><br><br><br>b+c\right)<br><br>{<br><br>s<br><br>}^2+\left(1-\frac{1}{3}<br><br>b\right),& <br><br>s<br><br>\le <br><br>1\\ \left(-\frac{1}{6}<br><br>b-c\right)<br><br>{<br><br>s<br><br>}^3+\left(b+5<br><br><br><br>c\right)<br><br>{<br><br>s<br><br>}^2+\left(-2<br><br><br><br>b-8<br><br><br><br>c\right)<br><br><br><br>s<br><br>+\left(\frac{4}{3}<br><br>b+4<br><br><br><br>c\right),& 1<<br><br><br><br>s<br><br>\le <br><br>2\\ 0,& \mathrm{Otherwise}\end{array}<br><br><br><br>h<br><br>\left(s\right)=\{\begin{array}{cc}1,& -d<<br><br><br><br>s<br><br>\le <br><br>d\\ 0,& \left(1-d\right)\ge <br><br>s><br><br>\left(1-d\right)\\ 2<br><br>{<br><br>\frac{s-d}{1-2<br><br>d}<br><br>}^3& -3<br><br>{<br><br>\frac{s-d}{1-2<br><br>d}<br><br>}^2+1,\end{array}<br><br><br><br>\begin{array}{c}{h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=0}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(s_x\right)}_{c=0.5}·<br><br>{h<br><br>\left(s_y\right)}_{c=0}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=\mathrm{\pi <br><br>}/2}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(s_x\right)}_{c=0}·<br><br>{h<br><br>\left(s_y\right)}_{c=0.5}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=\mathrm{\pi <br><br>}/4}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\frac{s_x+s_y}{2}\right)}_{c=0.5}·<br><br>{h<br><br>\left(\frac{s_x-s_y}{\sqrt{2}}\right)}_{c=0}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=3<br><br><br><br>\mathrm{\pi <br><br>}/4}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\frac{s_x+s_y}{\sqrt{2}}\right)}_{c=0}·<br><br>{h<br><br>\left(\frac{s_x-s_y}{2}\right)}_{c=0.5}\right\},\end{array}<br><br>$

and wherein h(s) is a modified cubic kernel, s_x=x/Δ

x and s_y=y/Δ

y are re-sampling distances in the horizontal and vertical directions, respectively, and ·

indicates matrix multiplication.

5. The method according to claim 1, wherein the first set of discrete sample values are at a different resolution to the second set of discrete sample values.

6. The method according to claim 1, wherein one or more of the indicators are processed using a morphological process.

7. A method of interpolating image data, said method comprising the steps of:

accessing a first set of discrete sample values of the image data;

identifying text regions in the first set of discrete sample values depending on a local contrast indicator associated with each of the discrete sample values of the first set;

identifying edge regions in the first set of discrete sample values depending on an edge strength indicator and an edge direction indicator for each of the discrete sample values of the first set;

combining the text regions and the edge regions to form a kernel selection map;

cleaning the kernel selection map by re-assigning orientations, of any edge regions having isolated edge directions occurring in an otherwise uniformly directed local region of the first set of discrete sample values, according to the uniform direction;

calculating kernel values for each of the discrete sample values using one of a plurality of kernels, wherein the one kernel is selected from the plurality of kernels using the cleaned kernel selection map; and

convolving the kernel values with the discrete sample values to provide a second set of discrete sample values.

8. The method according to claim 7, wherein the plurality of interpolation kernels are each derived from a universal interpolation kernel, h(s).

9. The method according to claim 7, wherein the plurality of kernels are given by:

${h<br><br>\left(s_x,s_y\right)}_{0\le <br><br>\mathrm{\theta <br><br>}\le <br><br>\mathrm{\pi <br><br>}/2}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\left(1-2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_y\right)}_{c=0.5}·<br><br>{h<br><br>\left(\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-1\right)<br><br>s_y\right)<br><br>w<br><br>\left(\mathrm{\theta <br><br>}\right)\right)}_{c=0}\right\}$ ${h<br><br>\left(s_x,s_y\right)}_{\mathrm{\pi <br><br>}/2<<br><br>\mathrm{\theta <br><br>}<<br><br>\mathrm{\pi <br><br>}}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-1\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-2\right)<br><br>s_y\right)}_{c=0.5}·<br><br>{h<br><br>\left(\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-2\right)<br><br>s_x+\left(1-2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_y\right)<br><br>w<br><br>\left(\mathrm{\theta <br><br>}\right)\right)}_{c=0}\right\},$

and wherein h(s) is a universal interpolation kernel, s_x=x/Δ

x and s_y=y/Δ

y are re-sampling distances in the horizontal and vertical directions, respectively, and ·

indicates matrix multiplication.

10. The method according to claim 7, wherein the plurality of kernels are given by:

$h<br><br>\left(s\right)=\{\begin{array}{cc}\left(2-\frac{3}{2}<br><br>b-c\right)<br><br>{<br><br>s<br><br>}^3+\left(-3+2<br><br><br><br>b+c\right)<br><br>{<br><br>s<br><br>}^2+\left(1-\frac{1}{3}<br><br>b\right),& <br><br>s<br><br>\le <br><br>1\\ \left(-\frac{1}{6}<br><br>b-c\right)<br><br>{<br><br>s<br><br>}^3+\left(b+5<br><br><br><br>c\right)<br><br>{<br><br>s<br><br>}^2+\left(-2<br><br><br><br>b-8<br><br><br><br>c\right)<br><br><br><br>s<br><br>+\left(\frac{4}{3}<br><br>b+4<br><br><br><br>c\right),& 1<<br><br><br><br>s<br><br>\le <br><br>2\\ 0,& \mathrm{Otherwise}\end{array}<br><br><br><br>h<br><br>\left(s\right)=\{\begin{array}{cc}1,& -d<<br><br><br><br>s<br><br>\le <br><br>d\\ 0,& \left(1-d\right)\ge <br><br>s><br><br>\left(1-d\right)\\ 2<br><br>{<br><br>\frac{s-d}{1-2<br><br>d}<br><br>}^3& -3<br><br>{<br><br>\frac{s-d}{1-2<br><br>d}<br><br>}^2+1,\end{array}<br><br><br><br>\begin{array}{c}{h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=0}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(s_x\right)}_{c=0.5}·<br><br>{h<br><br>\left(s_y\right)}_{c=0}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=\mathrm{\pi <br><br>}/2}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(s_x\right)}_{c=0}·<br><br>{h<br><br>\left(s_y\right)}_{c=0.5}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=\mathrm{\pi <br><br>}/4}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\frac{s_x+s_y}{2}\right)}_{c=0.5}·<br><br>{h<br><br>\left(\frac{s_x-s_y}{\sqrt{2}}\right)}_{c=0}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=3<br><br><br><br>\mathrm{\pi <br><br>}/4}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\frac{s_x+s_y}{\sqrt{2}}\right)}_{c=0}·<br><br>{h<br><br>\left(\frac{s_x-s_y}{2}\right)}_{c=0.5}\right\},\end{array}$

and wherein h(s) is a modified cubic kernel, s_x=x/Δ

x and s_y=y/Δ

y are re-sampling distances in the horizontal and vertical directions, respectively, and ·

indicates matrix multiplication.

11. The method according to claim 7, wherein the first set of discrete sample values are at a different resolution to the second set of discrete sample values.

12. An apparatus for interpolating image data, said apparatus comprising:

means for accessing a first set of discrete sample values of the image data;

text identifying means for identifying text regions in the first set of discrete sample values depending on a local contrast indicator associated with each of the discrete sample values of the first set;

edge region identifying means for identifying edge regions in the first set of discrete sample values depending on an edge strength indicator and an edge direction indicator for each of the discrete sample values of the first set;

kernel selection map means for combining said text regions and said edge regions to form a kernel selection map;

cleaning means for cleaning the kernel selection map by re-assigning orientations of any edge regions having isolated edge directions occurring in an otherwise uniformly directed local region of the first set of discrete sample values, according to the uniform direction;

calculator means for calculating kernel values for each of the discrete sample values using one of a plurality of kernels, wherein the one kernel is selected from the plurality of kernels using the cleaned kernel selection map; and

convolution means for convolving the kernel values with the discrete sample values to provide a second set of discrete sample values.

13. The apparatus according to claim 12, wherein the plurality of interpolation kernels are each derived from a universal interpolation kernel, h(s).

14. The apparatus according to claim 12, wherein the plurality of kernels are given by:

${h<br><br>\left(s_x,s_y\right)}_{0\le <br><br>\mathrm{\theta <br><br>}\le <br><br>\mathrm{\pi <br><br>}/2}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\left(1-2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_y\right)}_{c=0.5}·<br><br>{h<br><br>\left(\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-1\right)<br><br>s_y\right)<br><br>w<br><br>\left(\mathrm{\theta <br><br>}\right)\right)}_{c=0}\right\}$ ${h<br><br>\left(s_x,s_y\right)}_{\mathrm{\pi <br><br>}/2<<br><br>\mathrm{\theta <br><br>}<<br><br>\mathrm{\pi <br><br>}}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-1\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-2\right)<br><br>s_y\right)}_{c=0.5}·<br><br>{h<br><br>\left(\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-2\right)<br><br>s_x+\left(1-2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_y\right)<br><br>w<br><br>\left(\mathrm{\theta <br><br>}\right)\right)}_{c=0}\right\},$

and wherein h(s) is a universal interpolation kernel, s_x=x/Δ

x and s_y=y/Δ

y are re-sampling distances in the horizontal and vertical directions, respectively, and ·

indicates matrix multiplication.

15. The apparatus according to claim 12, wherein the plurality of kernels are given by:

$h<br><br>\left(s\right)=\{\begin{array}{cc}\left(2-\frac{3}{2}<br><br>b-c\right)<br><br>{<br><br>s<br><br>}^3+\left(-3+2<br><br><br><br>b+c\right)<br><br>{<br><br>s<br><br>}^2+\left(1-\frac{1}{3}<br><br>b\right),& <br><br>s<br><br>\le <br><br>1\\ \left(-\frac{1}{6}<br><br>b-c\right)<br><br>{<br><br>s<br><br>}^3+\left(b+5<br><br><br><br>c\right)<br><br>{<br><br>s<br><br>}^2+\left(-2<br><br><br><br>b-8<br><br><br><br>c\right)<br><br><br><br>s<br><br>+\left(\frac{4}{3}<br><br>b+4<br><br><br><br>c\right),& 1<<br><br><br><br>s<br><br>\le <br><br>2\\ 0,& \mathrm{Otherwise}\end{array}<br><br><br><br>h<br><br>\left(s\right)=\{\begin{array}{cc}1,& -d<<br><br><br><br>s<br><br>\le <br><br>d\\ 0,& \left(1-d\right)\ge <br><br>s><br><br>\left(1-d\right)\\ 2<br><br>{<br><br>\frac{s-d}{1-2<br><br>d}<br><br>}^3& -3<br><br>{<br><br>\frac{s-d}{1-2<br><br>d}<br><br>}^2+1,\end{array}<br><br><br><br>\begin{array}{c}{h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=0}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(s_x\right)}_{c=0.5}·<br><br>{h<br><br>\left(s_y\right)}_{c=0}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=\mathrm{\pi <br><br>}/2}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(s_x\right)}_{c=0}·<br><br>{h<br><br>\left(s_y\right)}_{c=0.5}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=\mathrm{\pi <br><br>}/4}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\frac{s_x+s_y}{2}\right)}_{c=0.5}·<br><br>{h<br><br>\left(\frac{s_x-s_y}{\sqrt{2}}\right)}_{c=0}\right\},\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=3<br><br><br><br>\mathrm{\pi <br><br>}/4}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\frac{s_x+s_y}{\sqrt{2}}\right)}_{c=0}·<br><br>{h<br><br>\left(\frac{s_x-s_y}{2}\right)}_{c=0.5}\right\},\end{array}$

and wherein h(s) is a modified cubic kernel, s_x=x/Δ

x and s_y=y/Δ

y are re-sampling distances in the horizontal and vertical directions, respectively, and indicates matrix multiplication.

16. The method according to claim 12, wherein the first set of discrete sample values are at a different resolution to the second set of discrete sample values.

17. A computer readable medium for storing a program for an apparatus which processes data, said processing comprising a method of interpolating image data, said program comprising:

code for accessing a first set of discrete sample values of the image data;

code for identifying text and regions in the first set of discrete sample values depending on a local contrast indicator associated with each of the discrete sample values of the first set;

code for identifying edge regions in the first set of discrete sample values depending on an edge strength indicator and an edge direction indicator for each of the discrete sample values of the first set;

code for combining the text regions and the edge regions to form a kernel selection map;

code for cleaning the kernel selection map by re-assigning orientations of any edge regions having isolated edge directions occurring in an otherwise uniformly directed local region of the first set of discrete sample values, according to the uniform direction;

code for calculating kernel values for each of the discrete sample values using one of a plurality of kernels, wherein the one kernel is selected from the plurality of kernels using the cleaned kernel selection map; and

code for convolving the kernel values with the discrete sample values to provide a second set of discrete sample values.

18. The computer readable medium according to claim 17, wherein the plurality of interpolation kernels are each derived from a universal interpolation kernel, h(s).

19. The computer readable medium according to claim 17, wherein the plurality of kernels are given by:

${h<br><br>\left(s_x,s_y\right)}_{0\le <br><br>\mathrm{\theta <br><br>}\le <br><br>\mathrm{\pi <br><br>}/2}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\left(1-2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_y\right)}_{c=0.5}·<br><br>{h<br><br>\left(\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-1\right)<br><br>s_y\right)<br><br>w<br><br>\left(\mathrm{\theta <br><br>}\right)\right)}_{c=0}\right\}$ ${h<br><br>\left(s_x,s_y\right)}_{\mathrm{\pi <br><br>}/2<<br><br>\mathrm{\theta <br><br>}<<br><br>\mathrm{\pi <br><br>}}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-1\right)<br><br>s_x+\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-2\right)<br><br>s_y\right)}_{c=0.5}·<br><br>{h<br><br>\left(\left(\left(2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}-2\right)<br><br>s_x+\left(1-2<br><br><br><br>\mathrm{\theta <br><br>}/\mathrm{\pi <br><br>}\right)<br><br>s_y\right)<br><br>w<br><br>\left(\mathrm{\theta <br><br>}\right)\right)}_{c=0}\right\},$

and wherein h(s) is a universal interpolation kernel, s_x=x/Δ

x and s_y=y/Δ

y are re-sampling distances in the horizontal and vertical directions, respectively, and ·

indicates matrix multiplication.

20. The computer readable medium according to claim 17, wherein the plurality of kernels are given by:

$h<br><br>\left(s\right)=\{\begin{array}{cc}\left(2-\frac{3}{2}<br><br>b-c\right)<br><br>{<br><br>s<br><br>}^3+\left(-3+2<br><br><br><br>b+c\right)<br><br>{<br><br>s<br><br>}^2+\left(1-\frac{1}{3}<br><br>b\right),& <br><br>s<br><br>\le <br><br>1\\ \left(-\frac{1}{6}<br><br>b-c\right)<br><br>{<br><br>s<br><br>}^3+\left(b+5<br><br><br><br>c\right)<br><br>{<br><br>s<br><br>}^2+\left(-2<br><br><br><br>b-8<br><br><br><br>c\right)<br><br><br><br>s<br><br>+\left(\frac{4}{3}<br><br>b+4<br><br><br><br>c\right),& 1<<br><br><br><br>s<br><br>\le <br><br>2\\ 0,& \mathrm{Otherwise}\end{array}<br><br><br><br>h<br><br>\left(s\right)=\{\begin{array}{cc}1,& -d<<br><br><br><br>s<br><br>\le <br><br>d\\ 0,& \left(1-d\right)\ge <br><br>s><br><br>\left(1-d\right)\\ 2<br><br>{<br><br>\frac{s-d}{1-2<br><br>d}<br><br>}^3& -3<br><br>{<br><br>\frac{s-d}{1-2<br><br>d}<br><br>}^2+1,\end{array}<br><br><br><br>\begin{array}{c}{h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=0}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(s_x\right)}_{c=0.5}·<br><br>{h<br><br>\left(s_y\right)}_{c=0}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=\mathrm{\pi <br><br>}/2}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(s_x\right)}_{c=0}·<br><br>{h<br><br>\left(s_y\right)}_{c=0.5}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=\mathrm{\pi <br><br>}/4}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\frac{s_x+s_y}{2}\right)}_{c=0.5}·<br><br>{h<br><br>\left(\frac{s_x-s_y}{\sqrt{2}}\right)}_{c=0}\right\}\\ {h<br><br>\left(s_x,s_y\right)}_{\mathrm{\theta <br><br>}=3<br><br><br><br>\mathrm{\pi <br><br>}/4}=\frac{1}{\sqrt{2}}<br><br>\left\{{h<br><br>\left(\frac{s_x+s_y}{\sqrt{2}}\right)}_{c=0}·<br><br>{h<br><br>\left(\frac{s_x-s_y}{2}\right)}_{c=0.5}\right\},\end{array}$

and wherein h(s) is a modified cubic kernel, s_x=x/Δ

x and s_y=y/Δ

y are re-sampling distances in the horizontal and vertical directions, respectively, and ·

indicates matrix multiplication.

21. The computer readable medium according to claim 17, wherein the first set of discrete sample values are at a different resolution to the second set of discrete sample values.