Image processing apparatus

• US 5,546,515 A
• Filed: 10/13/1995
• Issued: 08/13/1996
• Est. Priority Date: 07/08/1992
• Status: Expired due to Term

• Current Assignee: Matsushita Electric Industrial Company Limited (Panasonic Holdings Corporation)
• Original Assignee: Matsushita Electric Industrial Company Limited (Panasonic Holdings Corporation)
• Inventors: Mochizuki, Yoshiyuki
• Primary Examiner(s): Nguyen, Phu K.
• Application Number:
  US08/542,874
• Time in Patent Office: 305 Days
• Field of Search: 395/133, 395/118, 395/120, 395/126, 395/127, 395/129, 395/132, 345/118, 345/132
• US Class Current: 345/606
• CPC Class Codes: G06T 15/55 Radiosity

Abstract

In the radiosity calculation, the intersection decision is effected by the ray-casting method with respect to one portion of element of the polygon interior with the use of the coherence of the dodow so as to obtain the illumination intention and the interpolation domain of the form-factor from the intersection decision results. As the radiosity calculation is effected in accordance with the algorithm that the linear, non-linear interpolation equations are properly used in accordance with the precision to use the interpolation calculation in the interior portion the interpolation domain, the intersection decision frequency is sharply reduced so that the whole processing can be effected with higher speed.

Claims

1. An image producing apparatus for generating images using the distribution of illumination intensity values on a surface of an object consisting of quadrangle polygons, triangle polygons, or parallelogram polygons in a three dimensional space, said apparatus comprising:

a patch producing apparatus for producing two pairs of the same size parallelogram patches and the same size quadrangle patches from a quadrangle polygon, the same size triangle patches from a triangle polygon, or the same size parallelogram patches from a parallelogram polygon;

an element producing apparatus for producing two pairs of the same size parallelogram elements and the same size quadrangle elements from a quadrangle patch, the same size triangle elements and the same size parallelogram elements from a triangle patch, or the same size parallelogram elements from a parallelogram patch;

an illumination intensity interpolation domain deciding apparatus which selects sample elements in a patch and decides whether an element in the patch belongs to the interpolation domain which can be gotten the illumination intensity or the domain of shadows which can not be gotten the illumination intensity in accordance with the results of intersection decisions for said sample elements;

an illumination intensity interpolation calculating apparatus which calculates two radiosity values, B₁ and B₃ obtained from a light source for two elements in said interpolation domain with calculation representative points R₁ and R₃, decides whether a condition of ##EQU93## is satisfied or not, where "||·

||" represents the length of a vector, and calculates a radiosity value B₂ obtained from the light source for a element with a calculation representative point R₂ which has the relation of

space="preserve" listing-type="equation">R.sub.2 L=tR.sub.1 L+(1-t)R.sub.3 L

with using the roughly approximate interpolation equation if said condition is satisfied, or with using the accurately approximate interpolation equation if said condition is not satisfied;

a radiation method deciding apparatus which selects the patch with the maximum unshot energy, calculates the distance between the calculation representative point of the patch with the maximum unshot energy and the calculation representative point of a patch receiving the unshot energy from the patch with the maximum unshot energy and area values of the patch with the maximum unshot energy and the patch receiving the unshot energy, decides whether the condition that the ratio of said distance to the larger area value of said two area values is smaller than the threshold value is satisfied or not, and decides whether the radiation from the patch with the maximum unshot energy to the patch receiving the unshot energy is carried out with dividing the patch with the maximum unshot energy into elements or not dividing according to the consequence of the decision for said condition;

a form-factor interpolation domain deciding apparatus which selects sample elements in a patch receiving the unshot energy and decides whether an element in the patch belongs to the interpolation domain for form-factor calculation which can be gotten the unshot energy or the shadow domain for form-factor calculation which can not be gotten the unshot energy in accordance with the results of the intersection decisions for said sample elements;

a form-factor interpolation calculating apparatus which calculates two form-factor values, F_ij1 and F_ij3 for two elements in said interpolation domain for form-factor calculation, j₁ and j₃, with calculation representative points R₁ and R₃ that are obtained from the relation with patch (element) i with the calculation representative point S and the normal vector N_i, decides whether two conditions of ##EQU94## are satisfied or not, where "·

" represents inner product, calculates a form-factor value F_ij2 for a element j₂ with a calculation representative point R₂ which has the relation of

space="preserve" listing-type="equation">R.sub.2 S=tR.sub.1 S÷

(1-t)R.sub.3 S

with using one of 4 interpolation equations classified by approximation levels according to the consequence of the satisfactions for said conditions, and calculates the radiosity value for an element obtained from the patch with the maximum unshot energy with using the form-factor value for the element;

an illumination intensity mapping apparatus which calculates a coordinates value at a point on the illumination intensity distribution space that is the direct product space consisting of a position on a surface and a illumination intensity at the position obtained from the calculation on said illumination intensity interpolation calculating apparatus and said radiosity calculating apparatus from coordinates values of the point and vertices of the polygon including the point on the 3 dimensional space, and outputs the illumination intensity value of the coordinates value at the point on the illumination intensity distribution space to the displaying apparatus.

2. The apparatus as claimed in claim 1, wherein said patch producing apparatus for producing patches by the equal division of the respective sides of the quadrangle polygon and the diagonal lines with a decomposition number corresponding to calculation precision with respect to the quadrangle polygon, and by the division thereof with line segments for connecting equally dividing points on the respective sides with corresponding equally dividing points on the diagonal lines, for producing patches by the equal division of the respective sides of the triangle polygon with the decomposition number corresponding to the calculation precision with respect to the triangle polygon, by the division thereof with line segments for connecting the corresponding equally dividing points among the adjacent sides of the triangle polygon, for producing patches by the equal division of the sides of the respective sets, with two decomposition numbers corresponding to the calculation precision opposite two sets of respective sides of the parallelogram polygon with respect to the parallelogram polygon, and by the division thereof with the line segments for connecting the corresponding equally dividing points of the opposite sides, in obtaining the radiosity values on the polygon surfaces when a shape in the three dimensional space is defined with a quadrangle polygon, triangle polygon, a parallelogram polygon, where radiosity values are calculated in accordance with patches produced by the patch producing apparatus.

3. The apparatus as claimed in claim 1, wherein said element producing apparatus for producing elements, by the division of the polygon into quadrangle patches, by the equal division of the respective sides of the square polygon and one of the diagonal lines with a decomposition number corresponding to calculation precision with respect to the quadrangle patch, and by the division thereof with line segments for connecting the equally dividing points on the respective sides with corresponding equally dividing points on the diagonal lines, in obtaining the radiosity values on the polygon surfaces when a shape in the three dimensional space is defined with a polygon, where radiosity values are calculated in accordance with elements produced by the element producing apparatus.

4. The apparatus as claimed in claim 1, wherein said element producing apparatus for producing elements, by the division of the polygon into triangles, by the equal division of the respective sides of the triangle patch with the decomposition number corresponding to the calculation precision, by the division thereof with line segments for connecting the equally dividing points of the side opposite to the reference point with the equally dividing points corresponding to each of the remaining sides with the reference point from the triangle vertex being fixed by one, in obtaining the radiosity values on the polygon surfaces when a shape in the three dimensional space is defined with a polygon, where radiosity values are calculated in accordance with elements produced by the element producing apparatus.

5. The apparatus as claimed in claim 1, wherein said element producing apparatus for producing elements by the equal division of the respective sides of the square patch and the diagonal lines with a decomposition number corresponding to calculation precision with respect to the quadrangle patch, by the division thereof with line segments for connecting the equally dividing points on the respective sides with the corresponding equally dividing points on the diagonal lines, for producing elements by the equal division of the respective sides of the triangle patch with the decomposition number corresponding to the calculation precision with respect to the triangle patch, by the division thereof with line segments for connecting the equally dividing points of the sides opposite to the reference point with the equally dividing points corresponding to each of the remaining sides with the reference point from the triangle vertex being fixed by one, for producing elements by the equal division of the respective sides of the parallelogram patch with the decomposition number corresponding to the calculation precision with respect to the patch upon the parallelogram and by the division thereof with the line segments for connecting the corresponding equally dividing points of the opposite sides, in obtaining the radiosity values on the polygon surfaces when a shape in the three dimensional space is defined with a polygon, where radiosity values are calculated in accordance with elements produced by the element producing apparatus.

6. The apparatus as claimed in claim 1, wherein said illumination intensity interpolation domain deciding apparatus which selects a sample element for each group with respect to a group of a parallel quadrilateral element arranged in a triangle shape which can be made when the triangle patch has been element-divided, and a group of a triangle element arranged in a straight line shape, effects an interpolation decision upon the sample element, decides an element belonging to the interpolation domain of the illumination intensity and the domain of the shadow for each group in accordance with the intersection decision result, where a radiosity value is calculated with respect to the interpolation domain of the illumination intensity to be decided by the illumination intensity interpolation domain deciding apparatus.

7. The apparatus as claimed in claim 1, wherein said illumination intensity interpolation domain deciding apparatus which selects a sample element for each group with respect to a group of a quadrangle element arranged in a diagonal line which can be made when quadrangle patch has been element-divided, and a group of two parallel quadrilateral elements arranged in a triangle shape, effects an interpolation decision upon the sample element, decides an element belonging to the interpolation domain of the illumination intensity and an element belonging to the domain of the shadow for each group in accordance with the intersection decision result, where a radiosity value is calculated with respect to the interpolation domain of the illumination intensity to be decided by the illumination intensity interpolation domain deciding apparatus.

8. The apparatus of claim 1, wherein the roughly approximate interpolation equation in said illumination intensity interpolation calculating apparatus is given the equation below for a point light source:

space="preserve" listing-type="equation">B.sub.2 =tB.sub.1 +(1-t)B.sub.3 (

     4)

9. The apparatus of claim 1, wherein the roughly approximate interpolation equation in said illumination intensity interpolation calculating apparatus is given equations below for a direction dependent point light source:

##EQU95## where N is the normal vector of the element, c₀, c₁, and c₂ are constant values determined by the decrement of the space,

space="preserve" listing-type="equation">τ

.sub. = R.sub.1 L,

space="preserve" listing-type="equation">τ

.sub.2 =R.sub.2 L,

space="preserve" listing-type="equation">τ

.sub.3 =R.sub.3 L,

each G₁, G₂, and G₃ is the light intensity from the light source toward R₁, R₂, and R₃ and p is the reflection constant of an element.

10. The apparatus of claim 1, wherein the accurately approximate interpolation equation in said illumination intensity interpolation calculating apparatus is given the equation below for a point light source:

##EQU96## where c₀, c₁, and c₂ are constant values determined by the decrement of the space,

space="preserve" listing-type="equation">τ

.sub. = R.sub.1 L,

space="preserve" listing-type="equation">τ

.sub.2 =R.sub.2 L,

space="preserve" listing-type="equation">τ

.sub.3 =R.sub.3 L,

11. The apparatus of claim 1, wherein the accurately approximate interpolation equation in said illumination intensity interpolation calculating apparatus is given equations below for a direction dependent point light source:

##EQU97## where N is the normal vector of the element, c₀, c₁, and c₂ are constant values determined by the decrement of the space,

space="preserve" listing-type="equation">τ

.sub. = R.sub.1 L,

space="preserve" listing-type="equation">τ

.sub.2 =R.sub.2 L,

space="preserve" listing-type="equation">τ

.sub.3 =R.sub.3 L,

each G₁, G₂, and G₃ is the light intensity from the light source toward R₁, R₂, and R₃ and p is the reflection constant of an element.

12. The apparatus of claim 1, wherein said form-factor interpolation domain deciding apparatus which, when said condition in said radiation method deciding apparatus is satisfied, selects sample elements in the patch receiving the unshot energy from the patch with the maximum unshot energy, decides whether the sample elements can be obtained the unshot energy or not according to two values of a inner product between the vector, whose starting point is the calculation representative point of the patch with the maximum unshot energy and end point is the calculation representative point of the sample element, and the normal vector of the patch with the maximum unshot energy and a inner product between said vector and the normal vector of the patch receiving the unshot energy, decides whether there is an object interfering the arrival for the unshot energy between the patch with the maximum unshot energy and the sample element or not by the intersection decision, determines the interpolation domain and the shadow domain according to the results of said decisions, and decisions whether an element in the patch receiving the unshot energy belongs to the interpolation domain or the shadow domain.

13. The apparatus of claim 1, wherein said form-factor interpolation domain deciding apparatus which, when said condition in said radiation method deciding apparatus is satisfied, divides the patch with the maximum unshot energy into elements, selects sample elements in the patch receiving the unshot energy from an element in the patch with the maximum unshot energy, decides whether the sample element in the patch receiving the unshot energy can be obtained the unshot energy or not according to two values of a inner product between the vector, whose starting point is the calculation representative point of the element in the patch with the maximum unshot energy and end point is the calculation representative point of the sample element in the patch receiving the unshot energy, and the normal vector of the patch with the maximum unshot energy and a inner product between said vector and the normal vector of the patch receiving the unshot energy, decides whether there is an object interfering the arrival for the unshot energy between the element in the patch with the maximum unshot energy and the sample element or not by the intersection decision, determines the interpolation domain and the shadow domain according to the results of said decisions, and decides whether an element in the patch receiving the unshot energy belongs to the interpolation domain or the shadow domain.

14. The apparatus of claim 1, wherein said form-factor interpolation domain deciding apparatus which, when said condition in said radiation method deciding apparatus is satisfied, divides the patch with the maximum unshot energy into elements, selects sample elements in the patch with the maximum unshot energy, selects sample elements in the patch receiving the unshot energy from an element in the patch with the maximum unshot energy, for each fixed sample element in the patch receiving the unshot energy and every sample elements in the patch with the maximum unshot energy, repeats the operation below;

deciding whether the fixed sample element in the patch receiving the unshot energy can be obtained the unshot energy or not according to two values of a inner product between the vector, whose starting point is the calculation representative point of the element in the patch with the maximum unshot energy and end point is the calculation representative point of the fixed sample element, and the normal vector of the patch with the maximum unshot energy and a inner product between said vector and the normal vector of the patch receiving the unshot energy;

deciding whether there is an object interfering the arrival for the unshot energy between the element in the patch with the maximum unshot energy and the sample element or not by the intersection decision;

determining the interpolation domain and the shadow domain according in the patch with the maximum unshot energy to the results of said decisions;

and storing the results of the decision whether an element in the patch with the maximum unshot energy belongs to the interpolation domain or the shadow domain for the fixed sample element;

and decides whether an element in the patch receiving the unshot energy belongs to the interpolation domain or the shadow domain for every elements in the patch with the maximum unshot energy by said storing results.

15. The apparatus of claim 1, wherein one of 4 interpolation equations classified by approximation levels in said form-factor interpolation calculating apparatus is given the equation below:

if neither the upper condition in said form-factor interpolation calculating apparatus nor the lower condition in said form-factor interpolation calculating apparatus is satisfied, ##EQU98## where

space="preserve" listing-type="equation">τ

.sub. = R.sub.1 S,

space="preserve" listing-type="equation">τ

.sub.2 =R.sub.2 S,

space="preserve" listing-type="equation">τ

.sub.3 =R.sub.3 S.

16. The apparatus of claim 1, wherein one of 4 interpolation equations classified by approximation levels in said form-factor interpolation calculating apparatus is given the equation below:

if the upper condition in said form-factor interpolation calculating apparatus is satisfied but the lower condition in said form-factor interpolation calculating apparatus is not satisfied, ##EQU99## where

space="preserve" listing-type="equation">τ

.sub. = R.sub.1 S,

space="preserve" listing-type="equation">τ

.sub.2 =R.sub.2 S,

space="preserve" listing-type="equation">τ

.sub.3 =R.sub.3 S.

17. The apparatus of claim 1, wherein one of 4 interpolation equations classified by approximation levels in said form-factor interpolation calculating apparatus is given the equation below:

if the upper condition in said form-factor interpolation calculating apparatus is not satisfied but the lower condition in said form-factor interpolation calculating apparatus is satisfied, ##EQU100## where

space="preserve" listing-type="equation">τ

.sub. = R.sub.1 S,

space="preserve" listing-type="equation">τ

.sub.2 =R.sub.2 S,

space="preserve" listing-type="equation">τ

.sub.3 =R.sub.3 S.

18. The apparatus of claim 1, wherein one of 4 interpolation equations classified by approximation levels in said form-factor interpolation calculating apparatus is given the equation below:

if both the upper condition and the lower condition in said form-factor interpolation calculating apparatus are satisfied,

space="preserve" listing-type="equation">F.sub.ij2 =tF.sub.ij1 ÷

(1-t)F.sub.ij3, (17)

where

space="preserve" listing-type="equation">τ

.sub. = R.sub.1 S,

space="preserve" listing-type="equation">τ

.sub.2 =R.sub.2 S,

space="preserve" listing-type="equation">τ

.sub.3 =R.sub.3 S.

19. The apparatus of claim 1, wherein the calculation of a coordinates value at a point on the illumination intensity distribution space in said illumination intensity mapping apparatus is given below for a triangle polygon:

when coordinates values of vertices of a triangle polygon ABC on the 3 dimensional space are given A(a₁,a₂,a₃), B(b₁,b₂,b₃), and C(c₁,c₂,c₃), the coordinates value of an internal point of ABC on the 3 dimensional space is given P(p₁,p₂,p₃), and the position coordinates value of the internal point P on the illumination intensity distribution space is given (s,t), if

(b₁ -a₁) (c₂ -a₂)-(c₁ -a₁) (b₂ -a₂)≠

0,

(s,t) is calculated by ##EQU101## if (b₁ -a₁) (c₃ -a₃)-(c₁ -a₁) (b₃ -a₃)≠

0,

(s,t) is calculated by ##EQU102## if (b₂ -a₂) (c₃ -a₃)-(c₂ -a₂) (b₃ -a₃)≠

0,

(s, t) is calculated by ##EQU103## and otherwise there is no answer for (s,t), and the point is given the illumination intensity value at the closest position for (s,t) on the illumination distribution in ABC as the illumination intensity at the point or is given the value interpolated of illumination intensity values in the neighborhood for (s,t) on the illumination distribution in ABC as the illumination intensity at the point.

20. The apparatus of claim 1, wherein the calculation of a coordinates value at a point on the illumination intensity distribution space in said illumination intensity mapping apparatus is given below for a quadrangle polygon:

when coordinates values of vertices of a quadrangle polygon ABCD on the 3 dimensional space are given A(a₁,a₂,a₃), B(b₁,b₂,b₃), C(c₁,c₂,c₃), and D(d₁,d₂,d₃), the normal vector of ABCD is given N=(N_x,N_y,N_z), the coordinates value of an internal point of ABCD on the 3 dimensional space is given X(x₁,x₂,x₃), and the position coordinates value of the internal point P on the illumination intensity distribution space is given (s+t,t) if X exists in the triangle ABC, (s,s) if X exists on the line segment AC, or (s,s+r) if X exists in the triangle ACD;

deciding the maximum value of N_x, N_y, N_z ;

when N_x is the maximum value and c₂ -a₂ =0, deciding that X exists in the triangle ABC if (x₂ -a₂) (b₂ -a₂)>

0, X exists on the line segment AC if x₂ -a₂ =0, and X exists in the triangle ACD (x₂ -a₂) (d₂ -a₂)>

0 and when N_x is the maximum value and c₂ -a₂ ≠

0, deciding that X exists in the triangle ABC if ##EQU104## X exists on the line segment AC if ##EQU105## and X exists in the triangle ACD if ##EQU106## when N_y is the maximum value and c₁ -a₁ =0, deciding that X exists in the triangle ABC if (x₁ -a₁)(b₁ -a₁)>

0, X exists on the line segment AC if x₁ -a₁ =0, and X exists in the triangle ACD (x₁ -a₁)(d₁ -a₁)>

0, and when N_y is the maximum value and c₁ -a₁ ≠

0 deciding that X exists in the triangle ABC if ##EQU107## X exists on the line segment AC if ##EQU108## and X exists in the triangle ACD if ##EQU109## when N_z is the maximum value and c₁ -a₁ =0, deciding that X exists in the triangle ABC if (x₁ -a₁)(b₁ -a₁)>

0, X exists on the line segment AC if x₁ -a₁ =0, and X exists in the triangle ACD (x₁ -a₁)(d₁ -a₁)>

0, and when N_z is the maximum value and c₁ -a₁ ≠

0, deciding that X exists in the triangle ABC if ##EQU110## X exists on the line segment AC if ##EQU111## and X exists in the triangle ACD if ##EQU112## when X exists in the triangle ABC, if (c₁ -a₁)(b₂ -a₂)-(b₁ -a₁)(c₂ -a₂)≠

0,

s and t are calculated by ##EQU113## if (c₁ -a₁)(b₃ -a₃)-(b₁ -a₁)(c₃ -a₃)≠

0,

s and t are calculated by ##EQU114## if (c₂ -a₂)(b₃ -a₃)-(b₂ -a₂)(c₃ -a₃)≠

0,

s and t are calculated by ##EQU115## and otherwise there is no answer for s and t;

when X exists on the line segment AC, if c₁ -a₁ ≠

0, s is calculated by ##EQU116## if c₂ -a₂ ≠

0 s is calculated by ##EQU117## if c₃ -a₃ ≠

0, s is calculated by ##EQU118## and otherwise there is no answer for s;

when X exists in the triangle ACD, if

s and r are calculated by

(c₁ -a₁)(d₂ -a₂)-(d₂ -a₂)(d₁ -a₁)(a₂ -a₂)≠

0, ##EQU119## if (c₁ -a₁)(d₃ -a₃)-(d₁ -a₁)(c₃ -a₃)≠

0,

s and r are calculated by ##EQU120## if (c₂ -a₂)(d₃ -a₃)-(d₂ -a₂)(c₃ -c₃)≠

0,

s and r are calculated by ##EQU121## and otherwise there is no answer for s and r;

the point X is given the illumination intensity value at the closest position for X on the illumination distribution in ABCD as the illumination intensity at the point or is given the value interpolated of illumination intensity values in the neighborhood for X on the illumination distribution in ABCD as the illumination intensity at the point.

21. The apparatus of claim 1, wherein the calculation of a coordinates value at a point on the illumination intensity distribution space in said illumination intensity mapping apparatus is given below for a triangle polygon:

when local coordinates values of vertices of a triangle polygon ABC as a 2 dimensional manifold on the 3 dimensional space are given A(u_A,v_A), B(u_B,v_B), and C(u_C,v_C), the local coordinates value of an internal point of ABC is given P(u_P,v_P), and the position coordinates value of P on the illumination intensity distribution space is given (s,t), if

(u_B -u_A)(v_C -v_A)-(u_C -u_A)(v_B -v_A)≠

0,

(s,t) is calculated by ##EQU122## and otherwise there is no answer for (s,t,) and the point is given the illumination intensity value at the closest position for (s,t) on the illumination distribution in ABC as the illumination intensity at the point or is given the value interpolated of illumination intensity values in the neighborhood for (s,t) on the illumination distribution in ABC as the illumination intensity at the point.

22. The apparatus of claim 1, wherein the calculation of a coordinates value at a point on the illumination intensity distribution space in said illumination intensity mapping apparatus is given below for a quadrangle polygon:

when local coordinates values of vertices of a quadrangle polygon ABCD as a 2 dimensional manifold on the 3 dimensional space are given A(u_A,v_A), B(u_B,v_B), C(u_C,v_C), and D(u_D,v_D), the local coordinates value of an internal point of ABCD is given X(u_X,v_X), and the position coordinates value of the internal point X on the illumination intensity distribution space is given (s+t,t) if X exists in the triangle ABC, (s,s) if X exists on the line segment AC, or (s,s+r) if X exists in the triangle ACD;

when u_C -u_A =0, deciding that X exists in the triangle ABC if (u_X -u_A) (u_B -u_A)>

0, X exists on the line segment AC if u_X -u_A =0, and X exists in the triangle ACD (X_X -U_A)(U_D -U_A)>

0 and when u_c -u_A ≠

0, deciding that X exists in the triangle ABC if ##EQU123## X exists on the line segment AC if ##EQU124## and X exists in the triangle ACD if ##EQU125## when X exists in the triangle ABC, if

space="preserve" listing-type="equation">(u.sub.C -u.sub.A)(v.sub.B -v.sub.A)-(u.sub.B -u.sub.A)(v.sub.C -v.sub.A)≠

0,

s and t are calculated by ##EQU126## and otherwise there is no answer for s and t;

when X exists on the line segment. AC, if u_C -u_A +0 and v_C -v_A ≠

0, s is calculated by ##EQU127## if u_C -u_A ≠

0, s is calculated by ##EQU128## and otherwise there is no answer for s;

when X exists in the triangle ACD, if

space="preserve" listing-type="equation">(u.sub.C -u.sub.A)(u.sub.D -u.sub.A)-(u.sub.D -u.sub.A)(v.sub.C -v.sub.A)≠

0,

s and r are calculated by ##EQU129## and otherwise there is no answer for s and z;

the point X is given the illumination intensity value at the closest position for X on the illumination distribution in ABCD as the illumination intensity at the point or is given the value interpolated of illumination intensity values in the neighborhood for X on the illumination distribution in ABCD as the illumination intensity at the point.