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--- pp:l04 [2020/03/12 13:58]
pdmatei
+++ pp:l04 [2023/03/22 18:54] (current)
mihai.udubasa old revision restored (2021/03/28 20:51)
@@ Line 1: / Line 1: @@
-====== Working with matrices and images using higher-order functions ======
+====== 4. Working with matrices and images using higher-order functions ======
-===== Introduction =====
+===== 1. Introduction =====
-We shall represent matrices as \\lists of lists\\, i.e. values of type ''[ [Integer ] ]''. Each element in the outer list represents a line of the matrix.
+We shall represent matrices as //lists of lists//, i.e. values of type ''[ [Integer ] ]''. Each element in the outer list represents a line of the matrix.
 Hence, the matrix
@@ Line 14: / Line 14: @@
 which makes the type-name ''Matrix'' stand for ''[ [Integer] ]''.
-. Write a function ''parsem'' which takes a string and parses it to a ''Matrix''. In the string, the columns are separated by whitespace and lines - by '\n'.
+.1. Write a function ''parsem'' which takes a string and parses it to a ''Matrix''. In the string, the columns are separated by whitespace and lines - by '\n'.
      * (Hint: You need to perform two types of very similar splitting operations: one into lines, and another into column values, for each line)
@@ Line 22: / Line 22: @@
 </code>
-. Write a function that converts a matrix to a string encoded as illustrated in the previous exercise.
+.2. Write a function that converts a matrix to a string encoded as illustrated in the previous exercise.
      * (Hint: use folds)
      * (Hint: test the function ''show'' on several different values)
+     * (Hint: first make a function to display one line)
 <code haskell>
-toString :: Matrix -> String
+toString:: Matrix -> String
-toString =
+toString=
 </code>
-. Add the following to your code. Test the function ''displaymat''.
+.3. Add the following to your code. Test the function ''displayMatrix''.
 <code haskell>
-displaymat = putStrLn . toString
+displayMatrix = putStrLn . toString
 </code>
-===== Matrix operations =====
+===== 2. Matrix operations =====
-. Write a function that computes the scalar product with an integer:
-$math[ \displaystyle 2 * \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 1 & 4 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \\ \end{array}\right)]
+.1. Write a function that computes the scalar product with an integer:
+$math[ \displaystyle 2 * \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \\ \end{array}\right)]
 <code haskell>
@@ Line 46: / Line 48: @@
 </code>
-. Write a function which adjoins two matrices by extending rows:
+.2. Write a function which adjoins two matrices by extending rows:
 $math[ \displaystyle \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\\end{array}\right)  hjoin \left(\begin{array}{cc} 5 & 6 \\ 7 & 8\\\end{array}\right) = \left(\begin{array}{cc} 1 & 2 & 5 & 6 \\ 3 & 4 & 7 & 8\\\end{array}\right) ]
@@ Line 55: / Line 57: @@
 </code>
-. Write a function which adjoins two matrices by adding new rows:
+.3. Write a function which adjoins two matrices by adding new rows:
 $math[ \displaystyle \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\\end{array}\right)  vjoin \left(\begin{array}{cc} 5 & 6 \\ 7 & 8\\\end{array}\right) = \left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6\\ 7 & 8\\ \end{array}\right) ]
@@ Line 64: / Line 66: @@
 </code>
-. Write a function which adds two matrices.
+.4. Write a function which adds two matrices.
 <code haskell>
@@ Line 71: / Line 73: @@
 </code>
-. Write a function which computes the transposition of a matrix:
+.5. Write a function which computes the transposition of a matrix:
-$math[ tr \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 1 & 4 & 6 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \\ \end{array}\right) ]
+$math[ tr \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \\ \end{array}\right) ]
 <code haskell>
@@ Line 81: / Line 83: @@
 </code>
-. Write a function which computes the vectorial product of two matrices.
+.6. Write a function which computes the vectorial product of two matrices.
   * (Hint: start by writing a function which computes $math[a_{ij}] for a given line $math[i] and column $math[j] (both represented as lists))
   * (Hint: write a function which takes a line of matrix m1 and the matrix m2 and computes the respective line from the product)
@@ Line 90: / Line 92: @@
 </code>
-===== Image operations =====
+===== 3. Image operations =====
 We can represent images as matrices of pixels. In our example a pixel will be represented as a ''Char'', which can take the values: '*' (signifying that the respective pixel is //on//) and ' ' (the pixel is off).
@@ Line 109: / Line 111: @@
 </code>
-. Implement an image-displaying function:
+.1. Implement an image-displaying function. Use the image from the Appendix as a test.
 <code haskell>
 toStringImg :: Image -> String
@@ Line 118: / Line 120: @@
 Add the following function in order to view images in a nicer format:
 <code haskell>
-displaym = putStrLn . toStringImg
+displayImg = putStrLn . toStringImg
 </code>
-. Implement a function which flips an image horizontally:
+.2. Implement a function which flips an image horizontally:
 <code haskell>
 flipH :: Image -> Image
@@ Line 127: / Line 129: @@
 </code>
-. Implement a function which flips an image vertically:
+.3. Implement a function which flips an image vertically:
 <code haskell>
 flipV :: Image -> Image
@@ Line 133: / Line 135: @@
 </code>
-. Implement a function which rotates an image 90grd clockwise
+.4. Implement a function which rotates an image 90grd clockwise
 <code haskell>
 rotate90r :: Image -> Image
@@ Line 139: / Line 141: @@
 </code>
-. Implement a function which rotates an image -90grd clockwise
+.5. Implement a function which rotates an image -90grd clockwise
 <code haskell>
 rotate90l :: Image -> Image
@@ Line 145: / Line 147: @@
 </code>
-. Implement a function which returns a diamond of a specified //height//.  Example:
+.6. Implement a function which inverts an image: ' ' becomes * and * becomes ' ':
-<code>
+<code haskell>
-                          *
+invert :: Image -> Image
-                         ***
+invert =
-   diamond 3 =          *****
-                         ***
-                          *
-                         *
-   diamond 2 =          ***
-                         *
 </code>
+.7. Implement a function ''maskKeep'' which takes a mask and some image and only keeps the part of image which overlays the mask. Use the mask and the logo from Appendix to test it.
 <code haskell>
-diamond :: Integer -> Image
+maskKeep :: Image -> Image -> Image
+maskKeep =
 </code>
+$math[ maskKeep \left(\begin{array}{ccc}  &  & * \\  &  & * \\  &  & * \\ \end{array}\right) \left(\begin{array}{ccc} * &  & * \\ * &  &  \\ * &  & * \\ \end{array}\right) = \left(\begin{array}{ccc}  &  & * \\  &  &  \\  &  & * \\ \end{array}\right) ]
-. Implement a function with takes two images with the same dimensions. The second image is a mask. The output will be another image in which all pixels from the first image overlaying with a '*'-pixel from the mask will be displayed. All others will be deleted (made equal to ' '). Example:
+.8. Implement a function ''maskDiscard'' which takes a mask and some image and discards the part of the mask which overlays the image.
-<code>
+$math[ maskDiscard\left(\begin{array}{ccc}  &  & * \\  &  & * \\  &  & * \\ \end{array}\right) \left(\begin{array}{ccc} * &  & * \\ * &  &  \\ * &  & * \\ \end{array}\right) = \left(\begin{array}{ccc}  &  &  \\  &  & * \\  &  &  \\ \end{array}\right) ]
-   ******                ****     ****
+.9. Implement the ''union''' of two images.
-   **  **   `overlay`   ******   **  **
+<code haskell>
-   **  **                ****     *  *
+union' :: Image -> Image -> Image
-   ******                 **       **
+union' =
 </code>
+$math[ union \left(\begin{array}{ccc}  &  & * \\  &  & * \\  &  & * \\ \end{array}\right) \left(\begin{array}{ccc} * &  & * \\ * &  &  \\ * &  & * \\ \end{array}\right) = \left(\begin{array}{ccc} * &  & * \\ * &  & * \\ * &  & * \\ \end{array}\right) ]
+.10. Implement a function which takes a list of transformation and an image and applies all those transformations.
 <code haskell>
-overlay :: Image -> Image -> Image
+transformationSequence :: [Image -> Image] -> Image -> Image
+</code>
+Test it with the following sequence
+<code haskell>
+seq1 = transformationSequence [invert, union' mask, rotate90r]
 </code>
+===== 4. More matrices =====
+.1. Implement a function which places zeros **above** the diagonal of a square matrix. Use folds.
+.2. Implement a function which places zeros **below** the diagonal of a square matrix. Use folds.
+.3. Implement either of 4.1 or 4.2 using the function(s) ''take'' and ''drop''. Check the types of these and test them.
+.4. Implement a function which computes the **diagonal** matrix of a square matrix.
+.5. Implement a function which computes **recursively** the determinant of a matrix. What additional functions would be necessary?
 ===== Appendix =====
 <code haskell>
-l1="        ***** **            ***** **    "
-l2="     ******  ****        ******  ****   "
-l3="    **   *  *  ***      **   *  *  ***  "
-l4="   *    *  *    ***    *    *  *    *** "
-l5="       *  *      **        *  *      ** "
-l6="      ** **      **       ** **      ** "
-l7="      ** **      **       ** **      ** "
-l8="    **** **      *      **** **      *  "
-l9="   * *** **     *      * *** **     *   "
-l10="      ** *******          ** *******    "
-l11="      ** ******           ** ******     "
-l12="      ** **               ** **         "
-l13="      ** **               ** **         "
-l14="      ** **               ** **         "
-l15=" **   ** **          **   ** **         "
-l16="***   *  *          ***   *  *          "
-l17=" ***    *            ***    *           "
-l18="  ******              ******            "
-l19="    ***                 ***             "
 logo = [l1,l2,l3,l4,l5,l6,l7,l8,l9,l10,l11,l12,l13,l14,l15,l16,l17,l18,l19]
+    where l1 ="        ***** **            ***** **    "
+          l2 ="     ******  ****        ******  ****   "
+          l3 ="    **   *  *  ***      **   *  *  ***  "
+          l4 ="   *    *  *    ***    *    *  *    *** "
+          l5 ="       *  *      **        *  *      ** "
+          l6 ="      ** **      **       ** **      ** "
+          l7 ="      ** **      **       ** **      ** "
+          l8 ="    **** **      *      **** **      *  "
+          l9 ="   * *** **     *      * *** **     *   "
+          l10="      ** *******          ** *******    "
+          l11="      ** ******           ** ******     "
+          l12="      ** **               ** **         "
+          l13="      ** **               ** **         "
+          l14="      ** **               ** **         "
+          l15=" **   ** **          **   ** **         "
+          l16="***   *  *          ***   *  *          "
+          l17=" ***    *            ***    *           "
+          l18="  ******              ******            "
+          l19="    ***                 ***             "
 </code>
+<code haskell>
+mask = [l1,l2,l3,l4,l5,l6,l7,l8,l9,l10,l11,l12,l13,l14,l15,l16,l17,l18,l19]
+    where l1 ="                       *****************"
+          l2 ="                       *****************"
+          l3 ="                       *****************"
+          l4 ="                       *****************"
+          l5 ="                       *****************"
+          l6 ="                       *****************"
+          l7 ="                       *****************"
+          l8 ="                       *****************"
+          l9 ="                       *****************"
+          l10="                       *****************"
+          l11="                       *****************"
+          l12="                       *****************"
+          l13="                       *****************"
+          l14="                       *****************"
+          l15="                       *****************"
+          l16="                       *****************"
+          l17="                       *****************"
+          l18="                       *****************"
+          l19="                       *****************"
+</code>