行列演算と特異値分解（SVD）の理解を基に、これらの概念を実践的なシナリオである画像圧縮に適用します。SVDはデータの次元削減に優れた手法であり、画像処理分野では元の情報をできるだけ保持しつつ画像を圧縮するために広く利用されています。このチャレンジでは、`scipy.linalg.svd` を用いてグレースケール画像の行列を特異値で切り捨てて圧縮し、圧縮後のデータから画像を再構成します。この手法により、SVDが画像品質と保存効率のバランスをどのように実現するかを示します。


import unittest
import user_code
import ast
import re   
import unittest
import numpy as np
from scipy.linalg import svd
import importlib

class TestTask(unittest.TestCase):
    def test_return_shape(self):
        import user_code
        importlib.reload(user_code)
        image = np.random.rand(8, 6)
        k = 3
        compressed = user_code.compress_image_svd(image, k)
        _dynamic_test(
            self,
            isinstance(compressed, np.ndarray) and compressed.shape == image.shape,
            "Returned matrix has the same shape as input image.",
            f"Expected shape {image.shape}, got {getattr(compressed, 'shape', None)}."
        )

    def test_top_k_reconstruction(self):
        import user_code
        importlib.reload(user_code)
        np.random.seed(0)
        image = np.random.rand(5, 5)
        k = 2
        U, S, VT = svd(image, full_matrices=False)
        U_k = U[:, :k]
        S_k = np.diag(S[:k])
        VT_k = VT[:k, :]
        expected = U_k @ S_k @ VT_k
        actual = user_code.compress_image_svd(image, k)
        _dynamic_test(
            self,
            np.allclose(actual, expected),
            "Matrix reconstructed correctly using top k singular values.",
            "Reconstructed matrix does not match expected result using top k singular values."
        )

    def test_full_rank_reconstruction(self):
        import user_code
        importlib.reload(user_code)
        np.random.seed(1)
        image = np.random.rand(7, 7)
        U, S, VT = svd(image, full_matrices=False)
        full_rank = np.linalg.matrix_rank(image)
        compressed = user_code.compress_image_svd(image, full_rank)
        _dynamic_test(
            self,
            np.allclose(compressed, image),
            "Full-rank SVD reconstruction matches original image.",
            "Full-rank SVD reconstruction does not match original image."
        )

    def test_low_rank_approximation(self):
        import user_code
        importlib.reload(user_code)
        np.random.seed(2)
        image = np.random.rand(10, 10)
        k = 1
        compressed = user_code.compress_image_svd(image, k)
        # Should be a low-rank approximation, not equal to original
        _dynamic_test(
            self,
            not np.allclose(compressed, image),
            "Low-rank approximation does not exactly match original image (as expected).",
            "Low-rank approximation should not exactly match original image."
        )
        # Should be closer to original than a zero matrix
        zero_approx = np.zeros_like(image)
        orig_norm = np.linalg.norm(image - compressed)
        zero_norm = np.linalg.norm(image - zero_approx)
        _dynamic_test(
            self,
            orig_norm < zero_norm,
            "Low-rank approximation is closer to the original than a zero matrix.",
            "Low-rank approximation is not closer to the original than a zero matrix."
        )

    def test_handles_rectangular(self):
        import user_code
        importlib.reload(user_code)
        np.random.seed(3)
        image = np.random.rand(12, 8)
        k = 4
        compressed = user_code.compress_image_svd(image, k)
        _dynamic_test(
            self,
            isinstance(compressed, np.ndarray) and compressed.shape == image.shape,
            "Handles rectangular matrices and returns correct shape.",
            f"Expected shape {image.shape}, got {getattr(compressed, 'shape', None)}."
        )

def _dynamic_test(test_case, condition, success_message, failure_message):
    if condition:
        test_case._testMethodName = success_message
        test_case.assertTrue(True, success_message)
    else:
        test_case._testMethodName = failure_message
        test_case.fail(failure_message)

def normalize_text(text):
    text = text.lower()
    text = re.sub(r"\s{2,}", " ", text)
    text = re.sub(r"\s*([,:?])\s*", r"\1 ", text)
    return text.strip()

def change_var(code: str, var_name: str, value: str) -> str:
    tree = ast.parse(code)
    lines = code.splitlines()
    for i, node in enumerate(tree.body):
        if isinstance(node, ast.Assign):
            for target in node.targets:
                if isinstance(target, ast.Name) and target.id == var_name:
                    start_line = node.lineno - 1
                    line = lines[start_line]
                    indent = ' ' * (len(line) - len(line.lstrip()))
                    lines[start_line] = f"{indent}{var_name} = {value}"
                    next_line = len(lines)
                    for next_node in tree.body[i+1:]:
                        if hasattr(next_node, 'lineno'):
                            next_line = next_node.lineno - 1
                            break
                    if next_line > start_line + 1:
                        lines[start_line+1:next_line] = []
                    
                    return '\\n'.join(lines)
    return code

if __name__ == "__main__":
    unittest.main()


test_main.py

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チャレンジ：画像圧縮のためのSVD

解答